BBailout Stigma
Yeon-Koo Che Chongwoo Choe Keeyoung Rhee ∗ June 11, 2020
Abstract
We develop a model of bailout stigma where accepting a bailout signals a firm’sbalance-sheet weakness and worsens its funding prospect. To avoid stigma, a firm withhigh-quality legacy assets either withdraws from subsequent financing after receiving abailout or refuses a bailout altogether to send a favorable signal. The former leads to ashort-lived stimulation with subsequent market freeze even worse than if there were nobailout. The latter revives the funding market, albeit with delay, to the level achievablewithout any stigma. Strikingly, a bailout offer is most effective when many firms rejectit (to build a favorable reputation) rather than accept it.
Keywords:
Adverse selection, bailout stigma, secret bailout
JEL Codes:
D82, G01, G18
History is fraught with financial crises and large-scale government interventions, the latteroften involving a highly visible and significant wealth transfer from taxpayers to banks andtheir creditors. According to an IMF estimate based on 124 systemic banking crises fromaround the world during the period 1970-2007, the average fiscal costs associated with crisismanagement were around 13 percent of GDP (Laeven and Valencia, 2008). More recently,during the 2007-2009 Great Recession, the US government paid $125 billion for assets worth ∗ Che: Department of Economics, Columbia University (email: [email protected]); Choe:Centre for Global Business and Department of Economics, Monash University (email: [email protected]); Rhee: Department of Markets and Institutions, Korea Development Institute (KDI)(email: [email protected]). The views expressed in this paper are solely those of the authors and shouldnot be attributed to KDI. The authors are grateful to Olivier Armantier, Xuandong Chen, Keshav Dogra,Oliver Giesecke, Spencer Kwon, Haaris Maarten, Jean Tirole, as well as participants at numerous seminarsand conferences, for their comments. a r X i v : . [ q -f i n . GN ] J un At the time of writing this paper, an unprecedented scaleof bailouts is being organized to meet the challenges of the unfolding COVID-19 crisis. Thebenefits of such interventions are difficult to measure since they depend on the unobservablecounterfactual that would have played out in the absence of such interventions.Philippon and Skreta (2012) and Tirole (2012) develop a plausible counterfactual of amarket freeze and a rationale for how government interventions may improve welfare. Theessence of the argument is that the government can jump-start a market that would otherwisefreeze due to adverse selection. By cleaning up bad assets, or “dregs skimming,” through publicbailouts, the government can improve market confidence, thereby galvanizing transactions inhealthier assets. However, the flip side of such dregs-skimming is that bailouts can attachstigma to their recipients, and thus increase future borrowing costs. The fear of this stigmamay in turn discourage financially distressed firms from accepting bailout offers in the firstplace.Policy makers during the Great Recession were well aware of such a fear and took effortsto alleviate the stigma. At the now-famous meeting held on October 13, 2008, Henry Paulson,then Secretary of the Treasury, “compelled” the CEOs of the nine largest banks to be theinitial participants in the TARP, precisely to eliminate the stigma (“Eight days: the battle tosave the American financial system,”
The New Yorker , September 21, 2009). The rates at theFed’s discount window, usually set above the federal funds rate, were cut half a percentagepoint to counteract the stigma that using the window would signal distress (Geithner, 2015,p. 129). Despite these efforts, the stigma remained real and significant. Armantier et al. (2015)documented that the banks were willing to pay 44 basis points (bps) more for borrowingfrom the Term Auction Facility (TAF) than they would pay for using the discount window.Gauthier et al. (2015) further demonstrate that the banks that accessed the TAF in 2008paid approximately 31 bps less in interbank lending in 2010 than those that used the discountwindow. Given that the TAF was designed to hide the identities of its users, a possible expla-nation is that banks wanted to avoid stigma attached to using the discount window. There arealso anecdotes highlighting the presence of stigma. Ford refused rescue loans under the Auto Congressional Budget Office (2012) estimates the overall cost of the TARP at approximately $32 billion,the largest part of which stems from assistance to AIG and the automotive industry while capital injections tofinancial institutions are estimated to have yielded a net gain. For detailed assessments of the various programsin the TARP, see the
Journal of Economic Perspectives (2015) . See also Fleming (2012) who discusses howthe various emergency liquidity facilities provided by the Federal Reserve during the 2007-2009 crisis weredesigned to overcome the limitations of traditional policy instruments at the time of crisis. Tong and Wei(2020) provide international evidence on the effect of unconventional interventions during 2008-2010. Such a concern is echoed in a speech given by the former Federal Reserve chairman Ben Bernanke in 2009:“The banks’ concern was that their recourse to the discount window, if it became known, might lead marketparticipants to infer weakness—the so-called stigma problem.”
The New York Times , December 19, 2008). Such reluctance to receive government offers ofrecapitalization was also noted during the Japanese banking crisis of the 1990s (Corbett andMitchell, 2000; Hoshi and Kashyap, 2010). While differing in nature, a similar concern aboutbailout stigma is expressed in the wake of the current COVID-19 crisis (“Bernanke and Yellen,the Federal Reserve must reduce long-term damage from coronavirus,”
The Financial Times ,March 23, 2020).Examples such as the above raise questions about whether public bailouts are effectiveand, if so, how they should be designed in light of the associated stigma. We address thesequestions by analyzing a two-period model of bailouts that can address reputational concernsmost parsimoniously. There is a continuum of firms, each with one unit of a legacy asset ineach period. The quality of asset in both periods is identical for each firm and is its privateinformation unobserved by other parties. In each period, firms have access to profitableinvestment opportunities. However, the liquidity constraint and the lack of pledgeability ofprojects imply that firms need to sell assets in the market to fund those projects. As in Tirole(2012), we focus on the case where adverse selection leads to a market freeze and calls for apublic bailout. To understand the reputational consequence of accepting a bailout, we assumethe government runs a bailout program in the first period only, and firms must sell theirassets in the market to fund their projects in the second period (as well as in the first period).The sale of assets in the market is not publicly observed, but firms’ acceptance of the bailoutoffer is. The market updates its belief on the quality of assets based on the observation offirms’ decisions on asset sale made in the first period, and thus makes its second-period offeraccordingly.Bailout stigma is captured in our model by the worsenened terms that bailout recipientsface for the sale of their assets in the second period. As shown by Philippon and Skreta (2012)and Tirole (2012), a key function of public bailouts is dregs-skimming: by taking out theleft tail of the worst quality assets, the bailout improves the perceived quality of remaining The market initially perceived Ford’s refusal to accept a bailout as a risky move, which was re-flected in the rise in Ford’s CDS spreads relative to Chrysler’s. However, Ford’s profit and stockprice showed a remarkable turnaround in 2009, part of which is attributed to the respect Fordgarnered with customers and investors by refusing a bailout. ( , accessed Nov 17, 2015). In a similar vein, par-ticipants in the TARP were eager to exit the program early, often citing stigma as their main motivation.Signature Bank of New York was one of the first to repay its TARP debt of $120 million for this reason. Itschairman, Scott A. Shay, said, “We don’t want to be touched by the stigma attached to firms that had takenmoney.” (“Four small banks are the first to pay back TARP funds,”
The New York Times , April 1, 2009). Itis also well known that Jamie Dimon, CEO of JP Morgan Chase, wanted to exit TARP to avoid the stigma(“Dimon says he’s eager to repay ‘Scarlet Letter’ TARP,”
Bloomberg , April 16, 2009). Of course, the fear ofstigma is not the only reason for an early exit. Wilson and Wu (2012) find that early exit by banks is alsorelated to CEO pay, bank size, capital, and other financial conditions. short-lived stimulation and delayed stimulation .A short-lived stimulation equilibrium arises when high-quality firms strategically avoidthe bailout stigma by accepting a bailout in the first period but boycotting the market inthe second period. Since bailout recipients with high-quality assets are more likely to boycottthe market (which is a consequence of the usual single-crossing property), those that accepta bailout but are compelled to participate in the second period market suffer a severe stigma.This makes low-quality firms reluctant to accept a bailout, forcing them to sell their assets tothe market instead at a discounted price to avoid the stigma. In the end, however, no arbitragebetween selling to the government and selling to the market means that market sellers mustalso suffer a corresponding level of haircut. Effectively, the bailout stigma “spreads” frombailout recipients to market sellers. In turn, this contagion of stigma leads firms with high-quality assets to avoid selling to the market, opting instead to accept bailout offers whileboycotting the second-period market altogether to avoid the stigma. Finally, their withdrawalfrom the second-period market inflicts a severe stigma on those firms (with low-quality assets)that must participate in that market, thus completing the feedback loop.The presence of firms that accept a bailout but boycott the subsequent market thwarts thedregs-skimming role of bailouts and undermines the overall effectiveness of bailouts. Not onlydo these firms withdraw from the second-period market, but their withdrawal also exacerbatesthe stigma for those participating in the second-period market with a devastating consequence: the market freeze is even worse than if there were no bailout!
However, this does not necessarilymean that a bailout has no effect; it stimulates asset trade in the first period, and this effectmay outweigh the dampening effect in the second period. Nevertheless, stimulation will beshort-lived in this equilibrium. We show that the policy maker can avoid this equilibrium byoffering sufficiently generous bailout terms—i.e., high purchase prices for assets, in which casethe stigma will manifest itself in a different form: “delayed stimulation.”A delayed stimulation equilibrium arises when high-quality firms refuse to sell either tothe government or to the market in first period. They do so to build a good reputation ontheir assets so that they can sell their assets in the second period at favorable terms. Thisequilibrium is possible when the bailout offer is generous enough to attract a large fraction offirms with lower-quality assets. This allows firms to send a credible signal about their assetquality when they refuse the bailout offer. Buyers will then respond with a very attractive4rice offer in the second period—one that makes it worthwhile for high-quality firms to foregoasset sale in the first period. In sum, the equilibrium endogenously creates an opportunityfor high-quality firms to signal their financial strength by rejecting the government’s generousoffer.Such a favorable signaling opportunity offsets the adverse effect of bailout stigma, eventhough market rejuvenation is delayed to the second period. The presence of firms rejecting abailout means that the volume of asset trade is lower in the first period than in the one-periodbenchmark with the same bailout offer. In an extreme case, it is even possible that the bailouthas no stimulation effect in the first period relative to the laissez-faire economy. Such an initiallack of response may be seen as a policy failure. However, the policy is “quietly” strengtheningthe confidence (in the refusing firms) and bears dividend in the second period. In fact, theoverall trade volume is higher than in a short-lived stimulation equilibrium; remarkably, it isthe same as if there were no bailout stigma —that is, if the identities of bailout recipients wereconcealed successfully, which is often difficult to achieve in practice. Except for the delay,rejection of bailouts by high-quality firms could very well be a blessing in disguise.The conclusion that the policy maker may wish to offer a generous bailout term to inducea delayed stimulation equilibrium is further reinforced once the costs of bailouts are takeninto account. Even though the bailout term required for a short-lived stimulation equilibriumis more modest, the policy ends up being costly since it induces high-type firms which wouldnot otherwise require a bailout to accept a bailout at terms that would compensate theirstigma. For this reason, the delayed stimulation equilibrium, if it can be induced, achieves thesame stimulation effect as short-lived stimulation equilibrium at a lower cost. Our theory thusrecognizes the need for bailout terms to be sufficiently generous to yield a tangible benefit.This implication, although departing from the classical Bagehot’s rule, is consistent with theapproach taken by the policy makers in the Great Recession.The remainder of the paper is organized as follows. Section 2 presents our model whileSection 3 analyses several benchmark cases. In Section 4, we study various equilibria undergovernment intervention. Section 5 provides the welfare analysis of bailout policy. Section 6discusses related literature. Section 7 concludes. Proofs not contained in the main text aredeferred to Appendix A and Online appendix. Bagehot’s rule, orginating from the 1873 book,
Lombard Street , by William Bagehot, prescribes thatcentral banks should charge a higher rate than the markets to discourage banks from borrowing once the crisissubsides. Bailout stigma was not a serious issue in 1873, however, since the regulatory system in 1873 Britainensured concealment of the identities of emergency borrowers, as Gorton (2015) points out. Model
We adopt a model that extends Tirole (2012) to a setup which admits a bailout stigma. Thereis a continuum of firms each endowed with two units of legacy assets with the same value; oneunit of the asset becomes available in each of two periods ( t = 1 ,
2) for possible sale. Theasset value θ of each firm is privately known to that firm and distributed on [0 ,
1] accordingto cdf F with density f . For convenience, we hereafter call a firm with legacy asset θ a type- θ firm. Throughout, we assume that f is log-concave, i.e., d log f ( θ ) /dθ <
0. Log-concavityof f implies intuitive properities we shall use on truncated conditional expectation: for any0 < a < b <
1, 0 < ∂∂a E [ θ | a ≤ θ ≤ b ] , ∂∂b E [ θ | a ≤ θ ≤ b ] < b ∈ (0 , E [ θ | a < θ < b ] − E [ θ | θ ≤ a ]is increasing in a for any a ∈ (0 , b ). These properties, which hold for many well-knowndistributions, facilitate the characterization of our equilibria.In each period, an investment project becomes available to each firm. The project issocially valuable with net return S >
I >
0. The firm can financethe project by selling its legacy asset each period. As we will see, the outcome from this laissez-faire regime will typically be inefficient due to the adverse selection associated withuncertain asset value. This inefficiency rationalizes a government bailout in the form of anoffer to purchase legacy assets at some price p g . The timeline of our full game is depicted inFigure 1: t = 1 t = 2 Firms privatelylearn the value oftheir legacy assets. Government offersto buy one unit ofthe asset at p g . Buyers in the marketmake offers. Firms accept either agovernment offer, amarket offer, or none.Those who sell fundthe project.Buyers make offersfor t = 2 legacy as-sets. Firms either sell tothe buyers or holdout. Those who sellfund their projects. Project returns forboth periods are real-ized. Figure 1 – Timeline for the two-period modelTo focus our attention on the main issue—namely, bailout stigma—we make several sim-plifying assumptions. One can think of the assets as account receivable or the contract for (securitized) assets to be deliveredover two periods. Second, project returns are realized at the end of t = 2, so it is impossible to use thereturn from t = 1 project to finance the t = 2 project. This assumption is made primarilyto simplify the analysis but is well justified in many settings in which there are differencesbetween accrual and realization of cash flows. That is, we consider the case where the projectreturn accrues each period if it is funded but the final cash flow realizes in t = 2.Third, as is standard in the literature, we assume that asset buyers are competitive andmake purchase offers in Bertrand fashion. We assume that, in the event of an indifference,a buyer breaks a tie in favor of buying an asset rather than not buying. Buyers live forone period and make offers that would break even in expectation. Importantly, the buyers in t = 2 can make rational inference about firms’ types from their observable behavior in t = 1,in particular with their acceptance/rejection of a bailout offer.Fourth, we assume that the sale of assets to the market in t = 1 is private and therefore notrevealed to buyers in t = 2. This implies that buyers in the t = 2 market cannot distinguishbetween those that sold in the t = 1 market and those that did not. Again the primary reasonfor this assumption is to simplify the analysis by shutting off channels of dynamic informationrevelation. But this assumption is well justified given that many important financial and realassets are sold privately over the counter. The main thrust of our results extends to the caseof observable sale, as shown by the working paper version of our paper (see Che, Choe andRhee (2018).) Further, this assumption makes a comparison with Tirole (2012) transparent,thus helping to isolate the effect of stigma.Fifth, a government bailout is available only in the first period. This is consistent withthe observed practice: governments refrain from engaging in long-term bailouts and fromcomplete “nationalization” of distressed firms (which would be equivalent to purchasing twounits of the asset in our model). Further, our goal is to study the reputational consequenceof accepting a bailout, which can be studied most effectively when no bailout is available in Their insight appears to apply to our context, which suggests that debt contracts would be optimal in ourcontext as well. Since the stigma issue is separable from the issue of contract form, we abstract from it in thecurrent paper. If a project return from t = 1 can be used to fund the project in t = 2, then this will affect the amountof funding each firm demands based on the successful funding in the first period. This complicates analysisin a way that does not add any obvious new insight to our central theme. In particular, bailout stigma willremain relevant as long as the funding need is not fully met by the cash flow generated by an early project. This tie-breaking assumption is largely to simplify analysis and exposition. It has no material effect onthe substantive results obtained in the paper. t = 2 market observes which firmshave accepted the bailout. Not only do transparent bailouts highlight the stigma effect mostclearly, but they are also important from a practical perspective. While secret bailouts mayaddress the stigma problem, secrecy is often difficult to achieve in practice. Before proceeding, we study several benchmarks. They will facilitate comparison with, andprovide a context for, our main results, which will follow in Section 4.
Laissez Faire without Government Bailout
We first consider the benchmark without a bailout. The timeline is the same as Figure 1,except that the government’s bailout is absent. Since the sale of the assets is private andnot publicly revealed, there is no linkage between the market outcomes across two periods.Thus, the game reduces to a one-period game (repeated twice) whose equilibrium coincideswith that of Tirole’s game without bailout.Fix any period. The market equilibrium is understood best as a form of Akerlof’s lemonsproblem, which is depicted in Figure 2. QuantityPrice Average BenefitCurve ( E [ θ | θ ≤ x ] )Supply Curve( θ − S ) θ p Marginal BenefitCurve − S Figure 2 – Determination of the cutoff type θ in the laissez-faire economy8 type- θ firm benefits p + S from selling a unit of asset at price p if p ≥ I , but loses itsasset value θ . Hence, the firm sells asset if and only if θ < p + S , as described by the supplycurve in the figure. Meanwhile, the buyer of the asset realizes its value θ minus price p . Sinceif type- θ firm sells, then type θ (cid:48) -firm with θ (cid:48) < θ strictly prefers to sell—a property known as“single-crossing”—the buyer’s perceived value is given by the average benefit E [ θ | θ ≤ x ], asdepicted in the figure, if the higest type selling is x .The equilibrium is then characterized by the intersection of the supply and average benefitcurves, namely by a cutoff θ such that: θ − S = E [ θ | θ ≤ θ ] =: p . (1)First, buyers must break even, so the equilibrium price must be p = E [ θ | θ ≤ θ ], theaverage value of the asset, and this price must be at least I (or else trading would not occur).Next, the threshold type θ must find the price p at least worth the opportunity cost of sale θ − S (depicted by Figure 2). Given the log-concavity assumption, there is a unique threshold θ satisfying the requirements. Figure 3 summarizes the equilibrium configuration. θ = p + St = 1 the worst type the best type t = 2 p = E [ θ | θ ≤ θ ] Figure 3 – No bailout equilibriumAdverse selection means that the above outcome is typically inefficient. Specifically, if
S < − E [ θ ], then θ <
1, so not all firms sell and finance their projects. It is also possiblefor θ = 0, in which case the market freezes completely. To focus on the nontrivial case, weassume θ <
1. For expositional ease, it is also convenient to focus on the partial freeze case( θ >
0) in what follows. We will discuss the full freeze case ( θ = 0) later in Remark 2. Example 1.
Consider the uniform case, i.e., F ( θ ) = θ . In this case, the equilibrium price p is determined by p = E [ θ | θ ≤ θ ] = θ . Suppose θ ∈ (0 , . Then, from the indifference It is routine to check that if f is log-concave ( ∂ log f ( θ ) ∂θ < θ ), then there is a unique θ satisfying(1); see Tirole (2012). As mentioned, buyers cannot update their information since the market transactions are private. If markettransactions were observable, then trading decisions become dynamic, which makes analysis complicated; seeChe, Choe and Rhee (2018). ondition θ = p + S = θ + S , the cutoff type θ is uniquely determined as θ = 2 S , and p = S , if S ∈ [ I, / . If S < I , then the equilibrium price p = S cannot fund the project,so the market fully freezes, and hence θ = 0 . If S ≥ − E [ θ ] = 1 / , then θ ≤ E [ θ ] + S for all θ ∈ [0 , , and therefore, θ = 1 . We next consider another benchmark, the one-period bailout model by Tirole (2012) in whichthe government offers to purchase assets at price p g above the laissez-faire price p before themarket opens. Specfically, the timeline simply comprises t = 1 in Figure 1. Since there is noconsequence of accepting a bailout from the government in this one-period model, there is nobailout stigma—at least in the sense we will capture in our two-period model later. Thus,this benchmark will help to identify the role of bailout stigma later in our main analysis.A Perfect Bayesian equilibrium, or called simply an equilibrium from now on, of this gameis characterized as follows. Let µ g and µ m denote the fractions of types θ ≤ p g + S that sell tothe government and to the market, respectively, where µ g + µ m = 1. We can first argue that µ g >
0. If no firm accepts the government offer, then the laissez-faire equilibrium will prevail,with marginal type θ and equilibrium price p = E [ θ | θ < θ ]. Since p g > p , however, firmswill deviate to accept the government offer, a contradiction.Next, suppose µ m >
0, so the market is active in equilibrium. Then, the market price p m must equal the government price p g , or else a lower offer will not be accepted. Given this,firms must sell (either to the government or to the market) if and only if θ < p g + S . Let θ g and θ m denote the average values of assets sold to the government and the market, respectively( θ m can be arbitrary in case µ m = 0). Clearly, we must have µ g θ g + µ m θ m = E [ θ | θ ≤ p g + S ] . (2)Further, since the market buyers must break even in case µ m >
0, we have p m = θ m . Given An astute reader will notice that this timeline differs slightly from that considered by Tirole (2012),where the market opens after firms have decided on the government offer. We adopt the current timelinesince it is arguably more realistic, and also it permits equilibrium existence more broadly for our two-periodextension. For the one-period version, the difference is immaterial, since the equilibrium under the currenttimeline is payoff-equivalent to Tirole’s equilibrium for all players involved. In addition, we do not invoke anequilibrium refinement adopted in Tirole (2012), as the central feature of the equilibrium holds irrespective ofthe refinement. See Remark 1. Note Tirole (2012) and Philippon and Skreta (2012) do recognize “stigma” associated with the types offirms that accept a bailout, but they do not study its effect on the subsequent game as well as on the initialdecision to accept the bailout, the dual focuses of the current paper. Under our timeline, the market may not be active in equilibrium. To see how such an equilibrium canbe supported, suppose a buyer deviates and offers a price p (cid:48) > p g . Since firms have not yet accepted thegovernment offer by then (given our timeline), all types θ < p (cid:48) would accept the deviation offer, and thedeviating buyer will suffer a loss since p (cid:48) > p g > p . g = p m , this in turn implies θ m = p g .The central feature of the Tirole (2012) follows from these observations: Theorem 1 (dregs-skimming) .(i)
In any equilibrium of the one-period benchmark with government offer p g > max { p , I } ,firms sell assets (either to the government or to the market) at price p g if and only if θ < p g + S . Since p g > p , more firms finance their projects than without the governmentintervention. (ii) If µ m > so that the market is active, then we must have θ g < θ m ; i.e., on average lowervalue assets are sold to the government than to the market.Proof. We have already established (i). To prove (ii), suppose to the contrary θ m ≤ θ g .Then, by (2), θ m ≤ E [ θ | θ ≤ p g + S ]. Since θ m = p g , we have p g ≤ E [ θ | θ ≤ p g + S ], or p g + S ≤ E [ θ | θ ≤ p g + S ] + S . By the definition of θ , p g + S ≤ θ = p + S , whichcontradicts p g > p . The remaining characterizations follow from the observations precedingthe theorem. Q.E.D.
Figure 4 illustrates the outcomes with and without government bailout. By offering ahigher price p g than the laissez-faire price p , the government does indeed take out relativelylow-value assets, which in turn improves the perception of the assets sold to the market andthus alleviates adverse selection. θ ˆ θ g = p g + S the worst type the best typeEquilibrium without bailoutEquilibrium with bailout Note: the types selling to a market is depicted by blue and the types selling to the governmentis depicted by red.
Figure 4 – Effects of bailout in the one-period benchmarkMore importantly for our purpose, assets are sold to the government at the same price asthey are sold to the market. This reflects the absence of stigma associated with accepting a11ailout. Plainly, in the one-period problem, firms that accept the bailout do not have anyconsequences to worry about simply because the game ends after the bailout.
Remark 1 (The role of the equilibrium refinement in Tirole (2012)) . To obtain the “dregs-skimming” role of bailout, Tirole (2012) invokes an equilibrium refinement—that the marketsale collapses with an arbitrarily small probability. This refinement “forces” the equilibriumto have the dregs-skimming feature, since it implies single-crossing: namely, there exists ˜ θ ∈ (0 , p g + S ) such that types θ ≤ ˜ θ all sell to the government and types θ ∈ (˜ θ, p g + S ] all sellto the market. However, this refinement obfuscates the source of dregs-skimming: namely,whether it is an artifact of the refinement or something more fundamental. Without invokingthat refinement, Theorem 1 proves that dregs-skimming is fundamental (and not driven bythe refinement). Without the refinement, however, there are multiple equilibria that differ interms of µ g and the value θ g , but every such equilibrium exhibits the “dregs-skimming” feature. In order to identify the effects of bailout stigma, we need to understand what happens if thepolicy maker can eliminate the stigma altogether. Imagine that the policy maker “completelyand successfully” conceals the identities of the firms that accept the government offer. Thiskind of secrecy has been an important part of the bailout policy, precisely because of the stigmaissue. In this sense, secret bailouts are worth studying in their own right. Nevertheless, weprimarily regard secrecy as a benchmark against which transparent bailouts are compared,given our premise that “complete” secrecy has been so far difficult to achieve despite manyconcerted efforts. The equilibrium under complete secrecy is very easy to analyze. Since neither selling to the Suppose a market sale is subject to probability (cid:15) > θ firm prefers to sell to thegovernment, then p g + S ≥ (1 − (cid:15) )( p m + S ) + (cid:15)θ , where p m is the equilibrium market price. This means thatall types θ (cid:48) < θ must strictly prefer to sell to the government. Gorton and Ordo˜nez (forth) supports such a policy. During crises, debt contracts lose “informationinsensitivity” as investors scrutinize the downside risk of underlying collaterals, leading to an adverse selection.They argue that withhoding information on whether borrowers borrow from discount windows of central bankscan make debtors less information sensitive and alleviate adverse selection. As will be seen, secrecy has a morenuanced effect in our model. The identities of banks borrowing from the discount window facilities (DW) are occasionally leaked toeither the news media or the market participants through a number of channels. First, despite the apparentsecrecy attached to DW, the access to DW by borrowing firms has been detected by news media (Armantieret al., 2015; Berry, 2012). For instance, the Financial Times reported the news that Deutsche Bank hadborrowed from DW one day ago (see “Fed fails to calm money markets,”
The Financial Times , August 20,2007). Second, the market participants can identify DW borrowers from these banks’ market activities orthe information released by the Fed. On its weekly report, the Fed discloses whether there is an increase inaggregate DW borrowing. In addition, financial institutions can observe whether a bank did not borrow orlend at the federal funds market at that time. Combining all the information, one can easily identify a DWborrower (Haltom, 2011). t = 1 actions. Hence,the equilibrium in t = 1 coincides with that of Theorem 1. Given no informational leakage,the outcome of t = 2 coincides with the no-intervention benchmark. Theorem 2 (Secret bailouts) . Suppose the government offers to purchase assets at p g > p with full secrecy. Then, in equilibrium, firms accept the government offer in t = 1 if and onlyif θ < p g + S . In t = 2 , firms sell assets to the market at price p if and only if θ < θ . p g + Sθ = p + St = 1 the worst type the best type t = 2 Note: the types selling to a market is depicted by blue and the types selling to the governmentis depicted by red.
Figure 5 – Equilibrium with secret bailouts
We now turn to the two-period game whose timeline is depicted in Figure 1. We continue toassume that the government offer is above the laissez-faire price: p g > p . Otherwise, there isonly a trivial equilibrium in which the laissez-faire outcome prevails, with no firms acceptingthe government offer.We begin by analyzing the structure of a possible equilibrium. We focus on the equilibriumthat is obtained in the limit as the relative weight δ < t = 2 payoff approaches 1. Lemma 1.
In any equilibrium with p g > p , there are three cutoffs < ˆ θ ≤ ˆ θ g ≤ θ such thattypes θ ≤ ˆ θ sell assets in both periods, some measure of whom sell to the government and the The assumption of δ < for more details). This observation indicates that the banks, after having joined TARPduring 2007-2009 Great Recession, had little trouble in funding at the markets after the crisis was over. emaining measure sell to the market in t = 1 ; types θ ∈ (ˆ θ, ˆ θ g ] sell only to the government in t = 1 but do not sell in t = 2 ; types θ ∈ (ˆ θ g , θ ] sell only in t = 2 ; and types θ > θ never selltheir assets in either period. The structure of an equilibrium is depicted in Figure 6. ˆ θ ˆ θ g θ t = 1 the worst type the best type t = 2 Figure 6 – General structure of equilibriumLemma 1 rests on several observations. First, firms’ preferences satisfy the single-crossingproperty, implying that a lower type has more incentives to sell than a higher type in eitherperiod; so the total number of units sold in equilibrium across the two periods must be non-increasing in θ . Second, the fact that buyers (either the government or the market) neverration sellers means that the quantity traded for each firm must be either zero or one in eachperiod. Third, an arbitrarily small discounting of the second-period payoff, along with thefirst two observations, implies that, among those that sell only in one period, early sellers areof lower types than late sellers. These observations give rise to the stated cutoff structure,as depicted in Figure 6. We omit the formal proof since it follows from a standard argumentbased on these observations.In what follows, we limit attention to the case where p g is not so high—more precisely p g < − S —that all firms accept the bailout. Then, Lemma 1 implies that there are only twopossible types of equilibria: (a) short-lived stimulation equilibria and (b) delayed stimulation equilibria, depending on whether the stimulation effect of a bailout arises in t = 1 or delayedto t = 2. More formally, these two types of equilibria correspond to the cases of ˆ θ g = θ and of ˆ θ g < θ , respectively, in the cutoff structure characterized in Lemma 1. In the OnlineAppendix B, we formally show that these are the only possible types of equilibria. This type of equilibrium corresponds to the case of ˆ θ g = θ in Lemma 1, and is depicted inFigure 7. Importantly, the segment [ˆ θ g , θ ] of firms selling in t = 2 in Lemma 1 (i.e., delayed Given the condition, we will have ˆ θ g <
1. In case p g is higher so that all firms accept the bailout, thesecond-period would coincide with the laissez-faire outcome. Such a boundary case is unrealistic in additionto being very costly from a welfare perspective, as we discuss later. t = 1, but, as we will argue, its effect will be short-lived. ˆ θ ˆ θ g t = 1 the worst type the best type t = 2 Note: the types selling to a market is depicted by blue and the types selling to the governmentis depicted by red.
Figure 7 – Short-lived stimulation equilibriumFor the purpose of characterization, we suppose that an equilibrium of this type exists,and investigate its properties. Specifically, fix an equilibrium in which the cutoffs defined inLemma 1 satisfy 0 < ˆ θ ≤ ˆ θ g = θ <
1, as illustrated in Figure 7. First, consider types θ ≤ ˆ θ .Let µ g and µ m be the fractions of these firms selling to the government and to the market in t = 1, respectively, where µ g + µ m = 1. One can show that both fractions are strictly positivein equilibrium. Let θ g and θ m denote respectively the mean values of the assets sold by thetwo groups. First, p g = θ g ≥ I , or else these firms would not sell in t = 2, a contradiction toLemma 1. Also, by definition, µ g θ g + µ m θ m = E [ θ | θ ≤ ˆ θ ] . (3)Next, let p m denote the price firms receive from selling to the market in t = 1. The t = 2market price depends on whether or not a firm received the bailout in t = 1, as these eventsare observed by buyers in t = 2. Let p g and p m denote respectively the prices offered in t = 2to those that accepted the bailout and to those that did not in t = 1. In the short-livedstimulation equilibrium, no firms sell only in t = 2, so the latter group consists of only thosethat sold to the market in t = 1. Since buyers break even in expectation, we must have p g = θ g . Similarly, p m = p m = θ m , since those that sold to the market in t = 1 are alsobelieved correctly to of type θ m on average in both periods. Further, those firms selling in First, suppose µ m = 0. Then, all types θ ≤ ˆ θ g sell to the government in t = 1. But then, the holdoutsin t = 1 would be revealed in t = 2 to have types θ > ˆ θ g . Given ˆ θ g <
1, a positive measure of them willattract buyers offering price higher than ˆ θ g . This leads to θ > ˆ θ g , a contradiction to the type of equilibriumwe are considering. Next, suppose µ g = 0, hence all types θ ≤ ˆ θ sell to the market. We cannot have ˆ θ = ˆ θ g ,since then no firm accepts bailout, and the laissez faire cannot support such a cutoff ˆ θ g ≥ p g + S , where theinequality is obtained earlier, since p g + S > p + S = θ . Hence, ˆ θ < ˆ θ g . But then, bailout recipients arerevealed to have types θ ≥ ˆ θ , hence attract offers higher than ˆ θ in t = 2, which contradicts the definition of ˆ θ in Lemma 1. t = 1: p g + p g + 2 S = p m + p m + 2 S ⇐⇒ p g + θ g = 2 θ m . (4)Next, suppose further that ˆ θ < ˆ θ g . Then, the cutoff type ˆ θ must be indifferent betweenselling to the market in both periods and accepting the bailout in t = 1 and not selling in t = 2: 2 θ m + 2 S = ˆ θ + p g + S. (5)Lastly, either the cutoff ˆ θ g must be one or else the type ˆ θ g must be indifferent betweenaccepting the bailout in t = 1 and not selling in t = 2 and not selling in either period. Hence,ˆ θ g = ( p g + S ) ∧ . (6)From the necessary conditions on the cutoff types above, we can derive the followingproperties of short-lived stimulation equilibria. Theorem 3 ( Short-lived stimulation equilibria) . Suppose there exists a short-lived stimulationequilibrium given p g > p . Then, (i) θ g < θ m < p g ; (ii) ˆ θ < θ < ˆ θ g = p g + S ; (iii) p g < θ − E [ θ | θ ≤ θ ] .Proof. See Appendix A.
Q.E.D.
We highlight three features of short-lived stimulation equilibria. First, bailout recipientssuffer stigma. In the t = 2 market, bailout recipients are believed to be of type θ g (on average),while those that sell to the market in t = 1 are believed to be of type θ m > θ g . Thus, assetsheld by bailout recipients are sold at discount precisely equal to ∆ = θ m − θ g , since the marketcorrectly infers the difference in their average asset values. Of course, this stigma must becompensated in t = 1, or else no firms would accept the bailout. In particular, the governmentmust pay more than the market does for the asset in t = 1. Since the government offer is fixedat p g , this means that the market in t = 1 must clear at price p g − ∆. In other words, buyersdemand a haircut ∆ from firms selling to them for avoiding that stigma. Since the market iscompetitive, buyers cannot earn positive profit, so what this simply means is that the averagetype θ m of firms selling to the market must (endogenously) equal p g − ∆. It thus follows thatthe bailout premium must precisely compensate the stigma, as is stated in (4).16econd, the dregs-skimming by government bailout—featured prominently in Tirole’s model—does not occur here. Bailout stigma here creates the incentive for high-type firms to mitigateit or avoid it altogether. Selling to the market in t = 1 instead is one option, but it is subjectto a mark-down of asset price by ∆; effectively, bailout stigma has “spread” to market sellersin t = 1. Another way to avoid the collateral damage is to accept the bailout but simplywithdraw from the t = 2 market. Indeed, types (ˆ θ, p g + S ] find it strictly profitable to acceptthe bailout but refuse to sell assets in t = 2 to avoid the stigma. Presence of these firmsundercuts the government’s role to take out the most toxic assets and to boost the marketreputation of the remaining firms. This has a long term effect, as we now turn to.Third, as will be seen more clearly, the government bailout worsens the adverse selectionin the t = 2 market relative to no bailout. The government’s inability to dregs-skim—or totake out the worst assets—makes stimulation short lived. In particular, high-type withdrawalfrom the t = 2 market to avoid stigma exacerbates the reputation of firms that do participatein the t = 2 market; they are effectively revealed to be of type θ g on average. In fact, thisnegative effect is so severe that the t = 2 market freezes more than if there were no bailoutin t = 1: only types θ ≤ ˆ θ sell in t = 2, where importantly ˆ θ < θ . By comparison, alltypes θ ≤ θ would have traded in t = 2 in the absence of bailout (recall Figure 3). Clearly,transparency entails a strict loss of trade. Compared with a secret bailout, the volume oftrade (and investment) induced under the transparent bailout is the same in t = 1 but strictlysmaller in t = 2.The properties identified so far are necessary but not sufficient for short-lived stimulationequilibrium. For a short-lived stimulation equilibrium to exist, additional conditions must bemet. Specifically, buyers targeting bailout recipients should not gain from raising their offersto attract the boycotters (i.e., types θ ∈ [ˆ θ, p g + S ]), and the buyers targeting non-recipientsshould have no incentives to raise their offers to attract holdouts (i.e., types θ > p g + S )together with the market sellers. These conditions are formally stated and shown to besufficient in Online Appendix C.1.These conditions are not easy to check, so it is difficult to establish the existence of theequilibrium (or its sufficient condition) in a simple manner. Nevertheless, short-lived stimu-lation equilibrium exists for a range of p g ’s, for many common distribution functions F . Forexample, Figure 9-(a) in Section 4.3) shows (a continuum of) short-lived stimulation equilibriawhen F is uniform.On the other hand, a short-lived stimulation equilibrium does not exist if p g is sufficientlyhigh. More precisely, as stated in (iii) of Theorem 3, the short-lived stimulation equilibriumdisappears if p g ≥ θ − E [ θ | θ ≤ θ ]. Roughly speaking, if p g is sufficiently high, accepting the17overnment offer becomes very attractive and this breaks the no arbitrage condition (4). Oneimplication is that the policy maker can avoid triggering undesirable short-lived stimulationequilibria if she were to make the bailout offer sufficiently generous—a point that will becomeclear as we now turn to delayed stimulation equilibria.
A delayed stimulation equilibrium has the structure that ˆ θ ≤ ˆ θ g < θ in the characterizationin Lemma 1, and is illustrated in Figure 8. We call this delayed stimulation equilibrium sincemuch of the stimulation effect materializes in t = 2. In particular, the highest type thattrades does so in t = 2. Types θ < ˆ θ g act similarly as in the short-lived stimulation equilibria:nonnegative fractions µ g and µ m of types θ ≤ ˆ θ sell respectively to the government and to themarket, and types θ ∈ (ˆ θ, ˆ θ g ) sell only to the government, where ˆ θ ≤ ˆ θ g . ˆ θ = θ ˆ θ g θ = p g + Sθ g = θ m = E [ θ | θ ≤ θ ] t = 1 the worst type the best type t = 2 Note: the types selling to a market is depicted by blue and the types selling to the governmentis depicted by red.
Figure 8 – Delayed stimulation equilibriumWhat makes this equilibrium possible is the incentive that t = 2 buyers have to offer asufficiently high price to attract high-type firms who hold out in t = 1. Such an incentive wasabsent in short-lived stimulation equilibria due to a sizable fraction µ m of low-type marketsellers. These firms cannot be distinguished from high-type hold-out firms and, therefore,would inflict a loss to buyers if they were to raise offers to attract high-type hold-out firms.In a delayed stimulation equilibrium, the fraction µ m of low-type market sellers is sufficientlysmall, especially when p g is large, so that t = 2 buyers do have an incentive to attract high-typeholdouts, unlike in short-lived stimulation equilibria. To satisfy that condition—or equivalently, to support the sale at t = 1 market—the stigma ∆ = θ m − θ g must increase, which further requires that firms selling to the t = 1 market must be of types close to ˆ θ . Thisin turn reduces the fraction µ m of these firms. If this fraction shrinks sufficiently, it is no longer incentivecompatible for buyers to buy only from these firms; it becomes profitable for buyers to deviate by raising theiroffers to attract high-type firms that held out in t = 1 (i.e., types θ > p g + S ), thus breaking the short-livedstimulation equilibrium. See the proof of Theorem 3-(iii) in Appendix A for the supplementary analysis.
18e now provide characterization of delayed stimulation equilibria.
Theorem 4 ( Delayed stimulation equilibria) . In any delayed stimulation equilibrium, ˆ θ = θ , ˆ θ g ∈ [ˆ θ, p g + S ) and θ = p g + S . In particular, the following holds. (i) Types θ ≤ θ sell in both periods, a positive (possibly the entire) measure of which acceptthe bailout. (ii) Among types θ ≤ θ , those that sell to the market in t = 1 (if they exist) receive price p in t = 1 and p g from the t = 2 market, and those that accept the bailout sell assets atprice p in t = 2 . Furthermore, these two groups of firms have the same average valueof E [ θ | θ ≤ θ ] = p . (iii) Types θ ∈ ( θ , ˆ θ g ] sell only in t = 1 and to the government at p g . (iv) Types θ ∈ (ˆ θ g , p g + S ] sell only in t = 2 at price p g . Higher-type firms never sell in eitherperiod.Proof. See Appendix A.
Q.E.D.
We discuss below several features of delayed stimulation equilibria. First, just as in short-lived stimulation equilibria, firms suffer from accepting a bailout. In t = 2, the market offersprice p to bailout recipients but p g > p to those that did not accept a bailout. But, unlike inshort-lived stimulation equilibria, this differential treatment is not attributed to the differencein the average types between bailout recipients and market sellers in t = 1. Theorem 4-(ii)shows that the average asset value is precisely the same for these two groups of firms andequals p = E [ θ | θ ≤ θ ]. The differential treatment is instead due to the high-type holdoutsin t = 1 that participate in the t = 2 market. Since buyers in t = 2 cannot distinguish thesefirms from the t = 1 market sellers, the latter firms receive a better offer.Just as before, the differential treatment by the t = 2 market of bailout recipients vis-`a-vismarket sellers in t = 1 can sustain in equilibrium only if it is counterbalanced by the oppositetreatment of these two groups in t = 1. Indeed, the government must pay more for the assetthan the market in t = 1 to compensate for the (relative) loss bailout recipients will suffer in t = 2. As before, the payoffs of those who sell in both periods are equalized. In short-livedstimulation equilibria, this payoff equalization, or no arbitrage, meant a “contagion of stigma”to all firms selling in both periods, which resulted in the worsening of the adverse selection in t = 2 than in the absence of bailout. This does not occur in delayed stimulation equilibria. Itis because adverse selection is ameliorated in delayed stimulation equilibria due to high-typeholdouts selling only in t = 2. As a result, bailout recipients are offered p = E [ θ | θ ≤ θ ]in t = 2, exactly the same as the market offer that would prevail absent any bailout by thegovernment.The above discussions lead to the following key observation.19 orollary 1. Each delayed stimulation equilibrium is equivalent to an equilibrium under secretbailout (described in Theorem 2) in total volume of asset sales—and thus in total investmentsundertaken by firms. The total volume of asset sales in any delayed stimulation equilibriumexceeds that in short-lived stimulation equilibria.Proof.
By Theorem 4, the total volume of asset sales is F ( θ ) + F ( p g + S ) in any delayedstimulation equilibrium, which equals that under a secret bailout. It also exceeds the totalvolume of asset sales in a short-lived stimulation equilibrium, F (ˆ θ ) + F ( p g + S ), since ˆ θ <θ . Q.E.D.
One interesting, and perhaps surprising, implication of this result is that delays in theeffect of bailout should not be viewed as a policy failure, at least if one takes the secretbailout equilibrium as an ideal benchmark. Take a possible equilibrium where ˆ θ g is very low;in fact, one can show that ˆ θ g = θ can be supported if p g is not too high (again such anequilibrium arises due to a large mass of high type firms holding out in t = 1). In such anequilibrium, it may appear from the perspective of t = 1 commentators that bailouts have noimpact, since trading and investment activity have not changed after the government offer.Their impression would be that the government purchase “crowded out” private purchase; apositive measure of firms with θ ≤ θ sell to the government at a higher price p g than p , theprice they would have sold at in the absence of bailouts. Indeed, in the wake of the GreatRecession, such a sentiment prevailed following the apparent lack of response by banks to thefirst wave of stimulation policies. However, our result suggests that holdout by high-type firms (instead of taking the bailoutand boycotting the t = 2 market) is a blessing in disguise. The presence of these holdoutfirms is precisely what leads the market to make very attractive offers to those that did notaccept the bailout. Indistinguishable from these holdout firms, those that actually sold to themarket in t = 1 also receive very attractive offers, thus overcoming their adverse selection.The alleviation of adverse selection for these firms in turn creates “collateral benefits” tothose that accept the bailout, since no arbitrage means that they too can overcome stigma.Consequently, the increase in trade volume in t = 2 more than makes up for the initial lackof response, in comparison with short-lived stimulation equilibria.From the policy maker’s perspective, bailouts could very well be seen as “working” mys-teriously here. Paradoxically, the policy works here because it allows firms to send a strongsignal on their financial strength, by “rejecting” the bailout offer. This indirect signaling cre- In fact, as pointed out by Bolton, Santos and Scheinkman (2009), LIBOR-OIS spreads did not decreasefollowing the implementation of the Fed’s emergency lending programs (such as PDCF and TSLF) during the2007 – 2009 Great Recession. Bolton, Santos and Scheinkman (2011) also argued that the public liquidityprograms, if implemented at a bad timing, will only crowd out the private liquidity supplied from the financialmarket. As can be seen, our theory provides a different perspective on the same phenomenon. However, the equilibrium existsfor a range of p g ’s under general distribution functions. For example, the equilibrium existsunder the uniform distribution, as we show in Section 4.3Corollary 1 also shows that, if a p g leads to a delayed stimulation equilibrium under thetransparent bailout, it also yields the same total volume of trade as under a secret bailout.From this, one may conclude that bailout stigma need not be worrisome. However, there aretwo caveats to this conclusion. First, there is a possibility of multiple equilibria as we show inthe next section. That is, the same p g may also support a short-lived stimulation equilibrium,which is clearly undesirable as discussed previously. In order to avoid the selection of suchan equilibrium, a policy maker may have to raise p g beyond what she would otherwise offer.Second, a delay in and of itself may be undesirable for reasons not modeled in our theory.For instance, a prompt revitalization of economic activities often have external benefits forthe rest of the economy. In particular, if we consider financial institutions investing in thereal economy, a prompt restoration of their activities will have a positive spillover effect, andfrom this perspective delayed stimulation can be harmful. For these reasons, one may viewthe delay itself as a cost of stigma. Remark 2 (The “full” freeze case: θ = 0) . We have so far implicitly assumed θ > , for easeof exposition. But our equilibrium characterizations from Theorems 3 and 4 extend even to thecase of θ = 0 ; i.e., when the market would freeze completely absent any bailout. In this case,our characterizations imply that ˆ θ = 0 , and this leads to existence of a unique equilibrium.Given p g ≥ I , the equilibrium admits a threshold ˆ θ g ∈ (0 , p g + S ∧ such that types θ ∈ [0 , ˆ θ g ] sell to the government in t = 1 but do not sell in t = 2 , and types θ ∈ (ˆ θ g , p g + S ∧ holdout in t = 1 but sell in t = 2 at price p g , where ˆ θ g satisfies E [ θ | ˆ θ g ≤ θ ≤ p g + S ] = p g . Types θ > p g + S ∧ sell in neither period. One can think of this as a form of delayed stimulationequilibrium: a bailout does not revive market in t = 1 but induces delayed trading in t = 2 .In keeping with Corollary 1, the equilibrium induces the same trade volome as a secret bailoutwould. To gain better understanding about the two types of equilibria and their possible coexistence,it is useful to exhibit them in full detail for some concrete parameter values. Assume uniform Online Appendix C.2 presents these necessary conditions and show them to be sufficient for equilibrium. F ( θ ) = θ , along with S = 1 / I = 1 / p g ^ (a) ˆ θ ’s in short-lived stimulation equilibria p g Short-lived stimulation Delayed stimulation Laissez Faire (b)
Overall trade in both types of equilibria
Figure 9 – Equilibrium outcome in the uniform example ( S = 1 / I = 1 / θ supported in short-lived stimulation equilibria for various bailout terms p g . The corresponding threshold fordelayed equilibria is θ , depicted as dotted line. The right panel (b) of Figure 9 depicts thetotal volume of trade induced by short-lived stimulation equilibria (blue area) and delayedstimulation equilibria (red line) for differing levels of p g .There are several interesting observations. First, as can be seen clearly by the blue area,there is a continuum of short-lived stimulation equilibria that induce different threshold valuesˆ θ . Note also that short-lived stimulation equilibria exist for p g > p = S = 1 / p g is sufficiently high. Specifically short-lived stimulation equilibria exist only for p g < . θ − E [ θ | θ ≤ θ ] = 1 identified in Theorem 3-(iii).Although not seen in the figure, there are typical multiple delayed equilibria (when at leastone exists), but all of them induce the same threshold ˆ θ = θ and the same total trade volume,as formally stated in Theorem 4.Second, both types of equilibria coexist for a range of p g ’s, as can be seen in the figure.This multiplicity reflects the endogenous nature of equilibrium belief formation. Given a p g ,suppose a large measure of high-type firms accept a bailout but boycott the t = 2 marketto avoid stigma. This causes buyers to adjust down their offers in the t = 2 market, in turnvalidating the firms’ decision not to hold out in t = 1. A short-lived stimulation equilibrium22hen arises. By contrast, if a large measure of high-type firms hold out in t = 1 for the same p g , then buyers in t = 2 make high offers to attract them, which in turn validates their decisionto hold out, leading to a delayed stimulation equilibrium.Third, as can be seen from Figure 9-(b), the effect of bailout can be discontinuous withrespect to the terms of bailout. Suppose the policy maker raises the bailout term p g startingfrom a low value close to p = 1 /
3. At first, a short-lived stimulation equilibria may arise.As p g rises past 0 . p g > .
804 in this example) in order to avoid the selection of lessdesirable short-lived stimulation equilibria.Lastly, despite the stigma, a bailout with p g > p boosts overall trade regardless of thetypes of equilibria selected. As shown in Figure 9-(b), the total trade volume in short-livedstimulation equilibrium is strictly higher than 2 F ( θ ) = 4 /
3, the total trade volume in thebenchmark without bailout. This suggests that the positive effect of bailout on asset tradingin t = 1—i.e., every type θ ≤ p g + S sells in t = 1—outweighs the negative effect of stigmaon asset trading in t = 2. Further, as can be seen in Figure 9-(b), the total trade volumein short-lived stimulation equilibrium tends to increase with p g , so a more generous bailouttends to have a higher stimulation effect. In the preceding analysis, we have abstracted from the cost of bailouts although it is a crucialpart of policy debates. In this section, we evaluate alternative bailout equilibria and char-acterize the optimal bailout term p g , with the cost of bailouts explicitly accounted for. Tomodel the cost, we follow the literature (Tirole, 2012; Chiu and Koeppl, 2016) and assume thatraising a dollar of public funds used for the asset purchase program costs the society (1 + λ )dollars, where λ ≥ outcome of anequilibrium by a pair of mappings, ( Q, T ) : [0 , → { , , } × R , induced by that equilibrium,where Q ( θ ) ∈ { , , } is a type- θ firm’s total asset sale across the two periods and T ( θ ) is thetotal transfer it receives across the two periods in equilibrium. The transfer includes paymentfrom both the government (if the firm accepts a bailout) and private buyers (if it sells to the23arket in either period). For our analysis we can view these mappings as a direct mechanism that implements a social choice as a function of report on the firm’s type θ . For this mechanismto represent an equilibrium outcome, it must then satisfy the usual incentive compatibility andparticipation conditions. One can then invoke the celebrated revenue equivalance or envelopetheorem to characterize the welfare of a particular equilibrium via the trade volume and apayoff for a reference type (e.g., the highest type) induced by the equilibrium.The next lemma provides this characterization. Lemma 2.
Any equilibrium outcome ( Q, T ) arising from a bailout policy yields welfare: (cid:90) (cid:26) θ + SQ ( θ ) − λ (cid:20)(cid:18) F ( θ ) f ( θ ) − S (cid:19) Q ( θ ) + u − (cid:21)(cid:27) f ( θ ) dθ, (7) where u ≥ is the highest-type ( θ = 1 ) firm’s payoff across the two periods in the equilibrium.Proof. See Appendix A.
Q.E.D.
The welfare of an equilibrium outcome consists of three terms. The first term, 2 θ , is simplythe value of type- θ asset; recall that each firm owns 2 units of such asset, and its value doesnot depend on who eventually owns that asset. The second term, SQ ( θ ), corresponds to thereturn from the project enabled by the sale of type- θ asset represented by Q ( θ ). The last termaccounts for the cost of bailouts. In particular, the terms inside the square brackets correspondto the budget shortfall arising from the equilibrium. Recall that private buyers break evenin expectation in any equilibrium. Hence, each of this budget shortfall must be paid for bypublic funds and thus incurs the social cost of λ . The budget shortfall consists of three terms.The term, u −
2, is the rent that the highest-type ( θ = 1) firm enjoys in equilibrium above itsasset value 2. (By the standard envelope theorem reasoning, this rent must accrue to “all”firm types.) The next term, F ( θ ) f ( θ ) Q ( θ ), is the incentive cost that is required to induce type- θ firm to sell additional unit of its asset: since any sale by type θ can be mimicked profitably byall lower types, these types must be paid rents to prevent their mimicking. Since higher typescan be mimicked by more types, the incentive cost is increasing in θ . Last, project returns SQ ( θ ) act as an incentive for sale by type θ : they mitigate the budget shortfall and save thesubsidy needed to induce a sale.The fact that sale by a higher type incurs a higher social cost provides a key argument forcomparing equilibrium outcomes. To see this, fix an equilibrium, say A , with allocation Q A ( · ) Again, letting Θ g denote the types of firms that accept the bailout, the government suffers a loss of (cid:90) Θ g ( p g − θ ) f ( θ ) dθ = (cid:90) ( T ( θ ) − θQ ( θ )) f ( θ ) dθ = u − (cid:90) (cid:18) F ( θ ) f ( θ ) − S (cid:19) Q ( θ ) f ( θ ) dθ, where we used the envelope theorem to substitute for T ( θ ). p g . Suppose the policy maker, with some bailout term p (cid:48) g (cid:54) = p g , cantrigger another equilibrium, say B , with allocation Q B ( · ), such that (a) the aggregate tradevolume remains unchanged; i.e., E [ Q A ( θ )] = E [ Q B ( θ )] but that (b) B “reallocates” sales awayfrom high-type firms towards low-type firms; i.e., Q B ( θ ) (cid:82) Q A ( θ ) if θ (cid:81) ˇ θ for some ˇ θ . Then, ashift from A to B preserves the same stimulation effect, and thus the same investment returns,but incurs a lower budget deficit borne by the government, and thus a lower social cost.This reasoning establishes the social cost of bailout stigma as follows. Theorem 5 (Welfare cost of bailout stigma) . Fix any bailout offer p g that yields a short-livedstimulation equilibrium under a transparent bailout. Then, there exists a p (cid:48) g < p g such that asecret bailout with p (cid:48) g yields strictly higher welfare than the short-lived stimulation equilibrium.Proof. See Appendix A.
Q.E.D.
Intuively, the additional welfare cost of a short-lived stimulation equilibrium is attributedto the overall market-freezing effect caused by bailout stigma. In the equilibrium, bailoutstigma discourages firms not only from accepting a bailout but also from selling to the marketin t = 1. This means that, in order to induce the same trade volume as under a secretbailout, the government must raise its bailout term p g , which leads to a higher cost of publicintervention.Given the outcome equivalence between a delayed stimulation equilibrium under transpar-ent bailout and the equilibrium under secret bailout (Corollary 1), the following corollary isimmediate. Corollary 2.
A delayed stimulation equilibrium given any bailout term p g yields identicalwelfare as a secret bailout given the same bailout term. While the broad theme of this paper is related to an extensive literature on the benefitsand costs of government intervention in distressed banks, our work is most closely related toPhilippon and Skreta (2012) and Tirole (2012), who focus on adverse selection in asset markets The primary rationale for intervention is to prevent the contagion of bank runs whether it stems fromdepositor panic (Diamond and Dybvig, 1983), contractual linkages in bank lending (Allen and Gale, 2000),or aggregate liquidity shortages (Diamond and Rajan, 2005). The costs of anticipated bailouts due to thetime-inconsistency of policy are discussed by, among others, Stern and Feldman (2004).
25s a primary reason for government intervention. As mentioned previously, these studies donot explicitly study the dynamic consequence of receiving a bailout—the focus of the currentstudy. Even though these papers recognize that relatively low types accept bailouts, this doesnot translate into an adverse effect on subsequent financing in their models. Our dynamicmodel captures not only how bailout stigma affects firms’ financing behavior but also howthe stigma fundamentally alters the role of a bailout. In particular, its role in enabling firmsto send a favorable signal by “refusing” an attractive bailout offer is the single most strikingtakeaway that has no analogues in these or other antecedent studies.Banks’ reputational concerns are explicitly considered in Ennis and Weinberg (2013), La’O(2014), and Chari, Shourideh and Zetlin-Jones (2014). In Ennis and Weinberg (2013), to meettheir short-term liquidity needs, banks with high-quality assets use interbank lending whilethose with low-quality assets use the discount window. The resulting discount window stigmais reflected in the subsequent pricing of assets. In La’O (2014), financially strong banks usethe Federal Reserve’s Term Auction Facility since winning the auction at a premium signalsfinancial strength, which protects them from predatory trading. The main focus in Chari,Shourideh and Zetlin-Jones (2014) is on how reputational concerns in secondary loan marketscan result in persistent adverse selection. Since all three studies consider discrete types ofbanks and there is no government bailout, their results are not directly comparable to ours.Our paper is also related to studies on dynamic adverse selection in general (Inderst andM¨uller, 2002; Janssen and Roy, 2002; Moreno and Wooders, 2010; Camargo and Lester, 2014;Fuchs and Skrzypacz, 2015) and those with a specific focus on the role of information inparticular (H¨orner and Vieille, 2009; Daley and Green, 2012; Fuchs, ¨Ory and Skrzypacz, 2016;Kim, 2017). The key insight from the first set of studies is that dynamic trading generatessorting opportunities, which are not available in the static market setting. However, each sellerhas only one opportunity to trade in these studies, so signaling is not an issue. The secondset of studies relates to different disclosure rules and how they affect dynamic trading. Forexample, H¨orner and Vieille (2009) and Fuchs, ¨Ory and Skrzypacz (2016) show that secrecy(private offers) tends to alleviate adverse selection but transparency (public offers) does not.Once again, each seller has only one trading opportunity in these studies. Hence, althoughpast rejections can boost reputation, acceptance ends the game. In contrast, in our model, Regarding the optimal form of bailouts, Philippon and Skreta (2012) show that optimal interventionsinvolve the use of debt instruments when adverse selection is the main issue. With additional moral hazardbut limits on pledgeable income, Tirole (2012) justifies asset purchases. When there is debt overhang due tolack of capital, Philippon and Schnabl (2013) find that optimal interventions take the form of capital injectionin exchange for preferred stock and warrants. During the US subprime crisis, the EESA initially grantedthe Secretary of the Treasury authority to purchase or insure troubled assets owned by financial institutions.However, the Capital Purchase Program under TARP switched to capital injection against preferred stockand warrants. Others include dynamic extensions of Spence’s signaling model with public offers (Noldeke andVan Damme, 1990), private offers (Swinkels, 1999), and private offers with additional public informationsuch as grades (Kremer and Skrzypacz, 2007).
The current paper has studied a dynamic model of a government bailout in which firms have acontinuing need to fund their projects by selling their assets. Asymmetric information aboutthe quality of assets gives rise to adverse selection and a concommitant market freeze, whichprovides a rationale for a government bailout, just as in Tirole (2012). However, in contrastto Tirole (2012), markets stigmatize bailout recipients, which jeopardizes their ability to fundsubsequent projects. The presence of this bailout stigma and other dynamic incentives yieldsa much more complex and nuanced portrayal of how bailouts impact the economy than havebeen recognized in the extant literature.Our main findings can be summarized as follows. The bailout stigma necessitates thegovernment to pay a premium over the market terms to compensate for the stigma. Even so,market rejuvenation can be short-lived and adverse selection can worsen in subsequent markettrading, resulting in a market freeze even more severe than in the absence of a bailout. Thisrequires the government to further increase a premium. A more attractive bailout premium canbe effective in stimulating trade and investment, but its effects are delayed. Delayed benefitsmaterialize as bailouts provide firms with opportunities to boost reputation by “rejecting”bailout offers. This improves their ability to trade in the market in subsequent periods.Indeed, there is no welfare loss in this case relative to a secret bailout that does not entailstigma. Thus delayed effects of bailouts can be a blessing in disguise, subject to two importantcaveats: the government may need to run a large budget deficit to support delayed marketstimulation, and delay in and of itself may be undesirable for reasons not modeled in the27urrent paper.The central lesson from the current work is that, compared with the static setting, theeffects of bailouts are very different due to the interplay between the bailout stigma, themarket’s belief within and across periods, and rich signaling opportunities firms have in thedynamic context. To the best of our knowledge, the insights we develop and the forces weidentify are novel and have not been recognized in the previous literature and should be partof the framework for conducting future policy debates and empirical studies.
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AppendixA Proofs
A.1 Proof of Theorem 3
Proof of (i): θ g < θ m < p g . We first establish the following claim.
Claim 1. θ g < θ m . roof: Suppose to the contrary that θ g ≥ θ m , which in turn implies p g ≤ θ m from (4).Given Lemma 1, there are two possible cases to consider: ˆ θ < ˆ θ g or ˆ θ = ˆ θ g .Suppose first ˆ θ < ˆ θ g . Then, we have ˆ θ g ≤ p g + S from (6). Moreover, we have from (3)that θ m ≤ E [ θ | θ ≤ ˆ θ ] ≤ E [ θ | θ ≤ p g + S ]. Since p g > p , we must have E [ θ | θ ≤ p g + S ] < p g ,or else p g ≤ p (recall definition of p from (1) as well as Figure fig:akerlof). Hence, θ m < p g ,which, however, contradicts the earlier hypothesis p g ≤ θ m .Suppose next ˆ θ = ˆ θ g . Then, a type-ˆ θ firm must be indifferent between selling in bothperiods and selling in neither period if ˆ θ < θ = 1. Hence, we must have 2ˆ θ ≤ θ m + 2 S , where the inequality holds strictly onlyif ˆ θ = 1. We thus conclude that ˆ θ = ( θ m + S ) ∧
1. Moreover, we have from (3) that θ m ≤ E [ θ | θ ≤ ˆ θ ] ≤ E [ θ | θ ≤ θ m + S ], which implies θ m ≤ p (again by the definition of θ in (1)). Since p g > p , this again contradicts an earlier hypothesis that p g ≤ θ m . We thusconclude that θ g < θ m . (cid:3) .By (4), Claim 1 in turn implies that θ m < p g . We have thus proven (i). (cid:3) Proof of (ii): ˆ θ < θ < ˆ θ g = p g + S . We first establish the following two claims:
Claim 2. ˆ θ (cid:54) = 1 .Proof: Suppose to the contrary that ˆ θ = 1. Then, ˆ θ = 1 ≤ θ g + S . Otherwise, we will have p g + S + ˆ θ > p g + θ g + 2 S = 2 θ m + 2 S , where the equality is from (4), so a type-ˆ θ firm woulddeviate by accepting the bailout but boycotting the t = 2 market. Since θ g < θ m (by Claim1) and µ g θ g + µ m θ m = E [ θ | θ ≤ ˆ θ ] = E [ θ | θ ≤ ≤ θ g + S < µ g θ g + µ m θ m + S ≤ E [ θ | θ ≤
1] + S. But this contradicts θ <
1, which we assume throughout. (cid:3)
Claim 3. ˆ θ < ˆ θ g .Proof: Suppose to the contrary that ˆ θ = ˆ θ g . By Claim 2, we have ˆ θ = ( θ m + S ) ∧ p g > θ m . In equilibrium, each type θ > ˆ θ never sells in either period and obtains payoff 2 θ .Since p g > θ m , however, types θ ∈ (ˆ θ g , p g + S ) will have a strictly higher payoff than 2 θ byselling to the government in t = 1, a contradiction. (cid:3) We are now ready to prove (ii). We first show that ˆ θ g > θ . Since p g > p , it is straight-forward from (6) that ˆ θ g = ( p g + S ) ∧ > θ . We next prove ˆ θ < θ . Since ˆ θ < ˆ θ g by Claim3, we have ˆ θ = θ g + S < E [ θ | θ ≤ ˆ θ ] + S, where the equality follows from (4) and (5), and the strict inequality follows from θ g < θ m and (3). The definition of θ then implies ˆ θ < θ . (cid:3) roof of (iii): a short-lived stimulation equilibrium exists only if p g < θ − E [ θ | θ ≤ θ ] . Toprove this, observe from (4) that p g = 2 θ m − θ g . Fixing ˆ θ , the RHS is maximized when, for some threshold ˜ θ ∈ [0 , ˆ θ ], all types θ > ˜ θ sellto the market and all types θ < ˜ θ sell to the government so that θ m = E [ θ | θ ∈ (˜ θ, ˆ θ ]] and θ g = E [ θ | θ ≤ ˜ θ ]. Hence, p g = 2 θ m − θ g ≤ max ˜ θ ∈ [0 , ˆ θ ] , ˆ θ ∈ [0 ,θ ) E [ θ | θ ∈ (˜ θ, ˆ θ ]] − E [ θ | θ ≤ ˜ θ ] < max ˜ θ ∈ [0 ,θ ] E [ θ | θ ∈ (˜ θ, θ ]] − E [ θ | θ ≤ ˜ θ ]= 2 θ − E [ θ | θ ≤ θ ] , where the strict inequality follows from ˆ θ < θ and E [ θ | θ ∈ ( a, b )] is increasing in b for all0 ≤ a ≤ b ≤
1, and the last equality follows from the regularity condition that 2 E [ θ | θ ∈ ( a, b )] − E [ θ | θ ≤ a ] is increasing in a for all 0 ≤ a ≤ b ≤ (cid:3) A.2 Proof of Theorem 4
As before, let µ g and µ m respectively denote the fractions of types θ ≤ ˆ θ that sell to thegovernment and to the market in t = 1, and let θ g and θ m denote their average values.Obviously, (3) must continue to hold. Let p m be the market price for the asset in t = 1, andlet p g and p m respectively denote the t = 2 prices for those that sold to the government andthose that did not. Note that p m applies to those that sold to the market in t = 1 and tothose that held out, since t = 2 cannot distinguish them.We wish to prove that ˆ θ = θ , θ = p g + S , and µ g >
0. There are two possible cases:ˆ θ g > ˆ θ and ˆ θ g = ˆ θ , and we treat them separately. (Recall by definition ˆ θ g ≥ ˆ θ .) A.2.1 The case of ˆ θ g > ˆ θ . In this case, firms with θ ∈ (ˆ θ, ˆ θ g ] sell to the government in t = 1. Obviously, µ g ≥ (cid:82) ˆ θ g ˆ θ f ( θ ) dθ >
0. We only need to prove ˆ θ = θ and θ = p g + S .We first show θ = p g + S . Observe that a type-ˆ θ g firm must be indifferent between sellingonly in t = 1 to the government at p g and selling only in t = 2 at price p m . Hence, we musthave p m = p g . Since a type- θ firm must be indifferent between selling only in t = 2 at price p m and not selling in any period, we must have θ = p m + S = p g + S .We next show ˆ θ = θ . We restrict our focus on the case µ g > µ m > µ m = 0 is similar, so we omit the proof). Then, these firms sell in bothperiods, and thus must be indifferent between selling to the government and to the market in t = 1. This implies (4), or p g + p g = p m + p m = p m + p g ⇒ p m = p g . This, together with the zero-profit condition, implies that θ g = p g = p m = θ m . (8)It then follows from (3) that θ g = θ m = E [ θ | θ ≤ ˆ θ ] . (9)Next, a type-ˆ θ firm must be indifferent between selling in both periods and selling only in t = 1 to the government at price p g : p m + p g + 2 S = p g + S + ˆ θ ⇐⇒ p m + S = ˆ θ, (10)which, together with (8) and (9), implies that E [ θ | θ ≤ ˆ θ ] + S = ˆ θ. By definition of θ , or (1), we then have ˆ θ = θ . This in turn implies p m = p g = E [ θ | θ ≤ θ ] = p . A.2.2 The case of ˆ θ g = ˆ θ . The proof proceeds in several claims.
Claim 4. µ g > .Proof: Suppose to the contrary that µ g = 0. Then, we must have p m = p since buyers in t = 2 do not observe any action taken by firms in t = 1. This implies p = p m = p g > p , acontradiction. (cid:3) Claim 5. θ = p g + S .Proof: Since types θ ∈ (ˆ θ, θ ) must weakly prefer selling only in t = 2 at price p m toselling only in t = 1 to the government at price p g . This implies p m ≥ p g . We now prove that p m = p g . Suppose to the contrary that p m > p g .We know from Claim 4 that µ g >
0. Suppose µ m >
0. No arbitrage between selling to the34overnment and selling to the market in t = 1 means that p g + p g = p m + p m , so we have p g = ( p m − p g ) + p m > p m , (11)where the strict inequality follows from our hypothesis above that p m > p g . Furthermore,since type ˆ θ must be indifferent between selling in both periods and selling only in t = 2 at p m , we must have ˆ θ + p m + S = p m + p m + 2 S = ⇒ ˆ θ = p m + S. (12)By the zero profit condition, p g = θ g and p m = θ m . Hence, by (11), we have θ g > θ m . By (3),we must have E [ θ | θ ≤ ˆ θ ] > θ m = p m . (13)Then (13) and (12) imply E [ θ | θ ≤ ˆ θ ] + S > ˆ θ. (14)This means that ˆ θ < θ , by the definition of θ . By (13), this means that p m = θ m < E [ θ | θ ≤ ˆ θ ] < E [ θ | θ ≤ θ ] = p , where the last equality follows from the definition of p . Suppose a buyer deviates and offers p (cid:48) ∈ ( p m , p ). Since p (cid:48) + p m > p m + p m = p g + p g for any p (cid:48) ∈ ( p m , p ), all types θ ≤ ˆ θ will sellto this deviating buyer. Furthermore, since p g + S + θ ≤ p (cid:48) + S + p m + S = ⇒ θ ≤ p (cid:48) + S forany p (cid:48) ∈ ( p m , p ), types θ ∈ (ˆ θ, p (cid:48) + S ] will sell at the deviation price, too. Since p (cid:48) < p , wehave E [ θ | θ ≤ p (cid:48) + S ] − p (cid:48) >
0, so the deviating buyer will enjoy a strict profit. We have thusobtained a contradiction to the hypothesis that p m > p g . A similar conclusion is also obtainedwhen µ m = 0. We therefore conclude that p m = p g . Since θ = p m + S , this in turn impliesthat θ = p g + S . (cid:3) Claim 6. ˆ θ = θ .Proof: We know from Claim 4 that µ g >
0. There are two cases depending on whether µ m > µ m = 0. Consider the former first. No arbitrage for type θ ∈ [0 , ˆ θ ] between sellingto the government and selling to the market in t = 1 implies p g + p g = p m + p m , which inturn implies p g = p m since p g = p m . By the zero-profit condition, p g = θ g and p m = θ m , soby the zero-profit condition θ g = θ m . Hence, by (3), we get p m = E [ θ | θ ≤ ˆ θ ]. Combining thisequality with (12), we have ˆ θ = E [ θ | θ ≤ ˆ θ ] + S. By definition of θ , the above equality implies ˆ θ = θ . In this case, again the indifference for type ˆ θ gives p g + E [ θ | θ ≤ ˆ θ ] + S = ˆ θ + p m . Since p m > p g , thisimplies ˆ θ < θ , or E [ θ | θ ≤ ˆ θ ] < p . This creates an opportunity for buyers to profitably deviating by offeringa price p (cid:48) m ∈ ( E [ θ | θ ≤ ˆ θ ] , p ). µ m = 0. Type ˆ θ must be indifferent now between selling to the governmentin t = 1 followed by selling to the t = 2 market (with stigma) and selling only in t = 2 market.Hence, p g + p g + 2 S = ˆ θ + p m + S. (15)In the proof of Claim 5, we already proved that p m = p g . Hence, (15) reduces to p g + S = ˆ θ. (16)Further, by the zero profit condition, p g = θ m = E [ θ | θ < ˆ θ ] , (17)where the second equality holds since µ m = 0. Combining (16) and (17) givesˆ θ = E [ θ | θ < ˆ θ ] + S, which proves that ˆ θ = θ . (cid:3) A.3 Proof of Lemma 2
Fix a direct mechanism (
Q, T ). Define u (˜ θ | θ ) := T (˜ θ ) + θ (2 − Q (˜ θ )) + SQ (˜ θ ) as type- θ firm’spayoff when it reports its type as ˜ θ . Since the mechanism must be incentive compatible for alltypes, we have u ( θ | θ ) ≥ u (˜ θ | θ ) for all ˜ θ, θ ∈ [0 , u ( θ ) := u ( θ | θ ). Since the participationconstraint must be satisfied for all types, we also have u ( θ ) ≥ θ for all θ ∈ [0 , u = u (1)be the highest-type ( θ = 1) firm’s payoff in equilibrium. Then, one can apply the envelopetheorem to find u ( θ ) = u − (cid:82) θ (2 − Q ( s )) ds .To calculate the welfare, let Θ g denote the set of θ ’s that sell to the government in t = 1.The welfare is the sum of the payoffs for firms and buyers minus the deadweight loss from adeficit run by the government. Given the government offer p g , the welfare is then written as (cid:82) [ u ( θ ) + ( θQ ( θ ) − T ( θ ))] f ( θ ) dθ − λ (cid:82) θ ∈ Θ g ( p g − θ ) f ( θ ) dθ . Since buyers in the market mustbreak even, we have (cid:82) θ ∈ Θ g ( p g − θ ) f ( θ ) dθ = (cid:82) ( T ( θ ) − θQ ( θ )) f ( θ ) dθ . Hence, the welfare is (cid:82) [ u ( θ ) + ( θQ ( θ ) − T ( θ )) − λ ( T ( θ ) − θQ ( θ ))] f ( θ ) dθ. By plugging u ( θ ) = u − (cid:82) θ (2 − Q ( s )) ds into the welfare and integrating by parts, we obtain (7).36 .4 Proof of Theorem 5 Consider a transparent bailout with p g that yields a short-lived stimulation equilibrium andlet ( Q SL , T SL ) denote the corresponding outcome (expressed in a direct mechanism). Then,we have Q SL ( θ ) = 2 if θ ∈ [0 , ˆ θ ], Q SL ( θ ) = 1 if θ ∈ (ˆ θ, ( p g + S ) ∧ Q SL ( θ ) = 0otherwise. Hence, the total trade volume induced by the short-lived stimulation equilibriumis F (ˆ θ ) + F (( p g + S ) ∧ u SL ,is equal to 1 + (( p g + S ) ∨ p (cid:48) g and let ( Q S , T S ) denote the corresponding outcome. Then we have Q S ( θ ) = 2 if θ ∈ [0 , θ ], Q S ( θ ) = 1 if θ ∈ ( θ , ( p (cid:48) g + S ) ∧ Q S ( θ ) = 0 otherwise. Thus, the total trade volumein this case is F ( θ ) + F (( p (cid:48) g + S ) ∧ u S , is equal to 1 + (( p g + S ) ∨ p (cid:48) g yields the same totaltrade volume as the short-lived stimulation equilibrium under the transparent bailout with p g . Since ˆ θ < θ from Theorem 3, we must have ( p (cid:48) g + S ) ∧ < ( p g + S ) ∧
1, which implies p (cid:48) g < p g . This also implies u S = 1 + (( p (cid:48) g + S ) ∨ ≤ p g + S ) ∨
1) = u SL .We next show that the secret bailout with p (cid:48) g yields strictly higher welfare than the short-lived stimulation equilibrium arising from p g , if the secret bailout induces the same totaltrade volume as the short-lived stimulation equilibrium does. From (7), the welfare differencebetween the two types of equilibria is (cid:90) (cid:26) S ( Q S ( θ ) − Q SL ( θ )) − λ (cid:20)(cid:18) F ( θ ) f ( θ ) − S (cid:19) ( Q S ( θ ) − Q SL ( θ )) + ( u S − u SL ) (cid:21)(cid:27) f ( θ ) dθ =(1 + λ ) S (cid:90) ( Q S ( θ ) − Q SL ( θ )) f ( θ ) dθ + λ (cid:90) (cid:20) F ( θ ) f ( θ ) ( Q SL ( θ ) − Q S ( θ )) + ( u SL − u S ) (cid:21) f ( θ ) dθ = λ (cid:90) (cid:20) F ( θ ) f ( θ ) ( Q SL ( θ ) − Q S ( θ )) + ( u SL − u S ) (cid:21) f ( θ ) dθ ≥ λ (cid:90) F ( θ ) f ( θ ) ( Q SL ( θ ) − Q S ( θ )) f ( θ ) dθ>λ (cid:20) F ( θ ) f ( θ ) (cid:90) θ ( Q SL ( θ ) − Q S ( θ )) f ( θ ) dθ + F ( θ ) f ( θ ) (cid:90) θ ( Q SL ( θ ) − Q S ( θ )) f ( θ ) dθ (cid:21) = λ F ( θ ) f ( θ ) (cid:90) ( Q SL ( θ ) − Q S ( θ )) f ( θ ) dθ = 0 , where the second and last equalities follow from (cid:82) ( Q S ( θ ) − Q SL ( θ )) f ( θ ) dθ = 0, the weakinequality follows from u SL ≥ u S , and the strict inequality follows from the facts that F ( θ ) f ( θ ) isstrictly increasing in θ , (cid:82) θ ( Q SL ( θ ) − Q S ( θ )) f ( θ ) dθ <
0, and (cid:82) θ ( Q SL ( θ ) − Q S ( θ )) f ( θ ) dθ > nline AppendixB Possible Types of Equilibria In this section, we show that there are only two types of equilibria, short-lived stimulationand delayed stimulation types. To this end, recall first from Lemma 1 that every equilibriummust have a cutoff structure 0 ≤ ˆ θ ≤ ˆ θ g ≤ θ ≤ ≤ ˆ θ sell in bothperiods (either to the government or to the market in t = 1); types θ ∈ (ˆ θ, ˆ θ g ] sell in t = 1 butnot in t = 2; types θ ∈ (ˆ θ g , θ ] sell only in t = 2; types θ > θ do not sell in either period. Lemma 3.
In the presence of government bailouts, there are only two possible types of equi-libria, < ˆ θ < ˆ θ g = θ ≤ (short-lived stimulation type) or < ˆ θ ≤ ˆ θ g < θ ≤ (delayedstimulation type).Proof. For the purpose of exposition, let p m , p g , and p m denote equilibrium asset pricesoffered to all firms at the market in t = 1, the bailout recipients (i.e., the firms that sold tothe government in t = 1) in t = 2, and the bailout holdouts (i.e., the firms that did not sellto the government in t = 1) in t = 2, respectively. Note that there are no firms that sell tothe market in t = 1 but not in t = 2: if there were such firms, buyers in t = 2 would offer p m to attract these firms.It suffices to prove that ˆ θ > θ < max { ˆ θ g , θ } . We prove these claims in sequence. Step 1. ˆ θ > θ = 0; namely, no firms sellassets in both periods. There are two possible cases: ˆ θ g = 0 or ˆ θ g > θ g = 0, there will be no asset trading in t = 1. Then, there is no updatingon firms’ types in t = 2, so t = 2 buyers hold the prior belief about firms’ types, leading to θ = θ >
0. However, by definition of θ and p , a buyer in t = 1 can profitably deviate byoffering p (cid:48) ∈ ( I, p ). Hence, ˆ θ g > θ g >
0. Once again, there are two possible cases, ˆ θ g = θ and ˆ θ g < θ . Wefirst show that it is impossible to have 0 < ˆ θ g = θ <
1. If 0 < ˆ θ g = θ <
1, this means thatno firms with θ > ˆ θ g sell in t = 2. However, a buyer in t = 1 can profitably deviate by offeringa p (cid:48) > ˆ θ g − S : the strict log-concavity property of f ( · ) implies f ( · ) / (1 − F (ˆ θ g )) is also strictlylog-concave, and thus there exists a θ (cid:48) > ˆ θ g such that p (cid:48) = θ (cid:48) − S and E [ θ | ˆ θ g < θ ≤ θ (cid:48) ] − p (cid:48) > < ˆ θ g = θ = 1, we have ˆ θ g > θ . By definition of θ , however, a buyer in t = 2 canprofitably deviate by offering p (cid:48) = p − ε for a sufficiently small ε to the bailout recipients.Consider next 0 < ˆ θ g < θ ≤
1. Since a firm sells only in t = 1 to the government at p g or only in t = 2 to the market at p m (recall by hypothesis no firm sells in both periods), noarbitrage implies p g = p m . There are two possible cases: ˆ θ g ≥ θ or ˆ θ g < θ . If ˆ θ g ≥ θ , a buyer38n t = 2 can profitably deviate by offering p (cid:48) ∈ [ I, p ) to the firms that sold to the governmentin t = 1. If ˆ θ g < θ , a buyer in t = 1 can profitably deviate by offering p (cid:48) ∈ (max { ˆ θ g − S.I } , p ):since buyers in t = 2 will offer p m = p g to any firm that did not sell to the government, allfirms with types θ ≤ p (cid:48) + S will sell at p (cid:48) in t = 1 and the sale is profitable for the buyer (bydefinition of p ). Consequently, there cannot exist any equilibrium with ˆ θ = 0. Step 2. ˆ θ < max { ˆ θ g , θ } .Suppose to the contrary that there is an equilibrium with 0 < ˆ θ = ˆ θ g = θ . This meansthat no firm sells only in one period. We first show that there are positive measures of firmssell to the government and to the market in t = 1 (that is, in terms of our notation in thetext, µ g ∈ (0 , θ ≤ ˆ θ refuse the bailout, the equilibrium is same as that withoutbailouts, and thus ˆ θ = θ . Since p g > p , however, types θ ∈ ( θ , p g + S ] will deviate and sellto the government in t = 1. If all types θ ≤ ˆ θ accept the bailout, there are two possible cases,either ˆ θ > θ or ˆ θ ≤ θ . In the former case, we must have ˆ θ ≤ p g + S , which implies p g > p .However, such a price cannot break even for the t = 2 buyers. In the latter case, firms withtypes θ ∈ (ˆ θ, p g + S ] will deviate and sell to the government in t = 1. We have thus proventhat positive measures of firms sell to the government and to the market in t = 1.Since all firms that sold to the market in t = 1 also sell in t = 2, the zero-profit conditionand the supposition ˆ θ = ˆ θ g = θ imply p m = p m . For expositional convenience, we throughoutabuse the notation p m as the equilibrium price offered in both periods to the firms that didnot sell to the government. Since a type-ˆ θ firm is indifferent between selling in both periodsand not selling in any period, we must have 2ˆ θ = 2 p m + 2 S , or equivalently ˆ θ = p m + S . If p m < p g , types θ ∈ (ˆ θ, p g + S ] will deviate and sell to the government in t = 1. Hence, wemust have p m ≥ p g , and thus ˆ θ ≥ p g + S > θ . Since the no-arbitrage condition for types θ ≤ ˆ θ implies p g + p g = 2 p m , we must have p m ≤ p g . Since the equilibrium prices p m and p g must break even for the t = 2 buyers, we must have E [ θ | θ ≤ ˆ θ ] = µ g p g + (1 − µ g ) p m , whichimplies E [ θ | θ ≤ ˆ θ ] + S ≥ p m + S = ˆ θ. By the definitin of θ (see (1)), we must then have ˆ θ ≤ θ , but this contracts an earlierobservation that ˆ θ > θ . Therefore, an equilibrium with 0 < ˆ θ = ˆ θ g = θ cannot exist for any p g > p . Q.E.D.
C Detailed Analysis of Equilibria
In the text, we characterized the necessary conditions for both types of equilibria (short-lived stimulation and delayed stimulation types). In this section, we present all the necessaryconditions in detail and show them they are sufficient for equilibrium.39 .1 Short-lived Stimulation Equilibrium
In a short-lived stimulation equilibrium, firms with types θ ≤ ˆ θ sell in both periods: a fraction µ g of these firms sell to the government at p g in t = 1 and to the market at p g in t = 2, and theremaining fraction µ m = 1 − µ g of them sell to the market at p m in both periods. In addition,types θ ∈ (ˆ θ, ˆ θ g ] sell to the government at p g in t = 1 but do not sell in t = 2 (i.e., boycottthe t = 2 market), and types θ > ˆ θ g do not sell in either period. Let θ g denote the averagevalue of types of the bailout recipients that sell to the market in t = 2, and let θ m denotethe average value of types that sell to the market in both periods. The zero-profit conditionimplies that p g = θ g and p m = θ m .There are several necessary conditions for the existence of short-lived stimulation equilibria.First, recall from the text the necessary conditions associated with the optimality of firms’prescribed equilibrium strategies: p g + θ g = 2 θ m , (4)2 θ m + 2 S = ˆ θ + p g + S, (5)ˆ θ g = ( p g + S ) ∧ . (6)Second, there are feasibility conditions on the endogenous variables ( θ g , θ m , µ g , ˆ θ ). One ofthese conditions is that every price offered to firms in equilibrium must be sufficiently high tofund the cost of investment I . By Theorem 3, we have p g = θ g < p m = θ m < p g . (18)Thus, we must have θ g ≥ I. (19)Furthermore, by definition of θ g , θ m , and µ g , we must have µ g θ g + µ m θ m = E [ θ | θ ≤ ˆ θ ] , (20)which is same as (3) in the text. Lastly, feasibility requires that θ g cannot be too low:specifically, we must have θ g ≥ E [ θ | θ ≤ F − ( µ g F (ˆ θ ))] . (21)Lastly, there are two other necessary conditions. First, buyers in t = 2 should not havean incentive to offer a p (cid:48) (cid:54) = p g = θ g to the firms that received a bailout in t = 1. If a t = 2buyer offers a p (cid:48) > θ g to the bailout recipients, all bailout recipients with types θ ≤ ˆ θ and40ypes θ ∈ (ˆ θ, ( p (cid:48) + S ) ∧ ˆ θ g ] will sell at p (cid:48) . By (6), no buyer will want to offer p (cid:48) > p g , so we canfocus on p (cid:48) ≤ p g . Since a fraction µ g of firms with types θ ≤ ˆ θ receives a bailout in t = 1 andsince their average value is θ g , for such a deviating offer to be unprofitable, we must have µ g F (ˆ θ ) θ g + (cid:82) p (cid:48) + S ˆ θ θf ( θ ) dθµ g F (ˆ θ ) + ( F ( p (cid:48) + S ) − F (ˆ θ )) − p (cid:48) < ∀ p (cid:48) ∈ ( θ g , p g ] . (22)Next, buyers in t = 2 should not have an incentive to offer a p (cid:48) (cid:54) = p m to the firms that did notsell to the government in t = 1. For instance, if a buyer offers p (cid:48) > p g in t = 2, the firms thatrefused the bailout with types θ ≤ ˆ θ and the additional firms with types θ ∈ (ˆ θ g , ( p (cid:48) + S ) ∧ − µ g of the firms with types θ ≤ ˆ θ refused the bailout and their average value is θ m , the necessary condition for such a p (cid:48) not to be profitable is(1 − µ g ) F (ˆ θ ) θ m + (cid:82) p (cid:48) + S ˆ θ g θf ( θ ) dθ (1 − µ g ) F (ˆ θ ) + ( F ( p (cid:48) + S ) − F (ˆ θ g )) − p (cid:48) < ∀ p (cid:48) ∈ ( p g , − S ] . (23)The following proposition states that the necessary conditions listed above are indeedsufficient for the existence of the short-lived stimulation equilibrium. Proposition 1.
For a p g > p , there exists a short-lived stimulation equilibrium if there exist ( µ g , θ g , θ m , ˆ θ ) that satisfy (19) – (23) and (4) – (6) .Proof. We throughout focus on the case ˆ θ g <
1: the proof for the other case ˆ θ g = 1 is verysimilar.We first show that it is optimal for every type- θ firm to play the prescribed equilibriumstrategies over two periods. Consider t = 2 first. After accepting a bailout, firms with types θ ≤ ˆ θ find it optimal to sell to the market at p g = θ g since θ ≤ ˆ θ = θ g + S from (6). The samecondition implies that it is optimal for the firms with types θ ∈ (ˆ θ, ˆ θ g ] (after accepting thebailout in t = 1) not to sell at p g . Consider now the firms with types θ ≤ ˆ θ that sold to themarket in t = 1. It is optimal for these firms to sell at p m = θ m since θ ≤ ˆ θ = θ g + S < θ m + S by Theorem 3-(i). Theorem 3-(i) also implies θ m + S < p g + S = ˆ θ g < θ , so it is optimal forthe firms with types θ > ˆ θ g not to sell at p m .Consider t = 1 next. From (4) and (5), firms with types θ ≤ ˆ θ are indifferent betweenselling to the government in t = 1 and selling to the market in t = 1. Furthermore, (5)implies θ + p g + S ≤ p m + 2 S = p g + p g + 2 S for all θ ≤ ˆ θ , so it is optimal for all firmswith types θ ≤ ˆ θ to play the prescribed equilibrium strategy. The same condition also implies p g + S + θ > p m + 2 S for all θ > ˆ θ . Moreover, (6) implies 2 θ > p g + S + θ if and only if θ > ˆ θ g . Hence, it is optimal for a type- θ firm to sell to the government in t = 1 if θ ∈ (ˆ θ, ˆ θ g ]and for a type- θ firm not to sell in t = 1 if θ > ˆ θ g .41e next show that it is optimal for every buyer in t = 1 , t = 2 who make offers to the bailout recipients. Since all the otherbuyers offer p g = θ g , any offer p (cid:48) < p g will be rejected and thus will not be profitable. Supposea t = 2 buyer offers p (cid:48) > θ g . This offer will attract all of the bailout recipients with types θ ≤ ˆ θ and those with types θ ∈ (ˆ θ, ( p (cid:48) + S ) ∧ µ g F (ˆ θ ) θ g + (cid:82) ( p (cid:48) + S ) ∧ θ θf ( θ ) dθµ g F (ˆ θ ) + ( F (( p (cid:48) + S ) ∧ − F (ˆ θ )) − p (cid:48) . However, the payoff above is negative by (22), and therefore, no p (cid:48) > θ g is profitable. There-fore, it is optimal for a t = 2 buyer to offer p g = θ g to the bailout recipients.Consider next buyers in t = 2 who make offers to the firms that did not accept a bailoutoffer. By the same logic as above, one can easily find that it is not optimal for any buyer tooffer p (cid:48) < p m = θ m . Furthermore, it is not optimal for any buyer to offer p (cid:48) ∈ ( θ m , p g ] sincesuch an offer will attract the sellers to the t=1 market, whose average value is θ m , so the buyerwill earn θ m − p (cid:48) <
0. Lastly, suppose a t = 2 buyer offers p (cid:48) > p g . Such an offer will attractall firms that sold to the market in t = 1 plus all types θ ∈ (ˆ θ g , ( p (cid:48) + S ) ∧ − µ g ) F (ˆ θ ) θ m + (cid:82) ( p (cid:48) + S ) ∧ θ g θf ( θ ) dθ (1 − µ g ) F (ˆ θ ) + ( F (( p (cid:48) + S ) ∧ − F (ˆ θ g )) − p (cid:48) . By (23), however, the above payoff is negative, which implies such a deviation price p (cid:48) > p g isunprofitable. Hence, it is optimal for every t = 2 buyer to offer p m = θ m to the firms refusingthe bailout.Lastly, consider buyers in t = 1. We shall prove that it is optimal for a buyer of t = 1 tooffer p m . To this end, we first establish a property on θ m . Lemma 4. θ m ≥ p in any short-lived stimulation equilibrium.Proof. Suppose to the contrary that θ m < p = E [ θ | θ ≤ θ ]. Since θ g < θ m by Theorem 3,we observe θ g < p . Now, suppose a t = 2 buyer deviates and offers p to the firms thataccepted a bailout. These firms comprise a fraction µ g of firms with θ ≤ ˆ θ (whose averageis θ g ) and all firms with θ ∈ (ˆ θ, ˆ θ g ]. All of the former firms will sell to the deviator, andamong the latter, all firms with θ ∈ (ˆ θ, θ ] are now willing to sell to the deviator at p . (Recallˆ θ g = p g + S > p + S = θ .) Thus the deviating buyer will get the payoff µ g F (ˆ θ ) θ g + (cid:82) θ ˆ θ θf ( θ ) dθµ g F (ˆ θ ) + F ( θ ) − F (ˆ θ ) − p . µ g θ g + (1 − µ g ) θ m = E [ θ | θ ≤ ˆ θ ] by (20) and θ g < θ m , however, we have µ g F (ˆ θ ) θ g + (cid:82) θ ˆ θ θf ( θ ) dθµ g F (ˆ θ ) + F ( θ ) − F (ˆ θ ) = F ( θ ) E [ θ | θ ≤ θ ] − (1 − µ g ) F (ˆ θ ) θ m F ( θ ) − (1 − µ g ) F (ˆ θ ) > E [ θ | θ ≤ θ ] = p . This implies the deviation will be profitable, a contradiction.
Q.E.D.
We now show that it is optimal for every t = 1 buyer to offer p m . Since buyers in t = 2 willoffer p m to every firm refusing the bailout, any offer p (cid:48) < p m = θ m is not attractive to any firm.Suppose a buyer offers p (cid:48) > θ m . Since p (cid:48) + p m + 2 S > p m + 2 S = p g + p g + 2 S = ˆ θ + p g + S ,all firms with type θ ≤ p (cid:48) + S − ( p g − θ m ) will sell at p (cid:48) . This implies the deviating buyer willget the payoff E [ θ | θ ≤ p (cid:48) + S − ( p g − θ m )] − p (cid:48) . Since p (cid:48) > p m = θ m ≥ p by Lemma 4 and p g > θ m by Theorem 3, however, we have E [ θ | θ ≤ p (cid:48) + S − ( p g − θ m )] − p (cid:48) < E [ θ | θ ≤ p (cid:48) + S ] − p (cid:48) < , where the strict inequality follows from definition of θ . This implies any offer p (cid:48) > p m is notprofitable. Q.E.D.
C.2 Delayed Stimulation Equilibrium
In a delayed stimulation equilibrium, firms with types θ ≤ ˆ θ sell in both periods: a fraction µ g of these firms sell to the government at p g in t = 1 and to the market at p g in t = 2 and therest of them sell to the market at p m in t = 1 and p m in t = 2. Meanwhile, types θ ∈ (ˆ θ, ˆ θ g ] sellto the government at p g in t = 1 but withdraw from the t = 2 market, and types θ ∈ (ˆ θ g , θ ]sell at p m only in t = 2; types θ > θ do not sell in any period. Recall from Theorem 4 thatˆ θ = θ , p g = θ g = p m = θ m = p = E [ θ | θ ≤ θ ], and p m = p g in any delayed-stimulationequilibrium.Throughout, we focus on delayed stimulation equilibria in which 0 < µ g < θ < ˆ θ g —one can easily establish an equilibrium without this property by applying the same logic. Likewe did in Section C.1, we first list all necessary conditions for a delayed stimulation equilibrium.First, like in the short-lived stimulation equilibrium, there are conditions associated with theoptimality of firms’ prescribed equilibrium strategies as listed below: p g + p g = p m + p m , (24) p m + p m + 2 S = ˆ θ + p g + S, (25) θ = ( p m + S ) ∧ . (26)43y applying p g = p m = p to the conditions above, we have p g = p m , (27)ˆ θ = p + S = θ , (28) θ = ( p g + S ) ∧ . (29)Second, buyers must find it optimal to offer the equilibrium price on the equilibrium pathover two periods. First of all, buyers in t = 2 should obtain the highest payoff from offering p to the firms that received a bailout in t = 1. Specifically, the t = 2 buyers must not havean incentive to offer p (cid:48) > p to the bailout recipients. By definition of θ , such a p (cid:48) will attractthe bailout recipients with types θ ≤ θ and θ ∈ ( θ , p (cid:48) + S ]. Since the fraction µ g of firmswith types θ ≤ θ receive a bailout in t = 1 and their average value is θ g = E [ θ | θ ≤ θ ], thefollowing condition must hold for p (cid:48) to be unprofitable: µ g F ( θ ) E [ θ | θ ≤ θ ] + (cid:82) p (cid:48) + Sθ θf ( θ ) dθµ g F ( θ ) + ( F ( p (cid:48) + S ) − F ( θ )) − p (cid:48) < ∀ p (cid:48) ∈ ( p , p g ] . (30)Moreover, buyers in t = 2 must get the highest payoff from offering p to the firms that refuseda bailout offer in t = 1. As before, this means that a t = 2 buyer must not have an incentiveto offer p (cid:48) > p m = p g to these bailout holdouts. Since the fraction 1 − µ g of firms with types θ ≤ θ do not receive the bailout in t = 1 and their average value is E [ θ | θ ≤ θ ], we must havethe following condition:(1 − µ g ) F ( θ ) E [ θ | θ ≤ θ ] + (cid:82) p (cid:48) + S ˆ θ g θf ( θ ) dθ (1 − µ g ) F (ˆ θ ) + ( F ( p (cid:48) + S ) − F ( θ )) − p (cid:48) < ∀ p (cid:48) > p g . (31)Indeed, the following observation reveals that the above necessary conditions—as well as theproperties of Theorem 4—are sufficient for the existence of the delayed stimulation equilibrium. Proposition 2.
For a p g > p , there exists a delayed stimulation equilibrium if there exist µ g ∈ (0 , and ˆ θ g ∈ [ θ , p g + S ) that satisfy (24) – (31) .Proof. Just as we did in the proof of Proposition 1, we focus on the case θ <
1; the proof forthe case θ = 1 is very similar.We first prove that it is optimal for every type- θ firm to play the prescribed equilibriumstrategies over two periods. Consider t = 2 first. By definition of θ and p , one can easilyshow that after receiving the bailout in t = 1, firms with types θ ≤ θ would sell whereasfirms with types θ ∈ ( θ , ˆ θ g ] would refuse to do so in t = 2. Furthermore, since θ = p g + S by Theorem 4, it is optimal for every firm with type θ ≤ θ to sell at p m = p g in t = 2 afterrefusing the bailout in t = 1. Lastly, since θ = p g + S and p m = p g , it is optimal for the firms44ith types θ > θ not to sell at p m in t = 2.Consider t = 1 next. Since p g + p g = p m + p m from (27), firms with types θ ≤ θ areindifferent between selling to the government and selling to the market in t = 1. Furthermore,since θ + p g + S ≤ p g + p + 2 S for all θ ≤ θ from (28), it is optimal for the firms withtypes θ ≤ θ to sell in t = 1. The same condition (28) also implies p g + p + 2 S < θ + p g + S for all θ > θ , so firms with types θ > θ do not prefer selling in t = 1. Furthermore, since p m = p g > p , firms with types θ ∈ ( θ , θ ] are indifferent between accepting the bailout in t = 1 (but boycotting the market in t = 2) and not selling in t = 1 (but selling in t = 2).Moreover, since θ = p g + S , it is optimal for the firms with types θ ∈ ( θ , ˆ θ g ] to sell to thegovernment in t = 1 and for the firms with types θ > ˆ θ g not to sell in t = 1.Next, we prove that it is optimal for buyers to offer the stated equilibrium prices in eachperiod. First, consider the offers made to the firms that received a bailout in t = 1. In t = 1,a fraction µ g of firms with types θ ≤ θ and all firms with types θ ∈ ( θ , ˆ θ g ] receive the bailout.Since any offer p (cid:48) < p g = p is unattractive to these firms, no t = 2 buyer will deviate andoffer p (cid:48) < p . Suppose a t = 2 buyer offers p (cid:48) > p . Such an offer will attract the bailoutrecipients with types θ ≤ θ and θ ∈ ( θ , p (cid:48) + S ]. Since the average value of firms with types θ ≤ θ that sell to the government is θ g = E [ θ | θ ≤ θ ], the deviating buyer will get the payoff µ g F ( θ ) E [ θ | θ ≤ θ ] + (cid:82) p (cid:48) + Sθ θf ( θ ) dθµ g F ( θ ) + ( F ( p (cid:48) + S ) − F ( θ )) − p (cid:48) . However, such an offer p (cid:48) is not profitable by (30): if p (cid:48) ∈ ( p , p g ], the payoff from offering p (cid:48) is negative; the payoff is strictly negative for all p (cid:48) > p g .Consider next offers made to the firms that refused a bailout in t = 1. In t = 1, a fraction(1 − µ g ) of firms with types θ ≤ θ refuse the bailout. Furthermore, the firms with highertypes θ > ˆ θ g refuse the bailout in t = 1. Since any offer p (cid:48) < p is unattractive to the firmsrefusing the bailout in t = 1, no t = 2 buyer will offer such a p (cid:48) . Suppose a t = 2 buyerdeviates and offers p (cid:48) > p . If p (cid:48) ≤ ˆ θ g − S , only the firms with types θ ≤ θ will sell at p (cid:48) ,making p (cid:48) unprofitable since θ m − p (cid:48) = E [ θ | θ ≤ θ ] − p (cid:48) <
0. If p (cid:48) > ˆ θ g − S ], the firms thatsold to the market in t = 1 and the holdout firms with types θ ∈ (ˆ θ g , ( p (cid:48) + S ) ∧
1] sell to thedeviating buyer. Since a fraction (1 − µ g ) of firms with types θ ≤ θ sell to the market in t = 1 and their average value is E [ θ | θ ≤ θ ], the deviating buyer will get the payoff(1 − µ g ) F ( θ ) E [ θ | θ ≤ θ ] + (cid:82) ( p (cid:48) + S ) ∧ θ g θf ( θ ) dθ (1 − µ g ) F (ˆ θ ) + ( F (( p (cid:48) + S ) ∧ − F ( θ )) − p (cid:48) . However, the above payoff is negative by (31).Lastly, consider buyers in t = 1. If a buyer offers p (cid:48) (cid:54) = p and a type- θ firm sells at that45rice, this firm will receive the price offer p m = p g from buyers in t = 2: recall the t = 2 buyerscan only observe whether or not firms receive the bailout. Since p (cid:48) + p g + 2 S < p + p g + 2 S forany p (cid:48) < p , any offer p (cid:48) < p is unattractive to any type- θ firm, and thus no buyer in t = 1has an incentive to offer such a price. Suppose a buyer in t = 1 deviates and offers p (cid:48) > p .Since p (cid:48) + p g + 2 S > p + p g + 2 S , all firms with types θ ≤ θ will also sell at p (cid:48) . Furthermore,since p (cid:48) + p g + 2 S > θ + p g + S , firms with types θ ∈ ( θ , ( p (cid:48) + S ) ∧