Solvency Shocks, Liquidity Shocks, and Fire Sales Johan Hombert∗ May 1, 2009

Abstract This paper offers a welfare analysis of fire sales. I develop an equilibrium model of fire sales due to uninsurable idiosyncratic risk, and I study its welfare properties from a second-best perspective. When idiosyncratic shocks are mainly liquidity shocks, asset prices are too low in the private economy. By contrast, when idiosyncratic shocks are mainly solvency shocks, competitive asset prices are too high. I then design an empirical test to determine where the economy stands. I also extend my model to an open economy and show that fire-sale FDI is associated with a deviation from the second-best, but that imposing capital control would make a bad situation worse.

1

Introduction

As exemplified by the current crisis, recessions are generally accompanied by a collapse in asset prices. This presumably reflects the lower fundamental value of assets as well as the lower ability of firms to fund asset purchases. In turn, depressed asset prices weaken firms’ balance sheets and thus their ability to roll-over current liabilities and to raise new capital. Therefore, as emphasized by Shleifer and Vishny (1992), asset prices and debt capacity are jointly determined in equilibrium. In particular, financial distress and fire sales of assets are mutually reinforcing phenomena.1 Shleifer and Vishny provide no formal welfare analysis of this phenomenon, though they informally suggest that “fire sales can have substantial private and social costs” for at least two reasons. First, asset trades guided by financial muscle rather than real efficiency may result in capital misallocation. Second, fire sales reduce liquidation values, which may reduce firms’ debt capacity. In this paper, I investigate the welfare consequences of fire sales. Addressing the (in)efficiency of fire sales raises a related question. It is well-known that all investment decisions are distorted by financial frictions; this result applies in particular to asset acquisitions. Then why should fire sales deserve a specific analysis? The short answer is that an asset transaction ∗

ENSAE-CREST Pulvino (1998) provides evidence of fire sales in the airline industry. Acharya, Bharath and Srinivasan (2007) extend the analysis to all defaulted bond issuers in the US. There are also evidence of fire sales at the country-wide level (Aguiar and Gopinath, 2005) and in equity markets (Coval and Stafford, 2007). 1

1

involves the simultaneous investment decisions of two firms rather than one. The distorted choice of the buyer affects the seller and vice versa, which paves the road for an externality. I build a model in which credit constrained firms trade assets with each others in response to uninsurable idiosyncratic shocks, and ask whether the competitive equilibrium achieves the second-best allocation. In the model, entrepreneurs raise debt from investors to implement a project. They face a tradeoff between raising more debt to set up a larger firm, and avoiding debt overhang in the future. Then firms are hit by an uninsurable idiosyncratic shock. Depending on the nature and the intensity of the shock and on their remaining debt capacity, firms may be scaled up or scaled down by trading assets with each others. The equilibrium on the secondary asset market exhibits the basic feature of fire sales, namely, that assets are sold at discounts to fundamental values. Furthermore, the equilibrium discount is determined by firms’ initial leverage: the firms are the more leveraged, the lower the asset price. My main results are then the following. First, I show that the pecuniary externality arises because firms do not take into account the impact of their choice of leverage on asset prices. Fire sales, in turn, act as transfers from sellers to buyers. While the distribution of wealth is irrelevant for welfare when financial markets are frictionless, it matters in the presence of credit rationing. Fire sales reduce welfare if firms on the sell-side of the asset market have a higher shadow value of wealth than firms on the buy-side of the market. When this is the case, I show that a higher asset price can improve utilitarian welfare. From an ex ante standpoint this is also a Pareto improvement, since firms are identical before the realization of the idiosyncratic shocks. Conversely, if assets buyers have a higher shadow value of wealth than asset sellers, then fire sales are socially beneficial and an asset price even lower than its competitive level leads to a Pareto improvement. Second, the competitive asset price is too low when idiosyncratic shocks are mainly liquidity shocks, that is, income shocks. Firms hit by a positive income shock have more debt capacity. Thus, at equilibrium, they are on the buy-side of the asset market and expand their scale of operation. Decreasing returns to scale imply that these firms create less value from an additional unit of wealth. Therefore, a higher asset price, by transferring wealth from high liquidity firms towards low liquidity firms, increases welfare. Third, the competitive asset price is too high when idiosyncratic shocks are mainly solvency shocks, that is, productivity shocks, provided that returns to scale are not too decreasing. Firms hit by a positive productivity shock make a better use of assets. As a consequence, they are on the buy-side of the asset market. Whether they have a higher or lower shadow value of wealth depends on returns to scale. On the one hand, firms hit by a positive productivity shock can derive more income from an additional unit of wealth. On the other hand, they are also bigger firms, which tends to reduce their marginal productivity. If returns to scale are not too decreasing, the former effect dominates, and a lower asset price, by transferring wealth from low solvency firms towards high solvency firms, improves welfare. If returns to scale are strongly decreasing, the reverse holds. Forth, I suggest an empirical test to assess whether the asset price is socially too low or too high. The model unveils that the answer depends on which side of the asset market has

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the higher shadow value of wealth. However, this variable is a conceptual one and has a priori no obvious empirical counterpart. I show that the Tobin’s q can proxy for the shadow value of wealth, which allows to test empirically the direction of the pecuniary externality. This paper belongs to the literature that analyzes the general equilibrium feed-back between asset prices and debt capacity. Its contribution is to offer a welfare analysis of fire sales and to make a first step in the design of an empirical test of the efficiency of the competitive equilibrium. Shleifer and Vishny (1992) build the theoretical foundations of fire sales due to the combination of financial frictions and industry-specific assets, and focus on the possibility for multiple equilibria. Tirole (2006) shows that their model predicts an inefficiently low volume of transactions since firms have few incentives to sell assets at fire-sale prices. Kiyotaki and Moore (1997) explore the dynamic linkages between asset prices and credit rationing, and argue that they can greatly amplify credit cycles. Relatedly, but on financial markets, Shleifer and Vishny (1997) emphasize that money managers experience outflows of funds after price drops, which force them to unwind their positions and thereby amplifies the initial shock. Krishnamurthy (2003) shows that such propagation mechanisms of aggregate shocks depend crucially on the assumption that they cannot be insured. In this paper, I put aside the issue of the insurability and propagation of aggregate shocks, and focus on the consequences of uninsurable idiosyncratic shocks. Closer to this paper are Gromb and Vayanos (2002), Holmstr¨om and Tirole (2008) and Lorenzoni (2008). Gromb and Vayanos (2002) develop a dynamic model of financially constrained arbitrageurs and study the efficiency of the competitive allocation. They also find that competitive prices can be too low or too high. One of my contributions is to offer a simple characterization of the nature of the shocks that lead to too low or too high asset prices. In Holmstr¨ om and Tirole (2008) and in Lorenzoni (2008), the asset price is too low when the asset can be put into an inferior alternative use. In Holmstr¨om and Tirole (2008), the asset price can also be too high when firms build financial muscle to overbid other potential acquirers of assets. My model sheds light on the mechanisms underlying these results by endogenizing the reason why at equilibrium poor users of assets may end up on the buy-side of the market – because there are large idiosyncratic liquidity shocks – or why the sell-side of the market may not value liquidity – because there are large idiosyncratic solvency shocks. This paper also contributes to the literature on bank failures. Indeed, the riskiness of banks’ loan portfolio bears similarities with the firms’ leverage in my model: both determine the ability to withstand shocks and to finance the acquisition of failed assets. Perotti and Suarez (2002) consider an oligopolistic banking industry and show that risk attitudes are strategic substitutes. They also derive implications for competition policy. Acharya and Yorulmazer (2008) extend their analysis to bailout policy and liquidity provision policy. I also shed light on the fire-sale FDI phenomenon emphasized by Krugman (1998) and documented by Aguiar and Gopinath (2005). Acharya, Shin and Yorulmazer (2009) study the optimal bailout policy when domestic assets are taken at fire-sale prices by foreigners. I complement the analysis by showing why the competitive equilibrium is constrained efficient in the first place. I argue that there is underinsurance as in Caballero and Krishnamurthy

3

(2003), but for a different reason. Finally, I also suggest that imposing capital control to prevent fire-sale FDI makes a bad situation worse. The rest of the paper is organized as follows. Section 2 describes the model and firms’ capital structure. In Section 3, I solve for the equilibrium of the private economy, study welfare, and design an empirical test. I present several extensions in Section 4 and make some concluding remarks in Section 5.

2

Model

2.1

Set-Up

A mass one continuum of identical entrepreneurs and a set of competitive investors live for three periods 1, 2 and 3. Investors are endowed with a large amount of wealth at each date. Everyone is risk-neutral and the discount rate is zero. At date 1, each entrepreneur is born with wealth A > 0 and has a project. He selects freely its level of capital k ≥ 0. When k exceeds A, which will be the case in equilibrium, an amount of k − A must be raised from an investor. As detailed in Section 2.2, the entrepreneur issues debt with face value denoted by D1 . At the beginning of date 2, each firm is hit by a productivity shock v˜ and an income shock y˜. Shocks are idiosyncratic, although v˜ and y˜. Since there is a continuum of firms, the law of large numbers implies that there is no aggregate uncertainty. The realization of the productivity shock v˜ > 0 determines the date 3 payoff in a way that is specified shortly. The income shock is an intermediate cash flow equal to y˜ > 0.2 There exist v > 0, y > 0, h ≥ 0, σv ≥ 0, σy ≥ 0 and an i.i.d. random vector (˜ εv , ε˜y ) with mean (0, 0) and covariance matrix ((1, ρ), (ρ, 1)) such that v˜ = v + hσv ε˜v , y˜ = y + hσy ε˜y . Therefore, v˜ and y˜ have, respectively, mean v and y, variance h2 σv2 and h2 σy2 , and covariance ρh2 σv σy . I also assume that (˜ εv , ε˜y ) has a bounded support and I denote by φ its probability distribution function. In the following, I focus on small shocks, that is, I assume that h is not too large. This allows me to solve the linearized model and to obtain simple closed-form expressions. Together with the assumption that (˜ εv , ε˜y ) has a bounded support, a small h also ensures that v˜ and y˜ are always strictly positive. Once the shocks are observed, the firm can be rescaled to final size k 0 by trading assets at the (endogenous) asset price p with the other firms. The firm can be either scaled up, k 0 > k, or scaled down, k 0 < k. I define the resizing factor as s ≡ k 0 /k. The firm needs a net amount of cash of p(s − 1)k + D1 − y˜ to buy additional assets (if s > 1) and repay its debt. If this cash need is strictly positive, the firm can raise additional funds from investors 2

The analysis would be similar with negative realizations of y˜. In that case, the firm would require a cash infusion at date 2.

4

according to a financial arrangement described in Section 2.2. There are decreasing returns to scale with respect to the date 2 resizing factor: the firm pays off v˜f (s)k at date 3, where v˜ is the productivity shock realized at date 2, and f (0) = 0, f 0 > 0, f 00 < 0, f 0 (0) = +∞, f 0 (+∞) = 0. There are two financial frictions in the model. First, there is a wedge between NPV and pledgeable income. A fraction β ∈ (0, 1) of the final payoff cannot be transferred to outside investors. This closed-form expression for the entrepreneur’s private benefits can support several usual interpretations.3 I assume that a firm which always continues with the same size at date 2 has on average a strictly positive NPV vf (1) > 1,

(1)

(1 − β)vf (1) < 1.

(2)

and a strictly negative pledgeable value

These assumptions will ensure that projects are worth being implemented, but that they cannot be entirely funded by outside investors. The second financial friction is that there is imperfect hedging of idiosyncratic shocks. The choice variables, k and s, the random shocks, v˜ and y˜, and the final payoff, v˜f (s)k, are commonly observable, but not verifiable. In particular, the date 1 financial contract cannot be made contingent on the realization of the shocks. This assumption will prevent firms from smoothing completely idiosyncratic shocks.4

2.2

Debt Contracts

It is worth noting that although payoffs are not verifiable, the entrepreneur can nevertheless commit to repay investors as the final payoff is certain conditional on the realization of the shock. He can therefore commit to repay the pledgeable value of the firm corresponding to the worst state of nature. There are two financing stages: at the initial date t = 1 and after the realization of the shocks at t = 2. The date 1 contract can only specifies uncontingent repayments, since no verifiable information is released after t = 1. Moreover, since additional funds can be raised at date 2, there is no loss of generality in restricting all the repayment to take place at date 2. Therefore, 3

For instance, this closed-form expression can stand for the following moral hazard problem. The entrepreneur has to provide an unobservable effort between date 2 and date 3. If he does so, the project succeeds and pays off v˜f (s)k/γ with probability γ, or fails and yields 0 with probability 1 − γ; if the entrepreneur does not exert effort, he obtains private benefits β˜ v f (s)k, but the project always fails. The outcome of the project (success or failure) is verifiable although the value of the payoff in case of success is not. The entrepreneur is induced to work if he is given a wage of (at least) β˜ v f (s)k/γ in case of success. Given the probability γ of success, the expected payoff is v˜f (s)k and the expected wage is β˜ v f (s)k. 4 To clarify the consequences of imperfect hedging, I consider in Section 4.1 the case of perfect hedging, which corresponds to the alternative model in which all the variables are verifiable (but the wedge between NPV and pledgeable income still exists). In that case, I show that the competitive equilibrium achieves the second-best allocation.

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the date 1 contract is implemented with a short-term risk-free debt contract with face value denoted by D1 . At date 2, the realization of the shocks is commonly known, although not verifiable. The firm can then raise additional funds from a new investor, or, equivalently, from the date 1 investor. The repayment takes place at date 3 and is also deterministic. Therefore, the date 2 contract is implemented with a short-term risk-free debt contract. The face value depends on the realization of the shocks and is denoted by D2 (˜ v , y˜).

2.3

(Privately) Optimal Leverage

I now solve the problem of an individual entrepreneur. Since the entrepreneur is atomistic, he takes the asset price p as given. He chooses the date 1 investment k, the amount of debt D1 , and the date 2 resizing strategy s(˜ v , y˜) and the amount of debt D2 (˜ v , y˜) as functions of the realization of the shocks, to maximize the expected NPV max

−k + E [˜ v f (s(˜ v , y˜))k − p(s(˜ v , y˜) − 1)k] ,

(3)

k, s(.,.), D1 , D2 (.,.)

where the expectation is taken over (˜ v , y˜), subject to k − A ≤ D1 ,

(4)

p(s(˜ v , y˜) − 1)k + D1 − y˜ ≤ D2 (˜ v , y˜)

∀ (˜ v , y˜),

(5)

D2 (˜ v , y˜) ≤ (1 − β)˜ v f (s(˜ v , y˜))k

∀ (˜ v , y˜).

(6)

(4) is the break-even condition of the date 1 investor. For every realization of the shocks, (5) is the break-even condition of the date 2 investor. (6) is the constraint that only a fraction 1 − β of the final payoff is pledgeable to outside investors. The face values of debt can be dropped from the program by summing constraints (4), (5) and (6). When no ambiguity exists, I denote the contingent resizing factor s(˜ v , y˜) by s˜ to allege notations. The problem then rewrites max k, s˜

s.t.

E [−1 + p + v˜f (˜ s) − p˜ s] k, A + y˜ + [−1 + p + (1 − β)˜ v f (˜ s) − p˜ s] k ≥ 0

(7) ∀ (˜ v , y˜).

(8)

For each realization of the shocks, the break-even condition (8) states that the total pledgeable income must be nonnegative. These break-even constraints are always binding in equilibrium, provided that h is not too large. Indeed, consider the limit case of no shock: h = 0. (1) ensures that the NPV is increasing in k, since s˜ = 1 in equilibrium when h = 0, and (2) implies that the pledgeable income is decreasing in k. Therefore, if the break-even constraint (8) was not binding, it would be possible to increase the NPV by choosing a higher k. By continuity of the problem in h, the break-even condition is binding for h small. Note that, since (8) is binding, the date 1 break-even condition (4) is also binding. Therefore, the initial investment k is directly related to the leverage of the firm, and the debt-toequity ratio at date 1 is equal to D1 /A = k/A − 1. In the following, I shall refer equivalently 6

to k as the date 1 investment or, with a slight abuse of terminology, as the leverage of the firm. For given k and (˜ v , y˜), the resizing factor s˜ is thus taken to the highest level that can be funded at date 2. Equation (8) implies that this optimal s˜ is increasing in v˜ and y˜, and decreasing in k. It is increasing in v˜ and y˜, since date 2 investment is constrained by the amount of pledgeable income, which increases with v˜ and y˜. The fact that it is decreasing in k reflects a tradeoff between date 1 investment and date 2 investment. Both consume pledgeable income, therefore increasing investment in one period requires to reduce investment in the other period. Interpreting k as the leverage of the firm, this tradeoff is reminiscent of the usual debt overhang problem: the date 1 leverage is larger, the lower the debt capacity at date 2. Lemma 1. There exist unique optimal leverage k ∗ (p) and resizing policy s˜∗ (p) solution to (7)(8) when p is in the neighborhood of an equilibrium value. k ∗ (p), s˜∗ (p), and the corresponding expected NPV denoted by N P V ∗ (p), are continuous in p. Proof. See Appendix A.1. ˜ I denote by λ(p)φ(˜ v /h, y˜/h) the shadow price of the break-even condition when the asset price is p, where the argument (˜ v , y˜) is omitted. Its expression is obtained by writing the Lagrangian and the first order condition with respect to s˜: ˜ λ(p) =

v˜f 0 (˜ s∗ (p)) − p p − (1 − β)˜ v f 0 (˜ s∗ (p))

∀ (˜ v , y˜).

(9)

A unitary increase in pledgeable income in the state (˜ v , y˜) can be levered 1/(p − (1 − β)˜ v f 0 (˜ s∗ (p))) times. Since the shock (˜ v , y˜) occurs with probability φ(˜ v /h, y˜/h) and the 0 ∗ ˜ marginal return of date 2 investment is equal to v˜f (˜ s (p)) − p, then λ(p) is given by equation (9).

3 3.1

Welfare Analysis Competitive Equilibrium

To begin with, I define what a competitive equilibrium is. Definition 1. A competitive equilibrium consists of an asset price pCE , and a set of the investment strategies (k CE , s˜CE ) of every firm, such that (i) the asset market clears and (ii) investment strategies are privately optimal given the asset price. Since firms are ex ante identical, they all choose the same leverage k CE and resizing policy s˜CE : k CE = k ∗ (pCE ), s˜CE = s˜∗ (pCE ). 7

The asset market clearing condition at date 2 writes as E[˜ sCE − 1]k CE = 0. Combining these conditions, pCE is an equilibrium price if, and only if, the expected (privately) optimal resizing factor is equal to one E[˜ s∗ (pCE )] = 1.

(10)

A small h ensures that condition (10) defines a unique pCE . Indeed, when the asset price increases, the expected resizing factor is affected by a substitution effect and a wealth effect. The substitution effect is a modification of the tradeoff between date 1 investment and date 2 investment in favor of date 1 investment. Therefore, k increases and s˜ decreases. The wealth effect is a modification of the date 2 pledgeable income. From equation (8), it appears that a higher asset price transfers wealth from asset buyers to asset sellers. This affects the amount of funds they can raise at date 2, and thus their resizing factor s˜. However, when h is close to zero, the heterogeneity of firms is small at date 2, and the redistribution of wealth is small. Therefore, when the asset market is in equilibrium and shocks are not too large, the substitution effect dominates the wealth effect. This ensures the uniqueness of the equilibrium. Lemma 2. There exists a unique competitive equilibrium. The competitive asset price satisfies pCE < v˜f 0 (˜ sCE )

∀ (˜ v , y˜).

(11)

Proof. See Appendix A.2. The interpretation of equation (11) is that the competitive equilibrium exhibits fire sales, in the sense that the asset price is below the fundamental value. The fundamental value of the asset is the valuation of the marginal asset for the best user. Formally, it is equal to maxv˜,˜y v˜f 0 (˜ sCE ), which is strictly larger than pCE by equation (11). The wedge between the NPV and the pledgeable income prevents firms from bidding up to the fundamental value.

3.2

Pareto Improvement

I address the question of whether fire sales are associated with an inefficiency of the competitive equilibrium. As already emphasized, fire sales are the consequence of the wedge between NPV and pledgeable income. In terms of welfare analysis, this reflects a deviation from the first-best allocation; I shall come back on that point in Section 4.1. I now ask whether the competitive equilibrium achieves the second-best allocation, which satisfies the same constraints as the private economy. The reason why it may not be (and will not be) the case in that firms exert an externality on each other. When choosing their date 1 leverage, firms determine their date 2 debt capacity to purchase assets and thus affect the asset market equilibrium. However, atomistic firms do not take into account the impact

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of their financial decisions on the asset price. This paves the road for a pecuniary externality that operates through the date 2 asset price. To investigate this pecuniary externality, I ask whether a deviation from privately optimal behaviors together with a deviation of the asset price from its equilibrium value, can lead to a Pareto improvement. I look for a Pareto improvement that makes entrepreneurs strictly better off and leaves investors with the same utility. This is without loss of generality since it is always possible to transfer wealth from investors to entrepreneurs. Moreover, since entrepreneurs are ex ante identical and the technology exhibits decreasing returns to scale, it is straightforward to see that there is no point in implementing an asymmetric allocation. Therefore, a candidate Pareto improvement is an allocation (p, k, s˜) that strictly increases the expected NPV   E [−1 + p + v˜f (˜ s) − p˜ s] k > E −1 + pCE + v˜f (˜ sCE ) − pCE s˜CE k CE , that satisfies the break-even condition of investors for each realization of the shocks A + y˜ + [−1 + p + (1 − β)˜ v f (˜ s) − p˜ s] k ≥ 0

∀ (˜ v , y˜),

and that clears the asset market E[˜ s − 1]k = 0. I use the following terminology:5 Definition 2. (i) The competitive asset price is too low if there exists a Pareto improvement (p, k, s˜) such that p > pCE . (ii) The competitive asset price is too high if there exists a Pareto improvement (p, k, s˜) such that p < pCE . (iii) The competitive equilibrium is second-best optimal if there exists no Pareto improvement. I proceed in two steps to find a Pareto improvement that corrects the pecuniary externality. First, starting from the competitive equilibrium, I consider a slight modification of the asset price, say a price increase p > pCE to fix ideas. Firms react in a privately optimal fashion and choose date 1 leverage k ∗ (p) > k CE and resizing factor s˜∗ (p) < s˜CE . The expected NPV solution to program (7)-(8) is affected by a direct effect and an indirect effect that operates through the break-even conditions. The direct effect of a unit price increase is to increase the expected NPV by E[(1 − s˜CE )k CE ] = 0. Firms that experience a shock that place them on the sell-side of the asset market benefits from a price increase, while the buy-side side of the market is hurt. However, since in equilibrium firms are on average as much on the sell-side as on the buy-side of the asset market, the expected effect is zero. The indirect effect is to modify 5

Note that it is a priori not clear whether (i) and (ii) are mutually exclusive. In the following, I focus on small deviations from the competitive equilibrium. Except possibly for a knife-edge set of parameters, it will appear that there exist Pareto improvements with p < pCE only, or with p > pCE only, but not both.

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the pledgeable income by (1 − s˜CE )k CE for every realization of the shocks. This, in turn, affects the highest expected NPV that can be implemented, with a multiplicative factor equal to the shadow price of the corresponding break-even constraint. Finally, the total derivative of the expected NPV with respect to the asset price is equal to h i   dN P V ∗ CE ˜ CE = −Cov s˜CE , λ ˜ CE k CE . (p ) = 0 + E (1 − s˜CE )k CE λ (12) dp ˜ CE = λ(p ˜ CE ) is the shadow price of pledgeable income when the asset market is where λ competitive. Intuitively, a price increase transfers wealth from buyers to sellers, which has no welfare effect per se. However, if buyers and sellers have different valuations for pledgeable ˜ CE , then a wealth transfer can affect welfare. Note income, as measured by the shadow price λ that, at this stage, the asset market is no longer in equilibrium since E[˜ s∗ (p)] < 1. The second step consists in a slight modification of individual behaviors to restore the equilibrium on the asset market. Since there is excess supply, clearing the market requires to decrease date 1 leverage, k < k ∗ (p), and increase the resizing factor, s˜ > s˜∗ (p), along the break-even conditions. This slight deviation from the privately optimal tradeoff between date 1 investment and date 2 investment has no first order effect on the expected NPV by the envelop theorem. Therefore, the only first order effect of the proposed allocation (p, k, s˜) is given by equation (12). ˜ CE ) < 0, and too high if Lemma 3. The competitive asset price is too low if Cov(˜ sCE , λ ˜ CE ) > 0. Cov(˜ sCE , λ Lemma 3 states that a marginally higher asset price improves welfare when the correlation ˜ CE is negative. If firms that shrink at date 2 have a higher valuation between s˜CE and λ of pledgeable income, then a higher asset price transfer wealth towards these firms, which increases welfare. Conversely, if firms that expand at date 2 have a higher valuation of pledgeable income, then welfare is improved with a lower asset price. ˜ CE depends on the underlying source of The sign of the covariance between s˜CE and λ variation. The second order Taylor expansion of the covariance around h = 0 gives the welfare effect of a price increase " ˜ CE ˜ CE dN P V ∗ CE ∂˜ sCE ∂ λ ∂˜ sCE ∂ λ (p ) = − σy2 + σ2 dp ∂ y˜ ∂ y˜ ∂˜ v ∂˜ v v +

˜ CE ˜ CE ∂˜ sCE ∂ λ ∂˜ sCE ∂ λ + ∂˜ v ∂ y˜ ∂ y˜ ∂˜ v

!

# ρσy σv k CE h2 + rh , (13)

where rh is a negligible remainder. The first term in the bracket is the welfare effect of a higher asset price due to the heterogeneity of firms with respect to y˜. The second term comes from the heterogeneity with respect to v˜. The third term is the welfare effect that arises from the covariation between y˜ and v˜. Before investigating the welfare effect of the various types of shocks, it is worth noting that if there is no shock (h = 0), then the competitive equilibrium is second-best optimal. Since 10

traded volumes are zero in equilibrium, fire sales have no redistribution effect even though the asset price is below the fundamental value. Consider first the effect of liquidity shocks (the first term). Firms with a higher y˜ are able to finance a higher date 2 investment: ∂˜ sCE /∂ y˜ > 0. As a consequence, they value additional pledgeable income less than firms with a bad liquidity shock because of decreasing returns ˜ CE /∂ y˜ < 0. This implies that the underlying variation in y˜ generates a negative to scale: ∂ λ ˜ CE . The firms with the highest shadow values of pledgeable correlation between s˜CE and λ income are on the buy-side of the asset market. Therefore, when liquidity shocks are prevalent (σv /σy small), the competitive asset price is too low. Consider now the effect of solvency shocks (the second term). Firms with a higher v˜ have a larger resizing factor: ∂˜ sCE /∂˜ v > 0. The effect of v˜ on the shadow price of pledgeable income is ambiguous. On the one hand, high v˜-firms are more efficient and make a better use of additional pledgeable income. On the other hand, they are also larger, which tends to reduce their shadow price of pledgeable income because of decreasing returns to scale. ˜ CE /∂˜ Overall, the sign of ∂ λ v depends on the extent to which returns to scale are diminishing. ˜ CE /∂˜ If returns to scale are not too decreasing, then ∂ λ v > 0 and the solvency shocks generate CE CE ˜ a positive correlation between s˜ and λ . In that case, the firms with the highest shadow values of pledgeable income are on the sell-side of the asset market, and the competitive asset price is too high provided that solvency shocks are prevalent (σv /σy large). The reverse holds if returns to scale are strongly decreasing. The following Proposition summarizes the above discussion. It also provides a closed-form expression of the condition on the returns to scale for a simple specification of the production function (the condition in the general case is relegated in Appendix A.3).6 Proposition 1. • When liquidity shocks are prevalent, the competitive asset price is too low. • When solvency shocks are prevalent, the competitive asset price is too high if returns to scale are not too decreasing, and too low if returns to scale are strongly decreasing. If, in addition, the production function is f (s) = sα , with α ∈ (0, 1), then the asset price is too high if α > (1 − β)v, and too low if α < (1 − β)v. • When there is no shock, the competitive equilibrium is second-best optimal. Proof. See Appendix A.3. 6

The solvency shocks v˜ of the model are actually a combination of shocks on the NPV (a pure productivity shock) and of shocks on the wedge β˜ v between the NPV and the pledgeable value. It is possible to distinguish between these two types of shock by defining the random variables z˜ = v˜ and ˜b = β˜ v . z˜ is a pure productivity shock and ˜b a pure solvency shock. Proceeding as in the basic model with f (s) = sα , it is straightforward to show that there exist αb and αz such that pure solvency (pure productivity) shocks make the asset price too low if α < αb (if α < αz ), and too high if α > αb (if α > αz ). Pure solvency shocks are more likely to make the asset price too high than pure productivity shocks in the sense that αb > αv . This is consistent with the intuition of Proposition 1.

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Consider now that the solvency shock and the liquidity shock are positively correlated: ρ > 0. Coming back to equation (13), this adds the two terms in parentheses to the correlation ˜ CE . First, firms with a good solvency shock expands. They also have on between s˜CE and λ average a good liquidity shock and thus a low shadow value of pledgeable income. This effect ˜ CE and, therefore, to make the tends to generate a negative correlation between s˜CE and λ competitive asset price too low. Second, firms with a good liquidity shock are on the buy-side of the asset market. They also have on average a good solvency shock, which is associated with a high (low) shadow value of pledgeable income if returns to scale are not too decreasing (strongly decreasing). The direction of this second effect is the same than for solvency shocks. The overall effect is to make the competitive asset price too high if returns to scale are not too decreasing, and too low if returns to scale are strongly decreasing. Proposition 2. For given variances of the shocks, when the correlation between the solvency shock and the liquidity shock increases, the competitive asset price is more likely to be too high if returns to scale are not too decreasing, and too low if returns to scale are strongly decreasing. If the production function is f (s) = sα , with α ∈ (0, 1), then the corresponding threshold of returns to scale is α = 2(1 − β)v/(1 + (1 − β)v). Proof. See Appendix A.4. Another interpretation of y˜ and v˜ is that they are idiosyncratic shocks to current income and future income, respectively. With this interpretation, ρ should be understood as the persistence of shocks. When ρ = 0, current and future income are independent; when ρ > 0, the current income predicts the future income. Figure 1: Welfare effect of fire sales for f (s) = sα and h → 0 σv /σy

ρ>0

ρ=0 pCE too high

pCE too low 0

(1 − β)v

1

2(1−β)v 1+(1−β)v

α

Figure 1 offers a graphical representation of Propositions 1 and 2 for the power production function. The solid line is the threshold value of the ratio of the variances of the solvency-to12

liquidity shocks below which the competitive asset price is too low, when ρ = 0. The asset price is always too low when α < (1 − β)v. When α > (1 − β)v, the asset price is too high if σv /σy is large. As α increases to 1, the set of ratios σv /σy such that the competitive asset price is too high expands. The dashed line corresponds to ρ > 0. If α < 2(1 − β)v/(1 + (1 − β)v), the increase in correlation between the solvency shock and the liquidity shock expands the set of ratios σv /σy such that the asset price is too low. Conversely, when α is above that threshold, a positive correlation between v˜ and y˜ tends to make the competitive asset price too high. To conclude that section, I determine whether the proposed Pareto improvements are associated with more or less reallocation of assets. Specifically, I focus on the volume of transactions over the volume of assets: E[max{˜ s − 1, 0}]. In a Pareto improvement, the asset price is modified in order to transfer wealth to the side of the asset market with the higher shadow value of wealth. When the asset price is increased, asset sellers are made better off and liquidate fewer assets, while asset buyers are made worse off and purchase fewer assets. Hence, the volume of transactions decreases. Conversely, when the Pareto improvement features a price reduction, the volume of asset transactions increases. Proposition 3. When the competitive asset price is too low (high), the competitive volume of transactions is too high (low). Proof. See Appendix A.5. Eisfeldt and Rampini (2006) provide evidence that the amount of capital reallocation is procyclical, and the cross-sectional dispersion of productivity is countercyclical. They explain that pattern by time-varying (countercyclical) adjustment costs. Under this interpretation, the competitive equilibrium is socially optimal. By contrast, Propositions 1 and 3 imply that when the dispersion of productivity σv is large, the amount of capital reallocation is socially too low.

3.3

Empirical Test Design

In Propositions 1 and 2, I show that a key determinant of whether the competitive asset price is too low or too high is the nature of the idiosyncratic shocks. However, in practice, it is usually difficult for policy makers to assess the nature of shocks that hit firms. In the model, a firm that reduces its scale of operation at date 2 can be either hit by a negative solvency shock or a negative liquidity shock. In this section, I suggest an empirical test to distinguish situations in which the asset price is too low from situations in which the asset price is too high. The logic of the test is the following. As already emphasized, it is arguably difficult for the econometrician to observe the realization of idiosyncratic shocks. It is nevertheless possible to use the market values at date 2 to extract all the information relevant for a welfare analysis. The Tobin’s q of a firm is the sum of the value of all the financial securities issued by the firm, divided by the book value of its assets. Whether the book value of assets is computed

13

from historical costs (at price 1) or at market value (at price pCE ) does not matter since pCE is a constant; I compute the book value using historical costs to allege formulas. At the end of date 2, the Tobin’s q of a firm incurring a shock (˜ v , y˜) is equal to q˜CE =

(1 − β)˜ v f (˜ sCE ) . s˜CE

Besides, by equation (9), the shadow value of pledgeable income is an increasing function of v˜f 0 (˜ sCE ). Then, since f (s)/s and f 0 (s) are both decreasing in s, this implies that q˜CE and ˜ CE vary in the same direction in the cross-section of firms. The empirical counterpart of λ Lemma 3 is therefore: Proposition 4. The competitive asset price is too low if Cov(˜ sCE , q˜CE ) < 0, and too high if Cov(˜ sCE , q˜CE ) > 0. Note that my model is a static one, therefore it cannot distinguish between growth rates and levels. The correlation in Proposition 4 could also be computed using the variation in q˜CE between the end of date 1 and the end of date 2. In practice, if there are firm specific fixed effects, it might be preferable to work with the variation of the Tobin’s q. An important assumption of the test outlined in Proposition 4 is that the growth rate of the firm and the Tobin’s q are measured after the after the size of the firm has completely adjusted to the shock. This implies that variables should be computed at a frequency at which the scale of operation can be adjusted. In practice, this frequency is a function of the magnitude of shocks and of the sluggishness in capital adjustment, which, in turn, might depend on the industry and time period under consideration. Since it may be difficult to work at a frequency at which the scale of operation adjusts, I investigate what happens if the variables are measured before the size has completely adjusted to the shock. This requires to know the dynamic of the adjustment from initial size k CE to final size s˜CE k CE within date 2. The task of modelling this dynamic is outside the scope of the current paper; instead, I make the simple assumption that within date 2 there is a continuum of periods t ∈ [0, 1], and at period t every firm has made a fraction t of the adjustment: s˜(t) = 1 + t(˜ sCE − 1). With that assumption in hands, it is possible to compute the covariance between the Tobin’s q and the resizing factor for any t. To focus on a simple case, I report here the covariance at time t = 0+ , that is, just after the shock has been incorporated in the market value, and once the firm size has slightly begun to adjust. At that time, the market value is equal to the interim cash flow, plus the final payoff to investors, net of the price to adjust the firm size: q˜CE (0+ ) =

y˜ + (1 − β)˜ v f (˜ sCE )k CE − p(˜ sCE − 1)k CE k CE − A = . k CE + y˜ k CE + y˜

The liquidity shock generates a negative correlation between q˜CE (0+ ) and s˜(0+ ), while the solvency shock does not generate any variation in q˜CE (0+ ). Unless there is no heterogeneity in y˜, q˜CE (0+ ) and s˜(0+ ) are negatively correlated in the cross-section of firms. This suggests

14

that measuring the Tobin’s q too early after the shock might bias toward finding a negative correlation. This test is related to the empirical literature on the investment-cash flow sensitivity. Fazzari, Hubbard, Petersen (1988) argue that, after controlling for the Tobin’s q, the remaining correlation between investment and cash flow is a measure of financial constraints. That view is challenged by Kaplan and Zingales (1997). In this paper I take a different perspective. I suggest that the Tobin’s q (or its variation) might be directly used a proxy for financial constraints: a firm with a high q should invest, which would reduce its q because of diminishing returns to scale, but it cannot do so if is credit constrained. In the test outlined in Proposition 4, the current Tobin’s q is then correlated with past investment to determine which side of the asset market is the more constrained.

4 4.1

Extensions Perfect Hedging: Irrelevance of Fire Sales

The key ingredient of the model is that idiosyncratic shocks cannot be insured. To illustrate that point, I show that the competitive equilibrium achieves the second-best allocation if idiosyncratic shocks can be perfectly hedged. Assume that the date 2 repayment can be made contingent on the realization of the shock: D1 (˜ v , y˜). The date 1 break-even condition (4) becomes k − A ≤ E[D1 (˜ v , y˜)]. This implies that the overall break-even condition is now taken in expectation over all the realizations of the shock, and the individual program becomes max k, s˜

s.t.

E [−1 + p + v˜f (˜ s) − p˜ s] k, A + E[˜ y ] + E [−1 + p + (1 − β)˜ v f (˜ s) − p˜ s] k ≥ 0.

The firm is now able to transfer wealth across realizations of the idiosyncratic shock. The optimal financial contract, described by k ∗ (p) and s˜∗ (p), consists in equalizing the shadow price of pledgeable income across states of nature. Indeed, if the shadow price of pledgeable income is higher in state (˜ vH , y˜H ) than in state (˜ vL , y˜L ), then reducing D1 (˜ vH , y˜H ) and increasing D1 (˜ vL , y˜L ) such that the investor breaks even increases the expected NPV by an amount ˜H − λ ˜ L > 0. Therefore, in equilibrium, the firm has the same shadow value proportional to λ of pledgeable income for all the realizations of the shock. This implies that there is no pecuniary externality through the asset price. A modification of the asset price transfers wealth between buyers and sellers, but all of them make an equally efficient use of an additional unit of wealth. Besides, the wedge between NPV and pledgeable income implies that the equilibrium asset price is below the fundamental value. Indeed, denoting by λCE the (unique) shadow price of wealth, the equilibrium asset price is equal to   λCE CE p = 1− β v˜f 0 (˜ s) (˜ v , y˜). 1 + λCE 15

Proposition 5. When idiosyncratic shocks can be perfectly hedged, the competitive equilibrium features fire sales and though it is second-best optimal. Proof. See Appendix A.6. Proposition 5 sheds light on the potential inefficiency informally suggested in the literature, namely that fire sales can result in asset misallocation and reduce debt capacity. It is true that in the competitive equilibrium, some assets may be reallocated from a high-value user to a low-value user. This can be the case when the former faces a negative liquidity shock and the latter faces a positive liquidity shock. However, Proposition 5 makes it clear that it is not the cause of the inefficiency of fire sales. Instead, this reflects a deviation from the first-best allocation in which there is no friction at all, β = 0, to the second-best in which there is wedge between NPV and pledgeable value. Proposition 5 also sheds light on the effect of fire sales on debt capacity. It is true that fire sales reduces proceeds to claimholders in case of liquidation. However, one should not forget that fire sales also make asset acquisitions cheaper. If firms are ex ante identical and idiosyncratic are perfectly hedged, then these two effects exactly offset and fire sales do not affect debt capacity in equilibrium.

4.2

Aggregate Shocks and Fire-Sale FDI

As emphasized by Krugman (1998) after the East Asian crisis, financial crises are often accompanied by an inflow of foreign direct investment. Foreigners take advantage of depressed asset prices to engage in cheap acquisitions. Acharya, Shin and Yorulmazer (2009) emphasize that, during the credit crisis of 2007/8 in the United States and Europe, banks have raised capital at steep discounts and at least 15% of this capital-raising has been from the sovereign wealth funds from Asia. Since sovereign wealth funds have until now been passive investors, their investments in banks, but also in private equity funds, are perhaps also best characterized as fire-sale FDI. My model is naturally extended to the case of an open economy to provide a welfare analysis of fire-sale FDI. In this section, I first extend the model to include insurable aggregate shocks in the closed economy; I then consider the case of an open economy. 4.2.1

Closed Economy: Irrelevance of Insurable Aggregate Shocks

In the closed economy, I show that only the dispersion of idiosyncratic shocks matters, just as in the basic model, while the aggregate component of the shocks does not matter. This is reminiscent to the analysis of Krishnamurthy (2003), who shows that the amplification mechanism of Kiyotaki and Moore (1997) disappears when aggregate shocks can be hedged away by financial contracts. I denote by ω ∈ Ω the aggregate state of nature. In state ω, the idiosyncratic shocks are drawn from the probability distribution function φ(˜ v , y˜|ω). The date 2 repayment can be

16

made contingent on the aggregate state of nature, D1 (ω), as well as the resizing factor s˜(ω). Denoting by p(ω) the asset price in the aggregate state ω, the problem of the entrepreneur is max k, D1 (ω), s˜(ω)

s.t.

Eω [E [−1 + p(ω) + v˜f (˜ s(ω)) − p(ω)˜ s(ω)|ω]] k, k − A ≤ Eω [D1 (ω)], −D1 (ω) + y˜ + [p(ω) + (1 − β)˜ v f (˜ s(ω)) − p(ω)˜ s(ω)] k ≥ 0

∀ (˜ v , y˜; ω).

It is optimal to equalize the average shadow price of pledgeable income across aggregate ˜ H )|ωH ] > E[λ(ω ˜ L )|ωL ], then reducing D1 (ωH ) and states of nature. Indeed, if we had E[λ(ω increasing D1 (ωL ) such that the date 1 investor breaks even strictly increases the expected ˜ NPV. Therefore, in equilibrium, E[λ(ω)|ω] is independent from ω, and ˜ CE (ω) = λ

v˜f 0 (˜ sCE (ω)) − pCE (ω) − (1 − β)˜ v f 0 (˜ sCE (ω))

pCE (ω)

∀ (˜ v , y˜; ω).

The equilibrium price vector pCE = {pCE (ω)}ω∈Ω is uniquely defined by the market clearing condition E[(˜ s∗ (pCE ; ω) − 1)k ∗ (pCE )] = 0 for all ω. As in the basic model, the direction of the pecuniary externality that operates through the state ω asset price is given by the ˜ CE (ω)|ω). Therefore, the generalization of Proposition 1 to the case of sign of Cov(˜ sCE (ω), λ insurable aggregate shocks is: Proposition 6. In the closed economy, when there is insurable aggregate risk and uninsurable idiosyncratic risk, • the competitive asset price is too low in the aggregate states of nature in which idiosyncratic liquidity shocks are prevalent, • the competitive asset price is too high (low) in the aggregate states of nature in which idiosyncratic solvency shocks are prevalent, if returns to scale are not too decreasing (are strongly decreasing). 4.2.2

Open Economy: Underinsurance of Aggregate Shocks and Fire-Sale FDI

Consider now that asset markets are international and foreign firms can buy and sell assets on the secondary market. There are insurable aggregate shocks and the net demand for domestic assets arising from the rest of the world is D(p(ω); ω) ≶ 0 in state of nature ω when the asset price is p(ω). The unique equilibrium is characterized by E[(˜ s∗ (pCE ; ω) − 1)k ∗ (pCE )] = D(pCE (ω); ω) for all ω. The domestic economy as a whole can now be a net seller of assets in some states of nature, and a net buyer of assets in others. A variation of the asset price has therefore a direct effect on the expected NPV7 n    h i o dN P V ∗ CE ˜ CE (ω)|ω k CE + 1 + E λ ˜ CE (ω)|ω D(pCE (ω) P [ω]. (p ) = −Cov s˜CE (ω), λ dp(ω) 7

That equation is not rigorously defined when there is a continuum of ω. To simplify the exposition I write it as if each state ω had a strictly positive probability P [ω].

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The second term is new. In states of nature in which firms are on average sellers of assets, a higher asset price transfers wealth from the rest of the world to the domestic economy. In that case, there exists a Pareto improvement for the domestic economy with a higher asset price. As in the basic model, it is implemented by lowering the (contingent) face value of debt D1 (ω) in state ω below its privately optimal level. Since p(ω) increases, fewer assets are sold abroad. Conversely, it is Pareto improving to increase the debt repayment in states of nature in which the domestic economy is a net buyer of assets. This reduces its date 2 debt capacity and thus the equilibrium asset price, which benefits domestic firms. Proposition 7. When the asset market is international and there is insurable aggregate risk, • there is underinsurance of aggregate shocks, • the competitive asset price is too low (high) when the domestic economy is a net seller (buyer) of assets, • the net volume of transactions with the rest of the world is always too high. Proposition 7 indicates that fire-sale FDI is associated to an inefficiency of the competitive equilibrium. Asset prices are socially too low and the volume of FDI inflows too high during financial crises. Note that my underinsurance result differs from the one in Caballero and Krishnamurthy (2003). In their model, firms do not trade assets but claims on assets. Because of the wedge between the NPV and the pledgeable value, the buyer of a claim on an asset cannot obtain the whole value of the asset and the private valuation of debt capacity is less than its social value. Although the volume of fire-sale FDI is higher than in the second-best allocation, imposing capital control to prevent foreigners from acquiring domestic assets would make a bad situation worse. That would lower the demand for assets and thus reduce the asset price, whereas domestic firms benefits from price increases. Instead, the solution is to reduce the supply of assets by domestic firms by reducing their liabilities in these states of nature. Put differently, the fact that the volume of fire-sale FDI is too high is a consequence of the pecuniary externality, not its cause.

4.3

Endogenous First Period Asset Price

In the basic model, the supply of assets at date 1 is infinitely elastic at price 1. As a robustness check, I now envisage the opposite assumption that the asset is in fixed supply k. I contrast two cases. In the first case, the assets are initially owned by entrepreneurs. In the second case, they are supplied by competitive unconstrained agents, e.g., the investors. 4.3.1

Fixed Internal Supply of Assets

Each entrepreneur owns at the beginning of date 1 k assets. The (endogenous) date 1 asset price is denoted by q.

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First, I argue that the model must be augmented with another productive input, otherwise the competitive equilibrium becomes trivial. Indeed, in equilibrium it must be that k CE = k and sCE = 1, assuming no shock to make the point clear. Since firms already own the asset, they need no cash at date 1, and the date 2 break-even condition becomes A + y + (1 − β)vf (1)k ≥ 0. This condition features constant terms only, since the initial investment is no longer determined endogenously at equilibrium. If this condition is satisfied, then financial constraints are not binding and the competitive equilibrium achieves the first-best allocation. If this condition is not met, then firms have not enough cash to operate at all. This second possibility arises when firms are born with liabilities or experience negative interim cash flows: A+y < 0. The fundamental reason why the basic model is not well-behaved when firms initially own all the capital is that the equilibrium investment cost is zero. To avoid this undesirable feature of a zero equilibrium investment cost, I introduce another input that I call labor. At date 2, firms buy ˜lk units of labor at constant unit cost. The date 3 payoff is v˜(f (˜ s) + g(˜l))k, where g(0) = 0, g 0 > 0, g 00 < 0, g 0 (0) = +∞, g 0 (+∞) = 0, and a fraction 1 − β of that payoff is pledgeable to the investor. To simplify the formulas, I specify f (s) = sα and g(l) = lα , α ∈ (0, 1). I show that all the results derived with an infinitely elastic asset supply extend to that situation. The pecuniary externality that operates through the date 2 asset price works in the same way as before. Besides, there is no pecuniary externality that operates through the date 1 asset price, since the traded volume is zero at equilibrium. Proposition 8. Assume that there is fixed supply of assets owned by entrepreneurs and that there is another factor of production. • When liquidity shocks are prevalent, the competitive date 2 asset price is too low. • When solvency shocks are prevalent, the competitive date 2 asset price is too high if α > (1 − β)v, and too low if α < (1 − β)v. • For any feasible allocation (q, p, k, s˜), all the allocations (q 0 , p, k, s˜) with q 0 6= q are feasible and achieve the same level of welfare. Proof. See Appendix A.7. The way the Pareto improvements are implemented is different from the basic model. A Pareto improvement can no longer modify the tradeoff between k and s˜ to correct the externality, since k is now fixed at k. Instead, the adjustment is made through the tradeoff between s˜ and ˜l. For instance, if the planner aims at increasing the date 2 asset price, she shifts the date 2 demand for labor below its privately optimal level and the date 2 demand for assets above its privately optimal level.

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4.3.2

Fixed External Supply of Assets

I now consider that assets are supplied at date 1 by competitive unconstrained agents. I still assume that there exists another input ˜l. In that case, a pecuniary externality through the date 1 asset price arises. Indeed, a lower price transfers wealth from the unconstrained date 1 asset suppliers to the constrained entrepreneurs. This relaxes financial constraints, and the expected NPV of the projects increases with the transfer to entrepreneurs by more than one-for-one, hence utilitarian welfare improves. This pecuniary externality through the date 1 asset price can be interpreted as a destructive effect of competition. By competing for the scarce date 1 assets, firms engage in a costly arm race that consumes their pledgeable income. However, I argue that this destructive effect of competition is debatable for at least two reasons. First, correcting the date 1 pecuniary externality cannot result in a Pareto improvement. Lowering the date 1 asset price benefits to entrepreneurs, but hurt the asset suppliers. Second, and perhaps more fundamentally, the addition of unconstrained asset suppliers introduces an asymmetry in the model between buyers and sellers that makes the direction of the date 1 pecuniary externality somehow hardwired. Indeed, assuming exogenously that the buy-side of the date 1 asset market is constrained while the sell-side is not, it is not surprising that the price is too high. The internal consistency of the model would require to model these date 1 asset suppliers as constrained agents, and to endogeneize their shadow value of pledgeable income. That task is left for future research.

5

Concluding Remarks

I have developed an equilibrium model of fire sales, and show that the fire-sale price is socially to low when idiosyncratic shocks are mainly liquidity shocks, and too high when idiosyncratic shocks are mainly solvency shocks. To conclude this paper, I argue that my model can support several interpretations. The most natural one is that the assets of the model are plants and machinery, or real estate, and the entrepreneurs stand for firms in the corporate sector. v˜ should then be understood as idiosyncratic productivity shocks. For instance, Maksimovic and Phillips (2001) provide evidence that on the market for corporate assets, buyers tend to be more efficient than sellers. The entrepreneurs can also be commercial banks, and assets their loan portfolios, as in Acharya and Yorulmazer (2008). With this interpretation, the liquidity shock y˜ can come from their revenues in their other activities, such as investment banking. v˜ is the bank’s ability to monitor and restructure existing loans. Finally, the asset can be a financial security and the entrepreneurs money managers, as in Shleifer and Vishny (1997). In that case, y˜ are the profits and loss made on other asset markets. v˜ can stand for executions costs; it can also come from a (negative) correlation with other non insurable revenues of the manager or of its investors, as in Gromb and Vayanos (2002).

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References Acharya, Viral, Sreedhar Bharath and Anand Srinivasan, 2007, Does industry-wide distress affect defaulted firms? Evidence from creditor recoveries, Journal of Financial Economics 85, 787–821. Acharya, Viral, Hyun-Song Shin, and Tanju Yorulmazer, 2009, Fire-sale FDI, Working Paper. Acharya, Viral, and Tanju Yorulmazer, 2008, Cash-in-the-market pricing and optimal resolution of bank failures, Review of Financial Studies 21, 2705–2742. Aguiar, Mark, and Gita Gopinath, 2005, Fire-sale foreign direct investment and liquidity crises, The Review of Economics and Statistics 87, 439–452. Caballero, Ricardo, and Arvind Krishnamurthy, 2003, Excessive dollar debt: Financial development and underinsurance, Journal of Finance 58, 867–893. Coval, Joshua, and Erik Stafford, 2007, Asset fire sales (and purchases) in equity markets, Journal of Financial Economics 86, 479–512. Eisfeldt, Andrea, and Adriano Rampini, 2006, Capital reallocation and liquidity, Journal of Monetary Economics 53, 369–399. Fazzari, Steven, Glenn Hubbard, and Bruce Petersen, 1988, Financing constraints and corporate investment, Brookings Papers on Economic Activity 1988, 141–206. Gromb, Denis, and Dimitri Vayanos, 2002, Equilibrium and welfare in markets with financially constrained arbitrageurs, Journal of Financial Economics 66, 361–407. Holmstr¨om, Bengt, and Jean Tirole, 2008, Inside and outside liquidity, Wicksell Lectures. Kaplan, Steven, and Luigi Zingales, 1997, Do investment-cash flow sensitivities provide useful measures of financing constraints?, Quarterly Journal of Economics 112, 169–215. Kiyotaki, Nobuhiro, and John Moore, 1997, Credit Cycles, Journal of Political Economy 105, 211–248. Krishnamurthy, Arvind, 2003, Collateral constraints and the amplification mechanism, Journal of Economic Theory 111, 277–292. Krugman, Paul, 1998, Fire-sale FDI, http://web.mit.edu/krugman/www/FIRESALE.htm. Lorenzoni, Guido, 2008, Inefficient credit booms, Review of Economic Studies 75, 809–833. Maksimovic, Vojislav, and Gordon Phillips, 2001, The market for corporate assets: Who engages in mergers and asset sales and are they efficiency gains?, Journal of Finance 56, 2019–2065. Perotti, Enrico, and Javier Suarez, 2002, Last bank standing: What do I gain if you fail?, European Economic Review 46, 1599–1622. Pulvino, Todd, 1998, Do fire sales exist? An empirical investigation of commercial aircraft

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transactions, Journal of Finance 53, 939–978. Shleifer, Andrei, and Robert Vishny, 1992, Liquidation values and debt capacity: A market equilibrium approach, Journal of Finance 47, 1343–1366. Shleifer, Andrei, and Robert Vishny, 1997, The limits of arbitrage, Journal of Finance 52, 35–55. Tirole, Jean, 2006, The Theory of Corporate Finance, Princeton University Press.

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A A.1

Appendix Proof of Lemma 1

First, the pledgeable income is strictly decreasing in k when h is small and p is close to its equilibrium value. Indeed, when h goes to 0 and p goes to its equilibrium value, s˜ goes (uniformly since the support of the shock is bounded) to 1. (2) then implies that the term in bracket in the break-even condition (8) is strictly negative. A similar argument using (1) shows that the NPV is strictly increasing in k. Therefore, the break-even condition is binding, which defines s˜ = sIR (˜ εv , ε˜y , k; p, h). The pledgeable income is strictly decreasing in s˜ when s˜ is equal to its optimal value, otherwise increasing s˜ would increase the NPV and relax the break-even condition. Since the pledgeable is also strictly decreasing in k, then sIR (˜ εv , ε˜y , k; p, h) is strictly decreasing in k. The program rewrites as the maximization of a function of k only, with parameters h and p,   max V (k; p, h) ≡ E −1 + p + (v + h˜ εv )f (sIR (˜ εv , ε˜y , k; p, h)) − psIR (˜ εv , ε˜y , k; p, h) k. k

V is continuously differentiable, and strictly concave in k since sIR (˜ εv , ε˜y , k; p, h) is strictly decreas∗ ing in k. Therefore, it has a unique maximum kh (p) characterized by the first order condition ∂V /∂k(kh∗ (p); p, h) = 0. kh∗ (p) is continuously differentiable in p and h by the implicit function theorem. εv , ε˜y , kh∗ (p); p, h), where, as in the text, I omit the The optimal resizing policy is then s˜∗h (p) = sIR (˜ realization of the shock as an argument of s˜∗h . Since sIR (., ., .; ., .) is continuously differentiable in all the variables and k.∗ (.) is continuously differentiable in p and h, then s˜∗. (.) is continuously differentiable in p, h and (˜ εv , ε˜y ). Besides, (˜ εv , ε˜y ) has a bounded support, therefore ((˜ εv , ε˜y ) 7→ s˜∗. (.)) is continuous in p and h with respect to ||.||∞ . In the text I omit the subscript h in k ∗ and s˜∗ to allege notations.

A.2

Proof of Lemma 2

The first order condition that defines k ∗ (p) is     A + y˜ (v + h˜ εv )f 0 (˜ sIR ) − p = 0, E −1 + p + (v + h˜ εv )f (˜ sIR ) − p˜ sIR − E k p − (1 − β)(v + h˜ εv )f 0 (˜ sIR ) where s˜IR = sIR (˜ εv , ε˜y , k; p, h) is defined in the proof of Lemma 1. Differentiating with respect to p in the neighborhood of an equilibrium, i.e., E[˜ s ≈ 1], shows that k ∗ (.) is increasing in p and that s˜∗ (.) is decreasing in p. Therefore the equilibrium condition (11) defines a unique equilibrium asset price. The existence is ensured by the intermediate value theorem, since E[˜ s∗ (p)] is continuous in p ∗ (because s˜ (.) is continuous with respect to ||.||∞ ), strictly larger than 1 if p < (1 − β)(v − hσv εv )f 0 (1) and strictly smaller than 1 if p > (v + hσv εv )f 0 (1), where [εv , εv ] is the bounded support of ε˜v . s) for all (˜ v , y˜) is a consequence of (1), which implies that the NPV is The property pCE < v˜f 0 (˜ decreasing in k as shown in the proof of Lemma 1.

A.3

Proof of Proposition 1

Differentiating (8) with respect to y˜ and v˜ gives ∂˜ sCE ∂ y˜ ∂˜ sCE ∂˜ v

= =

1/k CE > 0, − (1 − β)˜ v f 0 (˜ sCE ) (1 − β)f (˜ sCE ) > 0. CE p − (1 − β)˜ v f 0 (˜ sCE ) pCE

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Differentiating (9) and using the above formulas, one obtains ˜ CE ∂λ ∂ y˜ ˜ ∂ λCE ∂˜ v

= =

−

pCE β˜ v |f 00 (˜ sCE )|/k CE

3 < 0, (pCE − (1 − β)˜ v f 0 (˜ sCE ))  0 CE
 pCE β s ) pCE − (1 − β)˜ v f 0 (˜ sCE ) − (1 − β)˜ v f (˜ sCE )|f 00 (˜ sCE )| , 3 f (˜ CE 0 CE (p − (1 − β)˜ v f (˜ s ))

which can be either positive or negative. When liquidity shocks are prevalent, i.e., for σv /σy not too large, Cov(˜ sCE , λCE ) < 0, therefore an infinitesimal price increase improves welfare. ˜ CE /∂˜ When solvency shocks are prevalent, we need to determine the sign of ∂ λ v . When h goes to CE CE 0, s˜ goes (uniformly since (˜ εv , ε˜y ) has a bounded support) to p and k are given by the first order condition A+y vf 0 (1) − pCE −1 + vf (1) − CE CE =0 k p − (1 − β)vf 0 (1) and the break-even condition A + y + [−1 + (1 − β)vf (1)] k CE = 0, which gives pCE → same sign as

f 0 (1) f (1)

˜ CE /∂˜ and k CE → (A + y)/(1 − (1 − β)vf (1)), and shows that ∂ λ v has the f 0 (1)2 (1 − β)vf (1) − . f (1)|f 00 (1)| 1 − (1 − β)vf (1)

This expression is strictly positive when returns to scale are not too decreasing, or, substituting ˜ CE /∂˜ f (s) = sα , when α > (1 − β)v. In that case, ∂ λ v > 0 for h not too large, implying that CE CE Cov(˜ s , λ ) < 0 and that an infinitesimal price reduction improves welfare. The reverse holds when α < (1 − β)v. It remains to show that the competitive equilibrium is second-best optimal when h = 0. Indeed, dN P V ∗ /dp = 0 only indicates that the competitive equilibrium is a local optimum, and I want to show that it is a global maximum. Consider by contradiction a Pareto improvement (p, k, s), where s is deterministic since there is no shock. The market clearing condition implies s = 1. Then, by the break-even condition, k cannot be larger than k CE , hence the NPV cannot be larger that in the competitive equilibrium.

A.4

Proof of Proposition 2

When f (s) = sα and in the limit of h → 0, equation (13) becomes β dN P V ∗ CE (p ) = 2 dp α (1 − (1 − β)v)2   (1 − α)v 2 (1 − β)((1 − β)v − α) 2 2(1 − β)v − α(1 + (1 − β)v) σ + σv + ρσy σv . (A + y)2 y (1 − (1 − β)v)2 (A + y)(1 − (1 − β)v) Therefore, the competitive asset price is too low if α < (1 − β)v. If α > (1 − β)v, the asset price is too high if the ratio σv /σy is above some threshold r. Some straightforward algebra shows that r is decreasing in α, and decreasing in ρ if α > 2(1 − β)v/(1 + (1 − β)v), which is between (1 − β)v and 1, and increasing in ρ otherwise.

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A.5

Proof of Proposition 3

The first step consists in making an infinitesimal price variation dp1 and letting the firms react optimally: dk1 = (∂k ∗ /∂p)dp1 and d˜ s2 = (∂˜ sIR /∂p)dp1 +(∂˜ sIR /∂k)dk1 . The second step consists in variations dk2 and d˜ s2 along the break-even condition in order to clear the asset market: d˜ s2 = (∂˜ sIR /∂k)dk2 such that E[d˜ s1 + d˜ s2 ] = 0, or  IR  ∂˜ s ∂˜ sIR E dp1 + (dk1 + dk2 ) = 0. ∂p ∂k Besides, ∂˜ sIR /∂p = (1 − s˜)/(p − (1 − β)˜ v f 0 (˜ s)) and ∂˜ sIR /∂k = −(A + y˜)/(k 2 (p − (1 − β)˜ v f 0 (˜ s))), IR IR therefore at equilibrium E[∂˜ s /∂p] is of an order of magnitude smaller than E[∂˜ s /∂k] since s˜ ≈ 1 when h is small, and dp1 is of an order of magnitude larger than dk1 + dk2 ... This implies that d˜ s1 + d˜ s2 has the sign of ∂˜ sIR /∂pdp1 . When the competitive asset price is too low, the Pareto improvement features dp1 > 0, and d˜ s1 + d˜ s2 < 0 if s˜ > 1. In that case, the Pareto improvement reduces the proportion of assets that are reallocated. When the competitive asset price is too high, the Pareto improvement features dp1 < 0 and the proportion of reallocated assets increases.

A.6

Proof of Proposition 5

The optimal k ∗ (p) and s˜∗ (p) is characterized by the first order conditions with respect to k     λ E −1 + p + 1 − β v˜f (˜ s) − p˜ s , 1+λ and the s˜

 1−β

λ 1+λ



v˜f 0 (˜ s) − p = 0,

and the break-even condition A + E[˜ y ] + E [−1 + p + (1 − β)˜ v f (˜ s) − p˜ s] k = 0. A competitive equilibrium is such that E[˜ s∗ (pCE )] = 1. Proceeding as in the imperfect hedging case, it ∗ is not difficult to show that E[˜ s (p)] is decreasing in p when E[˜ s∗ (p)] = 1, which ensures the uniqueness of the competitive equilibrium. I now establish that the program of the planner is equivalent to the program of an individual entrepreneur. The problem of the planner is max

p,k, s˜

s.t.

E [−1 + p + v˜f (˜ s) − p˜ s] k, A + E[˜ y ] + E [−1 + p + (1 − β)˜ v f (˜ s) − p˜ s] k ≥ 0, E[˜ s] = 1,

or, equivalently, max k, s˜

s.t.

  E −1 + pCE + v˜f (˜ s) − pCE s˜ k,   A + E[˜ y ] + E −1 + pCE + (1 − β)˜ v f (˜ s) − pCE s˜ k ≥ 0, E[˜ s] = 1,

since E[˜ s] = 1. This is the same program as for an individual entrepreneur, with the additional constraint E[˜ s] = 1. However, by definition of the equilibrium, the solution to the program without that constraint actually satisfies it, which concludes the proof.

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A.7

Proof of Proposition 8

At date 2, when initial investment was k and the shock is (˜ v , y˜), the firm maximizes h i v˜f (˜ s) − p˜ s + v˜g(˜l) − ˜l k in s˜ and ˜l, subject to the break-even condition h i v f (˜ s) − p˜ s + v˜g(˜l) − ˜l k ≥ 0 A + y˜ + qk + −q + p + (1 − β)˜

∀ (˜ v , y˜).

Similarly to the derivation of equation (9), the first order conditions show that v˜f 0 (˜ s) − p v˜g 0 (˜l) − 1 = , p − (1 − β)˜ v f 0 (˜ s) 1 − (1 − β)˜ v g 0 (˜l) which is also equal to the shadow price of the break-even condition. Using a same power function for f (.) and g(.), one obtains ˜l = p1/(1−α) s˜. Therefore, the date 1 program writes as  α
 v s˜α − p˜ s) 1 + p 1−α k, max qk + E −q + p + (˜ k

subject to  α
 v s˜α − p˜ s) 1 + p 1−α k ≥ 0 A + y˜ + qk + −q + p + ((1 − β)˜

∀ (˜ v , y˜).

Similarly to the basic model, the first order condition is    α
 A + y˜ + qk v˜α˜ sα−1 − p = 0, E −q + p + (˜ v s˜α − p˜ s) 1 + p 1−α − E k p − (1 − β)˜ v α˜ sα−1 where s˜ is given by the date 2 break-even condition. I denote k ∗ (q, p) and s˜∗ (q, p) the privately optimal leverage and resizing strategy as function of date 1 and date 2 asset prices. A competitive equilibrium (q CE , pCE , k CE , s˜CE ) is such that individual behaviors are privately optimal and the asset market clears at date 1 and date 2 k CE = k ∗ (q CE , pCE ), s˜CE = s˜∗ (q CE , pCE ), k CE = k, E[˜ sCE ] = 1. Starting from the competitive equilibrium, a Pareto improvement that modifies the date 2 asset price can be implemented in a similar fashion as in the basic model. The total derivative of the NPV with respect to the date 2 asset price as the same sign as   v˜f 0 (˜ s) − p −Cov s˜, , p − (1 − β)˜ v f 0 (˜ s) which is strictly positive when liquidity shocks are prevalent, and strictly negative when solvency shock are prevalent and α > (1 − β)v. Suppose, for instance, that it is strictly positive so that the planner aims at increasing the date 2 asset price. In a first step, the planner increases p and let the individual behaviors adjust their s˜ and ˜l (but not k since it must stay equal to k) in a privately optimal fashion; the expected NPV increases at the first order. In the second step, the planner increases the s˜ and reduces the ˜l along the break-even conditions, so that E[˜ s] goes back to 1; the expected NPV is not affected by that operation at the first order. Finally, since the expected NPV and the break-even conditions do not depend on q when the asset market clears at date 1, any feasible allocation (q, p, k, s˜), every allocation (q 0 , p, k, s˜) with q 0 6= q is feasible and achieves the same level of welfare.

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