Liability Design with Information Acquisition
aa r X i v : . [ ec on . T H ] D ec Liability Design with Information Acquisition
Francisco Poggi and Bruno StruloviciNorthwestern University ∗ December 10, 2020
Abstract
How to guarantee that firms perform due diligence before launching potentiallydangerous products? We study the design of liability rules when (i) limited liabilityprevents firms from internalizing the full damage they may cause, (ii) penalties are paidonly if damage occurs, regardless of the product’s inherent riskiness, (iii) firms haveprivate information about their products’ riskiness before performing due diligence. Weshow that (i) any liability mechanism can be implemented by a tariff that depends onlyon the evidence acquired by the firm if a damage occurs, not on any initial report bythe firm about its private information, (ii) firms that assign a higher prior to productriskiness always perform more due diligence but less than is socially optimal, and (iii)under a simple and intuitive condition, any type-specific launch thresholds can beimplemented by a monotonic tariff.
In 2019, a California court sentenced paint maker Sherwin-Williams to pay hundreds ofmillions of dollars to address the dangers caused by lead paint. The sentence was remarkablebecause even though lead paint became banned in 1978, the suit concerned damage causedduring the decades before the ban and centered on the accusation that paint makers wereaware of the dangers caused by lead paint long before the ban was passed. ∗ Emails: [email protected] and [email protected]. Strulovici gratefully acknowl-edges financial support from the National Science Foundation (Grant No.1151410).
1n essence, the court’s argument was that Sherwin-Williams and other paint makers knewor should have known the dangers caused by lead paint.While it is difficult for a regulator to guess a firm’s private information, it is perhaps easierto assess due diligence: did paint makers research the risks of lead paint sufficiently wellbefore marketing it?Formally, the problem is not just one of private information, but also one of informationacquisition: how can a regulator make sure that agents learn sufficiently well before takingactions?One may model this question as a delegated Wald problem (Wald (1945)): the principal is aregulator who relies on an agent (the firm) to acquire information before deciding betweenlaunching a product and abandoning it.If the regulator could unrestrictedly penalize a firm, she could force the firm to internalizeany damage caused by the product and implement the socially-optimal level of informationacquisition.For various reasons, liability may be capped, however, which precludes the full transfer ofdamages to the firm. Moreover, the regulator may punish the firm only if some damageoccurs, and choose a penalty that depends only on the information available to the regulatorafter the damage has occurred.We analyze this problem in a Brownian version of the Wald Problem: the firm observes anarithmetic Brownian whose drift depends on the state of the world, i.e., on the riskiness ofthe product. Information acquisition is costly. The first-best policy is to acquire informa-tion until the riskiness of the product becomes sufficiently clear, launch the product if thisriskiness is low, and abandon it if the riskiness is high.We characterize all incentive-compatible liability rules when (i) the firm has initial privateinformation, (ii) liability is capped, and the (iii) regulator can penalize the firm only whendamage occurs. In general, the regulator may wish to propose at the outset a menu ofcontracts to the firm in order to extract some of the firm’s private information. Indeed, thisis the approach suggested by the Revelation Principle. In the present context, however, thisapproach may be difficult to implement, because it requires that the firm contracts with theregulator long before launching the product and, in fact, even before knowing whether itwishes to launch the product. 2ortunately, our first main result is that it is without loss generality for the regulator to focuson tariff mechanisms , which are mechanisms for which the firm does not report its privateinformation and only pays a penalty if damage occurs. This result may be viewed as aTaxation Principle for situations in which transfers take place only after some contingencies(damage occurs), but not others, and builds on our companion paper (Poggi and Strulovici(2020)), which provides a general Taxation Principle with Non-Contractible Events.With a tariff mechanism, a firm’s decision to launch the product depends on its prior in-formation, which affects the probability that the product causes damage. Our second mainresult is that any incentive-compatible tariff mechanism has the following property: firmswhose initial private information assigns a higher probability of damage always acquire moreevidence before launching their product. This monotonicity property is not an immediateconsequence of incentive compatibility, and would in fact be violated if the regulator couldimpose evidence-based transfers to the firm regardless of whether a damage occurred.Our third main result is to show that any launch thresholds that induce the firm to performmore due diligence that it would under a fixed penalty can be implemented by a monotonictariff, i.e., a tariff whose penalty is decreasing in the strength of evidence acquired by thefirm before launching the product.We also show that for a general specification of the regulator’s objective function, setting thetariff at its uniform ceiling induces to little due diligence compared to the social optimum,even when the social benefit from launching the product exceeds the firm’s profit from doingso. This result holds under a cost-benefit ratio condition, which stipulates that the socialbenefit from the product relative to the harm it may cause is smaller that the firm’s profitrelative to the maximum liability that it may face.
A firm must decide between launching a product and abandoning its development. Iflaunched, the product may cause damage with positive probability. The firm has someprivate information about the product’s riskiness and can acquire additional information(“due diligence”), before making a final decision.A regulator wishes to encourage the launch of low-risk products and deter the launch of high-risk ones, as well as to encourage the firm to acquire sufficient information before making its3ecision.The regulator faces two constraints. First, the firm has limited liability: the social costcaused by product damage is
L > l < L . Second, the regulator can penalize the firm only if damage occurs. In particular, itcannot penalize firms that acquired too little information and took an overly risky decisionunless such risk results in actual damage.The timing of the game is as follows:1. The firm is endowed with a prior θ ∈ Θ ⊂ [0 ,
1] about the product’s riskiness y ∈ { , } , with θ = Pr( y = 1).2. The firm can acquire additional information about y according to a dynamic tech-nology to be described shortly.3. The firm decides between launching and abandoning the product.4. If the firm launches the product, it causes some damage if the product was risky( y = 1) and doesn’t if the product was safe ( y = 0).5. In case of damage, the firm pays a penalty ψ ≤ l set by the regulator.The assumption that a risky product causes damage with probability 1 is without loss ofgenerality: if this probability were less than 1, the same analysis would apply using expecteddamage and expected penalties. Information structure:
During the information-acquisition stage, the firms observes aprocess X given by X t = ( − y ) t + σB t where B is the standard Brownian motion. The drift of X depend symmetrically on theproduct’s riskiness y : the drift is +1 if the product causes damage and − X gradually reveals y . This revelation is progressive due to the stochas-tic component of X .The firm stops acquiring information at some time τ that is adapted to the filtration of X .The regulator observes nothing about X except if some damage occurs, in which case sheobserves the last value X τ taken by the process at the time of the firm’s decision. X τ isa measure of the firm’s due diligence to assess the product’s riskiness before launching it:4n this Brownian model, it is well-known (though not immediate) that for each t >
0, thevariable X t is a sufficient statistic for the information about y contained by the entire path { X s } s ≤ t of the process X until time t . Mathematically, the likelihood ratio of y associatedwith a path of X from time 0 to t is only a function of X t .Because the stopping time τ is chosen endogenously by the firm, which has private informa-tion about y , X τ is not a sufficient statistic for y once the firm’s strategic timing is taken intoaccount. Our assumption that the regulator observes X τ instead of the entire path { X t } t ≤ τ captures the idea that the regulator does not perfectly observe all the decisions made by thefirm during the information acquisition stage. Intuitively, the regulator observes the mostinformative signal about y contained by the path of X that is independent of the firm’sprivate information. Payoffs:
The firm incurs a running cost c from acquiring information, and a profit π if itlaunches the product. Let d = 1 if the firm launches the product and d = 0 if it abandonsit, and τ denote the time spent acquiring information. The firm’s realized payoff is u = d ( π − yψ ) − cτ where π is the firm’s profit from the launch in the absence of damage. The regulator’sobjective internalizes the entire damage caused by the product: v = d ( β − yL ) − cτ where β is the social benefit from the launch in the absence of damage.Throughout the paper, we make the following assumption: Assumption 1 (Ordered Cost-Benefit Ratios) l/π < L/β . This assumption captures the idea that the risk of damage is more severe for the regulatorrelative to the benefit of launching the product than it is for the firm. The assumption allowsthe social benefit from launching the product to exceed the firm’s profit (i.e., β > π ). First Best:
If the regulator knew the firm’s type θ and could dictate the firm’s strategy,the optimal strategy would consist in launching the product if the process X drops belowsome lower threshold x ∗ θ and abandoning it if X exceeds some upper threshold ¯ x ∗ θ ≥ x ∗ θ .5 ariffs: A tariff is a function ψ : R → R mapping evidence x to a penalty ψ ( x ) ≤ l . Givena tariff ψ , a firm with prior θ chooses a stopping time τ and a launch/abandonment decision d ∈ { , } to maximize its expected utility E [ d ( π − yψ ( X τ )) − cτ | θ ] . (1)It is straightforward to check that the solution to this problem consists of cutoffs x ¯ ψθ < ¯ x ψθ such that the firm acquires information until X reaches either of the cutoffs.Limited liability affects incentives in two ways. First, since the firm does not fully internalizedamages, it is willing to take riskier decisions than is socially optimal for a given belief aboutthe product’s safety. Second, the value of information is different. For example, if the tariff is ψ ≡
0, the firm has no incentive to acquire any information and always launches its productimmediately.To appreciate the consequences of limited liability, suppose that the regulator sets the tariffuniformly equal to the allowed maximum: ψ ( x ) ≡ l for all x ∈ R . In this case, the firmlaunches the product if X drops below some cutoff x ¯ lθ and abandons it if X reaches someupper cutoff ¯ x lθ .This maximum penalty may motivate the firm to perform due diligence before launching theproduct, but the amount of due diligence is always strictly suboptimal, as the next resultshows. Proposition 1 (recklessness) x ¯ ∗ θ < x ¯ lθ for all θ ∈ Θ .Proof. We fix some prior θ ∈ Θ throughout the proof and let x ∗ and x l denote thesocially-optimal and firm-optimal launch thresholds, respectively, when ψ ≡ l , given prior θ .Given a current evidence level x , the firm’s expected payoff if it launches the product at x is: u ( x ) = π − p ( x ) l where p ( x ) = Pr( y = 1 | x, θ ). The regulator’s expected payoff if the firm stops at x is: v ( x ) = β − p ( x ) L. Assumption 1 implies that v ( x ) = Ll ( u ( x ) − k ) (2) We allow negative tariffs, which amount to a subsidy for the firm and may be used to reward firms thatperformed unusually careful inspections before launching their products. k = π − βl/L > c before launching or abandoning theproduct and a payoff normalized to zero if the product is abandoned. Proposition 1 thenfollows from two observations: Observation 1:
Consider two launch-payoff functions ˆ u, u . If ˆ u = αu with α >
1, then theoptimal launch threshold for ˆ u is lower than the optimal launch threshold for u . Observation 2:
Consider two launch-payoff functions ˆ u, u . If ˆ u = u − ˆ k with ˆ k >
0, theoptimal launch threshold for ˆ u is lower than the optimal launch threshold for u .Once we justify these observations, Proposition 1 follows from (2) by applying Observation2 to u − k and u and Observation 1 to v = L/l ( u − k ) and u − k , using the fact that L/l > u = αu with α >
1, the dynamic optimization problemwith launch payoff ˆ u and running cost c is equivalent to the problem with launch payoff u and running cost ˆ c = c/α < c , since the problems become identical up to the scaling factor α .With a lower running cost ˆ c , the continuation interval ( x ¯(ˆ u ) , ¯ x (ˆ u )) contains the continuationinterval ( x ¯( u ) , ¯ x ( u )) with running cost c . In particular, the launch thresholds are ranked: x ¯(ˆ u ) ≤ x ¯( u ).To prove Observation 2, consider the optimal continuation interval ( x l , ¯ x ) when the launch-payoff function is u and let τ = inf { t : X t / ∈ ( x l , ¯ x ) } . Fixing any x ∈ ( x l , ¯ x ), acquiringinformation is optimal when starting at x , which means that u ( x ) ≤ f ( x ) u ( x l ) − cE x [ τ ] (3)where f ( x ) is the probability that X τ = x l (as opposed to ¯ x ) and E x [ τ ] is the expected valueof τ when the process X starts at x . For the launch-payoff function ˆ u = u − k with k >
0, (3)implies that ˆ u ( x ) < f ( x )ˆ u ( x l ) − cE x [ τ ] . This shows that stopping at x to launch the product is strictly dominated by the strategythat consists in launching the product if X reaches x l and abandoning it X reaches ¯ x . Thisimplies that the optimal launch threshold with ˆ u is lower than x l and proves Observation 2. (cid:4) Intuitively, Proposition 1 captures the idea that the regulator values more than the firm7aving a safer product conditional on launch. Remarkably, however, this result holds evenwhen the social benefit from launching the product exceeds the firm’s profit from doing so.Although the uniform tariff ψ ≡ l brings the firm closest to fully internalizing the damagethat its product might cause, the regulator might choose a different tariff, for example, toreward the firm if it acquired more information. The next section studies the firms’ incentivesin more details. Suppose that the regulator can contract with the firm after the firm has received its initialprivate information and before it takes any action, and that the regulator has full commit-ment power.
Definition 1 A direct liability mechanism is a menu M = ( { τ θ , d θ , ψ θ } θ ∈ Θ ) such that forall θ ∈ Θ :(i) The stopping time τ θ is measurable with respect to the filtration {F Xt } t ≥ generated by X ;(ii) The decision d θ is measurable with respect to the information at time τ , i.e., to the σ -algebra F Xτ θ ;(iii) The tariff ψ θ : R → R is uniformly bounded above by l . Since the regulator has full commitment power, the Revelation Principle guarantees that itis without loss of generality to focus on direct liability mechanisms.Given a direct liability mechanism, the firm chooses an item f ˆ θ = ( τ ˆ θ , d ˆ θ , ψ ˆ θ ) from the menu.Faced with the tariff ψ = ψ ˆ θ , the firm chooses a stopping time and a decision to maximizesits expected utility as given by (1). Definition 2
A direct liability mechanism M is incentive compatible if for each θ ∈ Θ itis optimal to chooses the item f θ from M and the strategy ( τ θ , d θ ) . In general, a direct liability mechanism may implement absurd policies: for example, thefirm could get a very high reward (i.e., a negative penalty) if it launches the product when8 t is very high (and, hence, the product is very risky). We rule out such a possibility andfocus on admissible mechanisms: Definition 3
An IC direct liability mechanism is admissible if each type θ ’s strategy ischaracterized by thresholds x ¯ θ ≤ ¯ x θ such that θ launches the product if X t drops below x ¯ θ and abandons it if X t exceeds ¯ x θ . In practice, it may be difficult for a regulator to contract with the firm at the outset andagree on penalties that depend finely on a firm’s private information before it launches aproduct and, even earlier, before the firm decides how much due diligence to perform beforedeciding whether to launch its product. It is therefore valuable to determine when a directliability mechanism can be implemented by a tariff that is independent of the firm’s privateinformation.
Definition 4
A direct liability mechanism is a tariff mechanism if the tariffs { ψ θ } θ ∈ Θ areindependent of θ . Theorem 1
Any admissible direct liability mechanism is outcome-equivalent to a tariffmechanism.Proof.
Consider any direct liability mechanism M and let x ¯ θ = x ¯ ψ θ θ and ψ θ = ψ θ ( x ¯ θ )denote the firm’s launch threshold and penalty in case of damage that are implementedunder mechanism M when the firm has type θ .We introduce a ceiling mechanism ˜ M as follows: for each θ , ˜ ψ θ gives the maximal penalty l for all x except at x ¯ θ , where it gives ψ θ . The ceiling mechanism ˜ M is IC and implementsthe same thresholds x ¯ θ , because under M the firm faces the penalty only when it launchesthe product and higher penalties at other levels can only reduce the incentive to deviate.If M prescribes the same threshold x ¯ to types θ = θ ′ , the penalties ψ θ and ψ ′ θ must beidentical. Otherwise, one type would want to misreport its type and M would not beincentive compatible.We define the tariff ψ as follows: ψ ( x ¯ θ ) = ψ θ for all θ ∈ Θ and ψ ( x ) = l M , as is easily checked. (cid:4) Theorem 1 shows that any admissible liability mechanism can be implemented by a tariff.From now on, we invoke Theorem 1 and focus without loss of generality on admissiblemechanisms that are implemented by tariffs, hereafter “admissible tariffs”.Given any admissible tariff ψ : x ψ ( x ), each type θ faces a Markovian decision problem inwhich the state variable at time t is X t . Therefore, there exist thresholds x ¯ ψθ ≤ ¯ x ψθ such thattype θ stops acquiring information when the process X leaves the interval ( x ¯ ψθ , ¯ x ψθ ), launchesthe product at x ¯ ψθ and abandons it at ¯ x ψθ .Our next result establishes a single-crossing property for the firm. Lemma 1
Consider any admissible tariff ψ , level x , and type θ ∈ Θ . If θ prefers acquiringinformation at x to immediately launching the product at x , then so does any type θ ′ ≥ θ .Proof. We fix a tariff function ψ and a level x , and suppose that X t = x at some time t thatwe normalize to 0 for simplicity. Suppose that some type θ prefers the strategy that consistsin launching the product at x ¯ < x and abandoning it at ¯ x > x , and let p = Pr( y = 1 | θ ).If θ launches the product at x , it gets: π − pψ ( x ) . (4)Let T g , f g denote the expected hitting time and the probability of hitting x ¯ if y = 0 (theproduct is good), and T b and f b be defined similarly if y = 1 (the product is damaged). If θ continues until hitting x ¯ or ¯ x , its expected payoff is p ( f b ( π − ψ ( x ¯)) − cT b ) + (1 − p )( f g × π − cT g ) . (5)Comparing (4) and (5), continuing is optimal if p ( f b ( π − ψ ( x ¯)) + ψ ( x ) − cT b ) + (1 − p )( f g π − cT g ) ≥ π. (6)The left-hand side is a convex combination of two terms: a = f b ( π − ψ ( x ¯)) + ψ ( x ) − cT b and b = f g π − cT g . The second term, b is less than π , because f g is a probability. Therefore, (6)can hold only if the first term, a , is greater than π .10ewriting (6), a firm that assigns probability p to y = 1 wishes to continue if p ( a − b ) ≥ π − b. Since a > b , the coefficient of p is strictly positive. This implies that any type that assignsprobability p ′ > p to y = 1 also prefers the continuation strategy to launching the productimmediately at x . (cid:4) Lemma 1 has the following intuition: If a firm knew that the product were safe, it wouldoptimally launch the product immediately. The return to acquiring more evidence is negativein this case. Given any liability function, if a type wants to acquire more evidence it must bethat doing so has a positive return conditional on the product being unsafe. The expectedreturn from acquiring more evidence is thus increasing in the probability that the firm assignsto the product being faulty.Lemma 1 immediately implies the following monotonicity result:
Proposition 2
For any admissible tariff ψ , the launch thresholds x ψ ( θ ) are decreasing in θ . This monotonicity result crucially hinges on the fact that the regulator can only charge thefirm if it causes some damage. The following example shows that if the regulator can chargethe firm even when the product causes no damage, the launch thresholds increase with thetype of the firm. Example: Monotonicity Violation with Damage-Independent Fee
Suppose that the assumptions of our model are maintained with one exception: if the firmlaunches its product, the regulator charges the firm a fee η ( x ) ≤ l that depends on theevidence x demonstrated when the product is launched, independently of any damage sub-sequently caused by the product.We assume that l > π , so that the regulator can deter the firm from launching the productat any given x by setting η ( x ) = l . An admissible direct revelation mechanism specifies, foreach type, a launch threshold x ¯ θ and a fee η θ = η ( x ¯ θ ). Without loss of generality we assumethat η ( x ) = l for all x / ∈ { x ¯ θ } . This example is partially inspired by the approval mechanisms in McClellan (2019) andHenry and Ottaviani (2019). bad firms with prior θ = 0 and good firms with prior θ = 1.We construct an IC mechanism for which x ¯ < x ¯ . We start by setting a launch threshold x ¯ < η = η ( x ¯ ) low enough that (i) a bad firm forced to launchthe product at x ¯ abandons the product at a threshold ¯ x > x > | x ¯ | and (ii)this strategy yields a strictly positive expected payoff to the bad firm. Such a constructionis always possible by choosing x ¯ close enough to 0.Next, we fix some launch threshold x ¯ ∈ (2 x ¯ , x ¯ ) for the good firm and choose η = η ( x ¯ )so that a good firm is indifferent between launching the product at x ¯ and at x ¯ . Such aconstruction is always possible by choosing x ¯ close enough to x ¯ and η slightly lower than η . By construction, a good firm is indifferent between the two items of menu { ( x ¯ , η ) and( x ¯ , η ) } .To demonstrate incentive compatibility, there remains to show that a bad firm prefers thesecond item on this menu. Let ¯ x d denote the optimal abandonment threshold of the badfirm if it launches its product at x ¯ . Suppose first that ¯ x d ≤
0, This means that the firmprefers to abandon immediately, starting from X = 0. This yields an expected payoff ofzero and is dominated by the item ( x ¯ , η ). Now suppose that ¯ x d >
0. By construction, x ¯ is closer to x ¯ than it is to ¯ x d . Lemma 2, then implies that at state x ¯ , a good firm gets astrictly higher payoff than the bad firm by adopting the strategy ( x ¯ , ¯ x d ). The good firm isby construction indifferent between the strategy ( x ¯ , ¯ x ) and stopping immediately at x ¯ , sothe good firm weakly prefers to stop immediately to adopt the strategy ( x ¯ , ¯ x d ). Moreover,both types get exactly the same payoff if they stop at x ¯ , since both firms pay the η ( x ).Therefore, the strategy ( x ¯ , ¯ x d ) must be strictly worse for the bad firm than stopping at x ¯ .This shows that the mechanism is incentive compatible for both types of firms, and that therisky firm ( θ = 1) launches the product with less evidence than the safe one ( θ = 0), since x ¯ < x ¯ . Lemma 2
Consider the strategy that consists in launching the product at x ¯ and abandoningit at ¯ x > x ¯ , and consider any x ∈ ( x ¯ , ( x ¯ + ¯ x ) / . If η ( x ¯ ) < π , the expected payoff from thestrategy, starting from X = x , is higher for the good firm than for the bad firm.Proof. We follow the notation used in the proof of Lemma 1. Starting from X = x , theOptional Sampling Theorem applied to the identity function X t X t and to type θ = 112mplies that E [ X τ | x, θ = 1] = x + E [ Z τ − dt ] = x − T b . Expressing the expectation on the left-hand side in terms of hitting probability f b andrearranging yields:( − · T b = ( x ¯ − x ) f b + (¯ x − x )(1 − f b ) = (¯ x − x ) − f b (¯ x − x ¯)Proceeding similarly for type θ = 0, we get: T g = ( x ¯ − x ) f g + (¯ x − x )(1 − f g ) = (¯ x − x ) − f g (¯ x − x ¯)Summing the last two equations yields T g − T b = 2(¯ x − x ) − ( f g + f b )(¯ x − x ¯) . (7)We have x ∈ ( x ¯ , ( x ¯ + ¯ x ) / x − x ) < ¯ x − x ¯ and that f g + f b > Therefore (7) is negative, which shows that T b > T g . The difference of the good and badtypes’ expected payoffs is given by:( f g ( π − η ) − cT g ) − ( f b ( π − η ( x ¯)) − cT b ) = [ f g − f b ] | {z } > ( π − η ) | {z } > + c [ T b − T g ] | {z } > . which is strictly positive. (cid:4) Proposition 1, shows that the regulator would like to implement lower thresholds than thefirm when the firm faces with a uniform penalty, regardless of the firm’s private information.The next proposition shows that under these circumstances, it is without loss of generalityto focus on tariffs that are nondecreasing functions of x , i.e., which impose a lower penalty,the more due diligence is demonstrated by the firm. Proposition 3
Suppose that Θ is finite and consider any thresholds { x θ } θ ∈ Θ that are (i)decreasing in θ and (ii) such that x θ ≤ x ¯ lθ for all θ ∈ Θ . Then, there exists a non-decreasing,piecewise-constant tariff ψ such that x ¯ ψθ = x θ for all θ ∈ Θ . For the latter inequality, notice that the drifts of X t are exact opposite for good and bad firms, so that ( f g + f b ) is the probability that the Brownian process X t with drift either 1 or -1 with equal probabilityhits x ¯ before ¯ x when starting from x . Since x is closer to x ¯ than it is to ¯ x , this probability is greater than1 /
2, which implies that f g + f b > roof. We index the elements of Θ from the smallest θ to the largest θ | Θ | and constructthe tariff ψ by moving from large values of x to lower ones. We start by setting ψ ( x ) ≡ l for all x ≥ x θ . At x θ , we lower the tariff to a level ψ that makes θ exactly indifferentbetween launching the product at x θ and at x ¯ lθ . We keep ψ constant at the level ψ for x ∈ ( x θ , x θ ]. Since a firm’s launch threshold when it faces a constant tariff ˆ l is decreasingin ˆ l , and since ψ < l , we have x θ < x ¯ lθ ≤ x ¯ ψ θ where x ¯ ψ θ is the launch threshold used by type θ when the tariff is constant and equal to ψ . This implies that type θ prefers threshold x θ to any level x ∈ ( x θ , x θ ).At x θ , we lower the tariff ψ to a level ψ that makes type θ exactly indifferent betweenlaunching the product at x θ and at its preferred level ˆ x among all x > x θ , given the tariff ψ constructed so far. By the single-crossing property established in Lemma 1, this impliesthat θ prefers ˆ x to any x θ and, combined with the previous paragraph, that θ prefers x θ to any x ≥ x θ .We set ψ equal to ψ for all x ∈ ( x θ , x θ ]. Since x θ ≤ x ¯ lθ ≤ x ¯ ψ θ , type θ prefers x θ to any x ∈ ( x θ , x θ ). Another application of Lemma 1 guarantees that type θ also prefers x θ toany x ∈ ( x θ , x θ ).Proceeding iteratively, we then lower ψ at x θ to a level ψ that makes type θ exactlyindifferent between launching the product at x θ and at its preferred level ˆ x > x θ given thetariff ψ constructed so far. Repeated applications of Lemma 1 guarantee that types θ , θ prefer their respective thresholds x θ , x θ to x θ . We extend ψ by setting it constant, equalto ψ for all x ∈ ( x θ , x θ ]. The proof is completed by induction. (cid:4) When an IC mechanism implements distinct thresholds for distinct types, the conclusionof Theorem 1 is a corollary of the Taxation Principle with Non-Contractible Events of ourcompanion paper (Poggi and Strulovici (2020)).According to that paper, a mechanism is identifiable if satisfies two conditions that wetranslate into the present setting. Let A denote the set of all possible strategies by the14rm. Each element of A consists of a pair ( τ, d ), where τ is a stopping time adapted to thefiltration of X and d is measurable with respect to F Xτ . For any subset A ′ of A , let X ( A ′ )denote the set of observable outcomes by the regulator if the firm chooses an action a ∈ A ′ and causes some damage. Definition 5
An IC mechanism M is identifiable if there exists a partition A = { A k } Kk =1 of A such that(i) X ( A k ) ∩ X ( A k ′ ) = ∅ for all k = k ′ .(ii) All types θ who choose an action in A k under the mechanism choose the sameaction of A k . Proposition 4 If M implements distinct launch thresholds for all types, then it is identi-fiable.Proof. For each θ , let A θ denote the set of firm strategies that use launch threshold x ¯ θ , andlet A = A \ ( ∪ θ ∈ Θ A θ ). By assumption on M , x ¯ θ = x ¯ θ ′ for all θ = θ ′ . Therefore, A θ and A ′ θ are disjoint for all θ = θ ′ and A = { A , A θ : θ ∈ Θ } forms a partition of A . Condition (ii) istrivially satisfied since for each cell of A there is at most one type taking action in that cell.Moreover Condition (i) is also satisfied by construction of the partition: X ( A θ ) = { x ¯ θ } forall θ ∈ Θ and, hence, X ( A θ ) ∩ X ( A θ ′ ) = ∅ for all θ = θ ′ . (cid:4) Corollary 1
If an IC mechanism M implements distinct launch threshold for all types, itcan be implemented by a tariff mechanism.Proof. Proposition 4 implies that M is identifiable. The result then immediately followsfrom Theorem 1 in Poggi and Strulovici (2020) (cid:4) eferences Henry, E. and Ottaviani, M. (2019). Research and the Approval Process.
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