NBER WORKING PAPER SERIES

HOW I LEARNED TO STOP WORRYING AND LOVE FIRE SALES

Pablo Kurlat

Working Paper 24752http://www.nber.org/papers/w24752

NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue

Cambridge, MA 02138June 2018

I am grateful to Adrien Auclert, Alex Bloedel, Sebastian Di Tella, Peter Kondor and Arvind Krishnamurthy for helpful comments. The views expressed herein are those of the author and do not necessarily reflect the views of the National Bureau of Economic Research.

NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications.

© 2018 by Pablo Kurlat. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source.

How I Learned to Stop Worrying and Love Fire SalesPablo KurlatNBER Working Paper No. 24752June 2018JEL No. D62,D82,G14

ABSTRACT

In canonical models with financial constraints, the possibility of fire sales creates a pecuniary externality that results in ex-ante overinvestment. I show that this result is sensitive to the microfoundations for fire sales. If they result from asymmetric information instead of misallocation, the overinvestment result is reversed. Macroprudential policy may therefore need to treat different types of investment differently.

Pablo KurlatDepartment of EconomicsStanford University579 Serra MallStanford, CA 94305and NBERpkurlat@stanford.edu

Fire sales, where distressed sales depress asset prices, are a recurring feature of financialcrises, and preventing or mitigating them is a central concern of macro-prudential policy.One common policy recommendation is to preventatively curb investment booms in order toreduce the scale of fire sales during crises. Lorenzoni (2008) formalizes one possible rationalefor such policies: investors don’t internalize the pecuniary externality that falling asset pricesimpose on financially constrained agents. In this paper I show that this argument dependson the exact mechanism that makes asset prices fall. It is valid when fire sales involvemisallocation of real assets; if instead they result from asymmetric information, a socialplanner would like to increase rather than reduce investment.

It seems intuitive that prices should fall when constrained agents need to sell assets, butit requires some explanation. Assets are just claims on future cash flows. If asset sales donot alter either cash flows or the discount factor of the marginal investor, there is no reasonfor asset prices to fall. In other words, one needs a theory of what makes asset demanddownward-sloping as opposed to perfectly elastic.

A standard way to microfound downward-sloping demand curves for assets is the misallo-cation mechanism proposed by Shleifer and Vishny (1992) and Kiyotaki and Moore (1997),and adopted by Lorenzoni (2008). Assets are assumed to have different productivity de-pending on who holds them. They have a high marginal product in expert hands but adiminishing marginal product in non-expert hands. If constrained experts sell marginalunits to non-experts, this lowers the marginal product in second-best use, and therefore theequilibrium price. This mechanism is an appealing microfoundation for applications wherethe assets in question are real assets that must be actively managed by whoever owns them.For assets that are not actively managed by the owner, such as securities backed by pools ofmortgages which have a designated servicer, the fit is less clear. To a first approximation,the cash flows from these assets are the same regardless of who owns them.1

Empirically, fire sale effects have been documented in several different markets. Theseinclude real assets like used aircraft (Pulvino 1998) and real estate (Campbell et al. 2011)where misallocation seems like a first-order concern and financial assets like equities (Covaland Stafford 2007), corporate bonds (Ellul et al. 2011), convertible bonds (Mitchell et al.2007) and residential mortgage-backed securities (Merrill et al. 2014) where it seems lessessential.

In Kurlat (2016), I propose an alternative model of fire sales that does not rely on cash1Although differences across investors in tax rates, in the correlation between a given asset and the rest

of their portfolio or in beliefs about the asset’s cash flows might all be sources of differences in valuationacross investors even for assets that do not require management.

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flows changing depending on who owns the asset. Instead, I assume that there is asymmetricinformation and potential buyers differ in their expertise in evaluating assets: more expertbuyers can detect bad assets more accurately and refrain from buying them. This setupresults in an equilibrium where only investors with sufficiently high expertise buy assets.The cash flows obtained by the marginal buyer depend on his expertise because expertisedetermines the composition of the pool of assets he purchases. Under certain conditions onthe joint distribution of buyers’ wealth and expertise, the equilibrium features a downward-sloping relationship between asset sales and asset prices: in order to clear the market whensales are high, less-expert investors need to be induced to buy so the price needs to fall.

I embed this model of fire sales into a simplified version of Lorenzoni’s three-period modeland ask the same normative questions that have been asked of this canonical framework.Would a social planner that can intervene ex-ante but not ex-post want to deviate fromthe level of investment in the competitive equilibrium? In what way? I find that in theasymmetric information model, the normative conclusions are reversed: the social plannerwants higher investment relative to the competitive equilibrium, even if this will deepen thefall in asset prices.

The reason for this reversal is that the two models imply very different types of inefficien-cies. In the canonical model, the inefficiency comes from what Dávila and Korinek (2017)label a distributive externality. Incomplete markets prevent constrained investors from sav-ing towards the state of the world where they have high marginal utility of wealth, andraising asset prices partially substitutes for this. Since the net present value of investment iszero (given how it will be misallocated in equilibrium) there is no first-order loss in reducingit relative to the competitive equilibrium in order to raise asset prices. In the model withasymmetric information, there are two important differences. First, redistributing wealth to-wards constrained investors to mitigate fire sales has no social benefit. Privately, they have ahigh marginal utility of wealth because it allows them to retain assets instead of selling themat an adverse selection discount, but since no real output is destroyed in a fire sale, this isjust a transfer. Therefore the force for preventing fire sales in the canonical model is absent.Second, the social net present value of marginal investment in the competitive equilibriumis positive. What stops agents from choosing higher investment is the understanding thatthey may have to sell it at an adverse selection discount. But since this discount is just atransfer, the social planner would like investors to ignore it and choose higher investment.

I then extend the basic framework to allow for investment in the intermediate period inaddition to the initial period. By lowering asset prices, fire sales raise the rate of return on

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experts’ wealth, leading to inefficiently lower intermediate-period investment. In the canoni-cal model, this just reinforces the argument for lowering initial investment: it prevents bothmisallocation and future underinvestment. In the asymmetric information model, allowingfor intermediate-period investment introduces a tradeoff: the social planner must balancepresent and future underinvestment. The direction in which the planner wants to inter-vene depends on the strength of fire sale effects and the sensitivity of intermediate-periodinvestment to asset prices.

Finally, I examine what happens in the asymmetric information model if bad assetsare deliberately created with the intent of being sold, and their quantity is endogenous. Ifind conditions such that this reinforces the argument for increasing ex-ante investment: ifadditional investment leads to lower asset prices, it also reduces incentives to waste resourcesin creating bad assets.

1 The Canonical Model

The model is a simplified version of Lorenzoni (2008), similar to the example studied byMoore (2013). Lorenzoni’s original model features aggregate uncertainty, with fire sales onlytaking place in bad states of the world. While this feature is appealing on empirical grounds,it is not essential for the main argument, so, like Caballero and Krishnamurthy (2004), Ifocus on a deterministic economy.

Technology, Preferences and Markets

There are three periods: t = 0, 1, 2; two groups of agents: households and entrepreneurs; andtwo goods: consumption goods and capital. Preferences are given by u = c0 +c1 +c2 for bothgroups of agents. Entrepreneurs are endowed with n consumption goods at t = 0; householdsare endowed with e0 and e1 consumption goods in periods t = 0 and t = 1 respectively.

Consumption goods are perishable and not storable. However, entrepreneurs have accessto a technology that converts consumption goods into capital one-for-one at t = 0. Capitalrequires maintenance at t = 1 in order to produce output at t = 2. Maintenance costs are zconsumption goods per unit of capital. If a unit of capital’s maintenance costs are not paid,then that unit of capital becomes useless. At t = 2, a maintained unit of capital will produceA consumption goods if it is operated by entrepreneurs. Instead, if households operate kH

units of capital, they will produce F(kH)consumption goods, where F (·) is a differentiable

concave function.

4

Overall, the economy’s resource constraints are:

cE0 + cH0 + k ≤ n+ e0 (1)

cE1 + cH1 + zαk ≤ e1 (2)

kE + kH ≤ αk (3)

cE2 + cH2 ≤ AkE + F(kH)

(4)

Equation (1) is the t = 0 resource constraint. The total endowment of consumption goodsn+e0 has to be split up between consumption of entrepreneurs cE0 , consumption of householdscH0 and investment k. Equation (2) is the t = 1 resource constraint: the endowment is splitbetween consumption and maintenance costs; α ∈ [0, 1] is the fraction of capital for whichmaintenance costs are indeed incurred. Equation (3) says that maintained capital is allocatedfor use by either households or entrepreneurs. Equation (4) is the t = 2 resource constraint.

No inter-temporal contracts are enforceable, so no agents can borrow or lend. It isalso impossible to enforce contracts whereby entrepreneurs manage capital on behalf ofhouseholds: each unit of capital will be managed by whoever owns its. There is, however,a competitive spot market for for capital (after maintenance) at t = 1. Denote the price ofcapital in this market by q.

Equilibrium Definition

The entrepreneurs solve the following problem:

maxcE0 ,c

E1 ,c

E2 ,k,s,k

E ,αcE0 + cE1 + cE2 (5)

s.t.

cE0 + k ≤ n (6)

cE1 + αzk ≤ sq (7)

kE ≤ αk − s (8)

cE2 ≤ AkE (9)

cEt ≥ 0 t = 0, 1, 2; k ≥ 0; kE ≥ 0; α ∈ [0, 1] ; s ∈ [0, αk]

Equation (6) is the t = 0 budget constraint: entrepreneurs simply face a choice betweenconsumption and investment. Equation (7) is the t = 1 budget constraint. They raise sq

5

consumption goods by selling s units of capital, and divide this between consumption andmaintaining a fraction α of their capital.2 Equation (8) says that the capital they can carryover into t = 2 is the part of their original capital that they have chosen to maintain minusthe amount that they have sold. Equation (9) says that at t = 2 they just consume.

Households solve the following problem:

maxcH0 ,c

H1 ,c

H2 ,k

HcH0 + cH1 + cH2 (10)

s.t.

cH0 ≤ e0 (11)

cH1 + qkH ≤ e1 (12)

cH2 ≤ F(kH)

(13)

cHt ≥ 0 t = 0, 1, 2; kH ≥ 0

Equations (11)-(13) are the budget constraints the household faces in periods 0, 1, and 2respectively. At t = 0 they just consume because they have no investment technology oraccess to financial markets. At t = 1 they decide how much to consume and how muchcapital to buy. At t = 2 they just consume.

Definition 1. A competitive equilibrium is an allocationcE0 , c

E1 , c

E2 , c

H0 , c

H1 , c

H2 , k, s, k

E, kH , α

and a price q such thatcE0 , c

E1 , c

E2 , k, s, k

E, αsolves the entrepreneur’s problem (5), taking

q as given,cH0 , c

H1 , c

H2 , k

Hsolves the household’s problem (10), taking q as given, and the

capital market clears, so (3) holds with equality.

Equilibrium Characterization

Assume that the following conditions hold:

Assumption 1.

1. A > 1 + z

2. F ′ (0) < A

3. F ′ (0) > AzA−1

2There is no intra-period financial constraint saying that entrepreneurs must obtain goods to meet main-tenance costs before being able to sell capital. This is equivalent to assuming that capital can be sold withoutmaintenance and maintained by the buyer.

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4. −F ′′(x)xF ′(x)

< 1 for all x

5. F ′(A−1An)< Az

A−1

6. e1 > zn

Assumption 1.1 just says that investing in capital (and then maintaining it) has positivenet present value. Assumption 1.2 implies that capital is always more productive in thehands of entrepreneurs. Assumption 1.3, on the other hand, says that households are notso unproductive that the price of capital can fall to the point where no investment takesplace (Assumption 1.1 ensures that the interval

[A, Az

A−1

]is nonempty so Assumptions 1.2

and 1.3 are not inconsistent). Assumption 1.4 ensures that there is a unique market-clearingprice of capital. Assumption 1.5 ensures that entrepreneurs’ endowment n is large enoughthat in equilibrium they do not invest all of it and consume at t = 0. Assumption 1.6 saysthat households’ endowment at t = 1 is sufficiently large to meet maintenance costs for themaximum possible level of investment.

Suppose the entrepreneur enters period t = 1 holding k units of capital, and the marketprice is q. His problem reduces to

V (k, q) ≡ maxcE1 ,c

E2 ,s,k

E ,αcE1 + cE2

s.t.

cE1 + αzk ≤ sq

kE ≤ αk − s

cE2 ≤ AkE

α ∈ [0, 1] ; cEt ≥ 0 t = 1, 2; kE ≥ 0; s ∈ [0, αk]

(14)

As shown below, Assumption 1 ensures that in equilibrium A > q > z. q > z means it’sbetter to maintain capital and sell it than to not maintain it, so constraint α ≤ 1 willbind. A > q means that entrepreneurs prefer to buy (or avoid selling) capital rather thanconsuming at t = 1, so cE1 ≥ 0 will bind, and kE ≥ 0 will not. The entrepreneur will incurthe maintenance cost on all his capital, will not consume at t = 1 and will carry as muchcapital as possible into t = 2. The solution to his problem is:

s (q, k) =zk

q(15)

so the entrepreneur sells just enough capital to meet maintenance costs, and can afford

7

kE = q−zqk. This gives the entrepreneur

V (k, q) =A

q︸︷︷︸rate of return on wealth

(q − z) k︸ ︷︷ ︸wealth at t=1

(16)

At t = 0, the entrepreneur’s problem is

maxcE0 ,k

cE0 + V (k, q)

s.t.

cE0 + k ≤ n

k ≥ 0; cE0 ≥ 0

The solution to this problem is

k

= 0 if q < Az

A−1

∈ [0, n] if q = AzA−1

= n if q > AzA−1

(17)

Turn now to the household’s problem. Assumption 1.6 ensures that e1 is large enough sothat cH1 ≥ 0 doesn’t bind, so households’ first order condition for choosing kH is

F ′(kH)

= q

Therefore, the market-clearing condition s = kH implies that in equilibrium the price ofcapital must satisfy:

F ′(zk

q

)= q (18)

For a general F (·) function, it’s possible for (18) to have more than one solution: a lowerprice makes entrepreneurs have to sell more units of capital to meet maintenance costs, whichlowers the marginal product obtained by households, justifying the lower price. Assumption1.4 (which is satisfied, for instance, with a Cobb-Douglas production function) ensures thatthis is not the case.

Lemma 1. If equation (18) has a solution, it is unique. The solution q (k) is a decreasingfunction.

Lemma 1 means that this model has “fire sale” effects. Higher t = 0 investment means

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that entrepreneurs will have to liquidate more capital at t = 1, so households will haveto absorb more units of capital, pushing down the marginal-product-in-second-best-use andtherefore asset prices.

Assumption 1.3 implies that for sufficiently low kH , F ′(kH)> Az

A−1. Using (17), this

implies that entrepreneurs will choose positive investment in equilibrium. On the otherhand, Assumption 1.5 implies that if k = n, then F ′

(kH)< Az

A−1; using (17), this would

imply that entrepreneurs choose zero investment, a contradiction. Therefore, in equilibriumthe entrepreneurs must choose an interior solution for the level of investment. Using (17),this requires:

q =Az

A− 1(19)

which leaves entrepreneurs exactly indifferent between investing and consuming at t = 0.Replacing (19) into (18):

Az

A− 1= F ′

(A− 1

Ak

)(20)

Equation (20) implicitly defines the level of investment k that entrepreneurs will choose inequilibrium. The rest of the equilibrium objects follow immediately.

Welfare

In line with the literature, I adopt the normative criterion of constrained efficiency, firstproposed by Hart (1975), Stiglitz (1982) and Geanakoplos and Polemarchakis (1986) in thecontext of incomplete-market models, and applied by Kehoe and Levine (1993) to economieswith financial constraints. Consider a constrained social planner that wants to maximizeentrepreneurs’ utility subject to delivering a minimum level of utility to households. Theplanner can dictate the level of investment that entrepreneurs carry out and can maketransfers across agents at t = 0, but cannot intervene in any way at time t = 1 or t = 2.Hence the planner is limited to choosing cE0 , cH0 and k.

In general it is not clear whether this type of normative exercise provides useful guidancefor the analysis of optimal policy with a limited set of instruments. However, as pointed outby Dávila and Korinek (2017), this particular planning problem is equivalent to a Ramseypolicy problem with a tax on investment and lump-sum redistribution, both applied at t = 0.

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The planner solves:

maxcH0 ,c

E0 ,kcE0 + V (k, q (k))

s.t.

cE0 + cH0 + k ≤ n+ e0

cH0 + e1 − q (k) s (q (k) , k) + F (s (q (k) , k)) ≥ Π

cH0 ≥ 0; cE0 ≥ 0; k ≥ 0

(21)

Unlike individual entrepreneurs, the planner takes into account how the choice of k willdetermine asset prices at t = 1. The first constraint is just the t = 0 resource constraint. Thesecond constraint imposes a minimum level of utility for households. Households consumecH0 at t = 0; at t = 1 they consume their endowment e1 minus what they spend buyings (k, q (k)) units of capital at a unit price q (k); at t = 2 they consume the output of thecapital they purchased. Replacing the constraints into the objective, the planner’s problemreduces to:

maxkW (k) ≡ n+ e0 + e1 − zk + F (s (q (k) , k))− Π− k + V (k, q (k))

s.t.

Π− e1 + q (k) s (q (k) , k)− F (s (q (k) , k)) ≥ 0

n+ e0 + e1 − q (k) s (q (k) , k) + F (s (q (k) , k))− Π− k ≥ 0

At an interior point where the nonnegativity constraints on t = 0 consumption don’tbind, the marginal value of investment is:

dW (k)

dk= −z − 1 + F ′ (s (q, k))

[∂s (q (k) , k)

∂qq′ (k) +

∂s (q (k) , k)

∂k

]+dV (k, q (k))

dk(22)

Using (16) and (15), this reduces to

dW (k)

dk= −z − 1 + F ′

(zk

q (k)

)zq (k)− q′ (k) k

q (k)2 +A (q (k)− z)

q (k)+

Azk

q (k)2 q′ (k)

At the equilibrium point, where F ′(

zkq(k)

)= q (k) = Az

A−1, this further reduces to:

dW (k)

dk=

(A− 1− z) (A− 1)

Azkq′ (k) < 0

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The inequality follows from Lemma 1 and Assumption 1.1. This establishes the followingresult:

Proposition 1. (Lorenzoni 2008) In the canonical model, the social planner can obtain aPareto improvement by lowering investment relative to the competitive equilibrium.

The equilibrium allocation is constrained inefficient: a constrained planner would be ableto make everyone better off by lowering investment. The reason for this inefficiency has beendiscussed by Lorenzoni (2008) and Dávila and Korinek (2017). In equilibrium, householdsconsume at both t = 0 and t = 1, so their marginal rate of substitution between wealth inboth periods is 1. Entrepreneurs, instead, consume at t = 0, so their marginal utility is 1,but are constrained at t = 1, when their marginal utility of wealth is A

q> 1. Hence marginal

rates of intertemporal substitution are not equalized across entrepreneurs and households. Ifintertemporal contracts were enforceable, entrepreneurs would want to save and householdswould want to borrow. Lower investment, by raising the price of capital, shifts wealth fromhouseholds to entrepreneurs at t = 1 and therefore partially substitutes for the missing creditmarket.3 At the equilibrium, the direct effect of marginal investment (taking q as given) issecond-order. Using (16), (18) and (19):

∂W

∂k= −1− z + F ′

(zk

q

)z

q+A (q − z)

q= 0 (23)

so the only first-order effect is the redistribution of wealth at t = 1.The literature has given different labels to this effect. Typically, it is described as a pe-

cuniary externality: individual entrepreneurs do not take into account that their investmentdecisions affect other entrepreneurs’ constraints by changing prices. In the terminology ofDávila and Korinek (2017), this is a distributive externality: prices affect the distributionof wealth across agents and dates in an environment where marginal rates of intertemporalsubstitution are not equalized. It is also often referred to as a “fire sale” externality: addi-tional investment means that additional units of capital will have to be sold, which lowersits price.

Caballero and Krishnamurthy (2003, 2004), Farhi et al. (2009), Bianchi (2011), Benignoet al. (2011), Bianchi and Mendoza (2013), Jeanne and Korinek (2010, 2016), Hart and Zin-gales (2015), Korinek (2017) and Di Tella (forthcoming) also analyze constrained efficiency

3Note that entrepreneurs are savings-constrained at t = 0 but borrowing-constrained at t = 1. They wantto saving towards t = 1 precisely because they know that they will not be able to borrow.

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in economies with financial constraints. They show examples where ex-ante decisions af-fect future constraints through aggregate consumption, which affects prices of goods and/orthe stochastic discount factor. The inefficiencies identified in this class of models are alsosometimes described as “fire-sale externalities”, but work through a different channel as themisallocation mechanism. Asriyan (2016) analyzes an example where misallocation interactswith dispersed information.

A central ingredient of Lorenzoni’s fire-sale argument is that the demand for capital isindeed downward sloping, so that additional sales lead to lower prices. In this canonicalmodel, this happens because the marginal product of capital in its second-best use is dimin-ishing. This assumption has been made by a large literature following Shleifer and Vishny(1992) and Kiyotaki and Moore (1997). In the next section, I revisit the normative analysisin a model with an alternative microfoundation for fire-sale effects.

2 An Asymmetric Information Model

The entrepreneurs’ side of the economy is the same as in the canonical model. The householdside follows Kurlat (2016) in assuming that there are households with different degrees ofexpertise for evaluating assets.

Technology, Preferences, Markets and Information

Relative to the canonical model, there are two differences. First, F(kH)

= AkH , so capitalis just as productive in the hands of households as in the hands of entrepreneurs. Second,there is a specific form of asymmetric information. In addition to real entrepreneurs, thereis a unit measure of fake entrepreneurs. They are endowed with nF consumption goods att = 0 plus a measure λ of “lemons”, completely useless pieces of fake capital, indexed by iand distributed uniformly in the interval [0, 1].

Households differ in their ability to distinguish lemons from real capital; their expertiseis indexed by θ ∈ [0, 1] and is exogenously given. When analyzing any asset, a householdwill observe a binary signal x ∈ 0, 1. If the asset is real capital, then the household willalways observe x = 1. If the asset is a lemon, the signal will depend on the lemon’s index andthe household’s expertise: a household of expertise θ will observe a signal x (i, θ) = I (i ≥ θ)

when analyzing lemon i. Hence, lemons in the interval [θ, 1] will be indistinguishable fromreal capital for household θ. Higher θ means that a given household is more expert, since itmistakes fewer lemons for real capital.

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Household θ is endowed with e0 goods at t = 0 and e1 (θ) goods at t = 1, where e (·) is acontinuous function. As in the canonical model, no intertemporal contracts are enforceableand there is a competitive market for capital at t = 1.

Equilibrium definition

The equilibrium concept is derived from Kurlat (2016). There is a single price q at whichreal capital and lemons trade at t = 1. Households can be selective in what they buy in thecapital market, to the extent that their expertise allows them to tell lemons and real capitalapart. Therefore household θ will buy real capital plus lemons with indices in the intervali ∈ [θ, 1], pro-rata relative to their respective supply. All the real capital offered on sale willindeed sell, but lemons will be rationed, since some of the households will reject them. Letµ (i) be the fraction of lemons of index i put on sale that indeed sell.4

The problem real entrepreneurs face is exactly the same as problem (5); what marketmechanism results in the price q does not affect the problem they face.

The fake entrepreneurs solve:

maxcF0 ,c

F1 ,s

F (i)cF0 + cF1 (24)

s.t.

cF0 ≤ nF (25)

cF1 ≤ q

1ˆ

0

sF (i)µ (i) di (26)

sF (i) ≤ λ (27)

At t = 0, their consumption is simply constrained by their endowment nF . At t = 1, theirconsumption is constrained by the revenue they obtain from selling lemons. If they put sF (i)

lemons of type i on sale, they actually sell µ (i) sF (i) of them, at a price q each. (26) resultsfrom integrating over their portfolio.

4Kurlat (2016) derives these equilibrium conditions from the assumption that there are markets at everypossible price and with every possible rule for ordering buyers’ trades, and buyers and sellers choose whereto trade. See Kurlat (2016) and Kurlat (2017) for details.

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Household θ, for its part, solves the following problem:

maxcH0 ,c

H1 ,c

H2 ,δ

cH0 + cH1 + cH2 (28)

s.t.

cH0 ≤ e0 (29)

cH1 + qδ ≤ e1 (θ) (30)

cH2 ≤ δs

s+´ 1

θsF (i) di

A (31)

cHt ≥ 0 t = 0, 1, 2; δ ≥ 0

where s is the amount of real capital sold by entrepreneurs and δ is the number of assets(capital plus lemons) that the household buys at t = 1. The only difference between thisproblem and problem (10) is in constraint (31). The pool which household θ draws fromincludes all the s units of real capital that entrepreneurs put on sale plus the sF (i) units oflemons indexed i ∈ [θ, 1] that fake entrepreneurs put on sale and the household is incapableof filtering out. Assets are divisible and the law of large numbers holds, so the fraction of realcapital among the δ assets bought by household θ is s

s+´ 1θ s

F (i)di; it obtains A consumption

goods from each of them at t = 2.

Definition 2. A competitive equilibrium is an allocation cE0 , cE1 , cE2 , cH0 (θ) , cH1 (θ) , cH2 (θ) ,

cF0 , cF1 , k, k

E, s, δ (θ) , α, sF (i), a price q and fractions of lemons sold µ (i) such that: cE0 , cE1 ,cE2 , k, s, k

E, α solves the entrepreneur’s problem (5), taking q as given; cH0 (θ) , cH1 (θ) , cH2 (θ) ,

δ (θ) solves household θ’s problem (28), taking q, s and sF (i) as given; cF0 , cF1 , sF (i) solvesthe fake entrepreneur’s problem, taking q and µ (i) as given; all the real capital put on saleindeed sells: ˆ

s

s+´ 1

θsF (i) di

δ (θ) dθ = s (32)

and the fraction of lemon i sold is:

µ (i) =

ˆ

θ≤i

1

s+´ 1

θsF (i) di

δ (θ) dθ (33)

Condition (32) is a market clearing condition. Household θ buys s

s+´ 1θ s

F (i)diδ (θ) units of

real capital, so adding up over households gives the total demand for real capital, which isequated to the supply s. Instead, the market for lemons does not clear, since lemon i is only

14

accepted by households with θ ≤ i. Hence, the fraction that indeed sells is given by (33).

Equilibrium Characterization

Assume that the following condition holds:

Assumption 2.´e1 (θ) dθ > zn+ λA

Assumption 2 says that, in total, households’ t = 1 endowment is large enough to pay forthe maintenance costs of the maximum level of investment, and to pay for all the lemons asthough they were real capital. It is enough to ensure than in equilibrium some householdsconsume at t = 1.

It is immediate that the solution to the fake entrepreneur’s problem is cF0 = nF , sF (i) = λ

and cF1 = qλ´ 1

0µ (i) di. Fake entrepreneurs have no value of keeping their lemons so they

always put them on sale, and at t = 1 they consume the proceeds of all the sales they are ableto make. This implies that

´ 1

θsF (i) di = λ (1− θ). Therefore the solution to the household’s

problem (28) is:

δ (θ)

= 0 if s

s+λ(1−θ)A < q

∈[0, e1(θ)

q

]if s

s+λ(1−θ)A = q

= e1(θ)q

if ss+λ(1−θ)A > q

(34)

Conditions

s+ λ (1− θ∗)A = q (35)

defines a cutoff level of expertise θ∗ such that households with θ > θ∗ spend their entireendowment of t = 1 goods to buys assets while households with θ < θ∗ do not buy any assetsat all because they understand that they would take in too many lemons and make losses.

Replacing this in the market clearing condition (32) and rearranging:

1ˆ

θ∗

1

s+ λ (1− θ)e1 (θ) dθ = q (36)

Also, as in section 1, the entrepreneur’s optimal investment decision is given by (17) andsales of real capital satisfy (15). Using (33), sF (i) = λ and (15), the fraction of asset i that

15

can be sold in equilibrium is:

µ (i) =1

q

iˆ

θ∗

1

s+ λ (1− θ)e1 (θ) dθ (37)

It is immediate that there always exists an equilibrium with q = 0, k = 0 and θ∗ = 1. Ifentrepreneurs expect the price of capital to be zero, they know that they will be unable toafford the maintenance costs of any capital they created, so they do not invest. As a result,only lemons are on sale at t = 1, which justifies q = 0, and households do not buy anything.In addition, depending on parameters, there can be equilibria with q = Az

z−1and an interior

level of investment k ∈ [0, n] and/or an equilibrium with q > AzA−1

and investment k = n.

Lemma 2.

1. Taking k as given, the system of equations (35), (36), (15) has a unique solutionq (k) , θ∗ (k) , s (k) for any k ≤ n

2. q′ (k) < 0 if and only if

e1 (θ∗ (k)) <q (k)λ

´ 1

θ∗1

[s(k)+λ(1−θ)]2 e1 (θ) dθ

A−q(k)s(k)+λ(1−θ∗(k))

(38)

Lemma 2 is the analogue of Lemma 1. It says that, depending on the magnitude of e1 (θ∗),the model with asymmetric information may or may not have the types of fire-sale effectsthat are present in the canonical model.5 Higher investment has two opposing effects on assetprices. First, it improves the overall mix of assets on sale at t = 1 since, by assumption, thenumber of lemons is fixed and more investment means more real capital needs to be sold tomeet maintenance needs. Other things being equal, this pushes asset prices up rather thandown, the opposite of a fire-sale effect.6 On the other hand, absorbing the extra supply ofassets on sale requires drawing less-expert households into the market, since the more-experthouseholds exhaust their wealth. These households are rationally aware of their lower abilityto filter out lemons, so other things being equal they are only willing to buy at lower prices.The net effect of higher investment on q depends on which of these two effects dominates.

5Kurlat (2016) shows a version of this result in a model where sales of real capital are exogenous, asopposed to being dictated by the need to meet maintenance costs.

6Eisfeldt (2004) and Uhlig (2010) point out that simple models of trading with asymmetric informationdo not produce fire sales precisely for this reason.

16

If e1 (θ∗) is high, this means that a small drop in the cutoff level of expertise is enough todraw a lot of wealth into the market. In this case, the first effect dominates and asset pricesrise. Conversely, when e1 (θ∗) is low, the marginal level of expertise must fall more, and thefire sale effect where q falls results. This possibility is important for the comparison withthe canonical model, since it shows that it’s possible to obtain the same positive predictionsfor the relation between investment and asset prices from both microfoundations.

Welfare

Consider again a constrained social planner that wants to maximize entrepreneurs’ utilitysubject to delivering a minimum level of utility to households and to fake entrepreneurs. Asbefore, the planner can dictate the level of investment that entrepreneurs carry out and canmake transfers across agents at t = 0, but cannot intervene in any way at time t = 1 ort = 2. Hence the planner is limited to choosing cE0 , cH0 (θ), cF0 and k. Denote by q (k), s (k) ,θ∗ (k) and µ (i, k) the solution to the system of equations (35), (36), (15), (37), taking k asgiven. Also, let

ρ (k) ≡ q (k)

1ˆ

0

µ (i, k) di

be the average revenue that fake entrepreneurs obtain for each lemon they own. Replacing(37) and simplifying:

ρ (k) =

1ˆ

θ∗(k)

1− θs (k) + λ (1− θ)

e1 (θ) dθ (39)

The planner solves:

maxcH0 (θ),cE0 ,c

F0 ,kcE0 + V (k, q (k)) (40)

s.t.

cE0 +

ˆcH0 (θ) dθ + cF0 + k ≤ n+ nF + e0 (41)

cH0 (θ) + e1 (θ) max

1,

s (k)

s (k) + λ (1− θ)A

q (k)

≥ Π (θ) (42)

cF0 + λρ (k) ≥ ΠF (43)

cH0 (θ) ≥ 0 cE0 ≥ 0 cF0 ≥ 0 k ≥ 0

17

Constraint (41) is the t = 0 resource constraint. Implicit in it is the assumption thatthe planner can redistribute consumption across real entrepreneurs, fake entrepreneurs andhouseholds at t = 0. Constraint (42) imposes a minimum level of utility on householdθ. At t = 0, the planner just assigns this household consumption cH0 (θ). At t = 1, thehousehold has the choice of either consuming its endowment e1 (θ) or spending it to buy e1(θ)

q(k)

and obtaining an average of s(k)s(k)+λ(1−θ)A consumption goods from each of them at t = 2.

Constraint (43) imposes a minimum level of utility for fake entrepreneurs.Define:

cH0 (θ, k) ≡ max

Π (θ)− e1 (θ) max

1,

s (k)

s (k) + λ (1− θ)A

q (k)

, 0

cF0 (k) ≡ max

ΠF − λρ (k) , 0

Given a choice of k and the t = 1 asset market equilibrium that results from it, cH0 (θ, k) andcF0 (k) are the minimal levels of t = 0 consumption that the planner must assign to householdθ and fake entrepreneurs respectively to ensure that they satisfy constraints (42) and (43).

Replacing the constraints into the planner’s objective results in:

maxkW (k) ≡ n+ nF + e0 − k − cF0 (k)−

ˆcH0 (θ, k) dθ + V (k, q (k))

k ≤ n+ nF + e0 − cF0 (k)−ˆcH0 (θ, k) dθ (44)

At an interior point where the nonnegativity constraint on cF0 (k) and cH0 (θ, k) don’t bind:

dW (k)

dk= −1 +

d

dk

(λρ (k) +

ˆe1 (θ) max

1,

s (k)

s (k) + λ (1− θ)A

q (k)

dθ

)+dV (k, q (k))

dk

= −1 +d

dk

1ˆ

0

e1 (θ) dθ − s (k) q (k) + As (k)

+dV (k, q (k))

dk

= A− 1− z (45)

The first step follows from rearranging, using (39) and using the market clearing condition(36) and the last step uses (16) and (15). As long as Assumption 1.1 (which states thatinvesting in capital has positive net present value) holds, then dW (k)

dk> 0. This establishes

the following result:

Proposition 2. In the model with asymmetric information, the social planner can obtain a

18

Pareto improvement by raising investment relative to the competitive equilibrium.

Proposition 2 says that this model has the opposite normative implications as the canon-ical model. The competitive equilibrium has underinvestment rather than overinvestment.Furthermore, this property holds regardless of whether condition (38) holds, i.e. regardlessof whether higher investment leads to fire-sale effects.

In both the canonical model and the asymmetric information model, the social planner’sinvestment decision takes into account the present value of investment and the indirect effectof investment through the effect of asset prices on t = 1 budget constraints. However, theseeffects work very differently in the two models.

In the canonical model, the net present value of investment is given by equation (23), andis equal to zero. Even though the net present value in best use is positive, only a fractionq−zq

of the marginal unit of capital will be used by entrepreneurs; a fraction zqwill be used

by households, and the net present value of building marginal capital and assigning it tohouseholds is negative. Taking into account how capital will be allocated, the overall netpresent value of investment is zero. Conversely, there is social value in reallocating wealthacross agents and periods. Shifting entrepreneurs wealth from t = 0 to t = 1 allows them toafford more capital and allocate it to its best use, which creates social gains.

Instead, in the asymmetric information model, the net present value of investment isalways positive and equal to A− 1− z, since capital will have the same marginal product nomatter who holds it. The reason why the equilibrium does not produce as much investmentas the social planner would like is that entrepreneurs do not capture the full value of invest-ment at the margin. They know that they will have to sell a part of their capital to meetmaintenance costs in a market where it will be partially pooled with lemons and therefore notbe paid its full value. The presence of lemons in the market acts like a tax on investment, andthe planner wants to counteract the effects of this tax on the level of investment.7 Conversely,given a level of investment, there is no social value in reallocating wealth across agents andperiods in any direction. This is despite the fact that marginal rates of substitution arenot equalized. Entrepreneurs’ marginal rate of substitution is A

q, just like in the canonical

model, while household θ’s marginal rate of substitution is max

ss+λ(1−θ)

Aq, 1< A

qand

fake entrepreneurs’ marginal rate of substitution is 1. As in the canonical model, the reasonentrepreneurs have a high marginal value of wealth at t = 1 is that having higher wealth

7Kurlat (2013) shows an equivalence result between asymmetric information taxes on trading. Here thepresence of maintenance costs and financial constraints implies that investment requires subsequent tradingto meet maintenance costs, so asymmetric information acts like a tax on investment.

19

lets them avoid selling capital for q < A. However, unlike the canonical model, this has afirst order negative effect on households and fake entrepreneurs, who benefit from trading ina market where real capital is on sale. Overall, since total output does not depend on thedistribution of wealth at t = 1, the benefit to entrepreneurs of higher wealth exactly cancelsthe total loss to households and fake entrepreneurs. Therefore the social planner only caresabout the positive net present value of investment and increases k until it hits nonnegativityconstraints on t = 0 consumption.

Dávila and Korinek (2017) show that in a class of models of which the canonical modelis a special case, the differences between the social planner’s problem and the competitiveequilibrium can be summarized by three sufficient statistics: the differences in the marginalrates of substitution across agents, their trading positions, and the sensitivity of asset pricesto investment. The model with asymmetric information does not fall within this class, andtheir result does not extend to it. Entrepreneurs have a higher marginal rate of substitution,are net sellers of capital, and if condition (38) holds asset prices fall when investment rises.Nevertheless, the social planner wants higher investment than in the competitive equilibrium.

Note that one premise of the normative exercise is that the planner can redistribute acrossagents at t = 0. If there are fire sale effects, this requires taking goods from households (andpossibly fake entrepreneurs) to compensate entrepreneurs for the lower prices that will resultfrom higher investment.8 If this redistribution is not feasible, then higher investment doesnot result in a Pareto improvement, and its desirability will depend on the welfare weightsthe planner places on each kind of agent. In the extreme case where the planner only caresabout entrepreneurs (for instance, because households and fake entrepreneurs are foreigners),then avoiding fire sales justifies limiting investment irrespective of the microfoundation.

3 New Investment Margin

In this section I ask how the normative conclusions in the canonical model and the asym-metric information model change if entrepreneurs have an additional opportunity to investat t = 1. Suppose entrepreneurs can convert ψ (x) goods at t = 1 into x units of capital thatwill yield Ax consumption goods at t = 1.

Assumption 3.

1. ψ (·) is increasing and convex, with ψ′ (0) < z

8In the canonical model, it is the other way around: entrepreneurs need to compensate households forthe profits they forgo when lower investment mitigates fire sales.

20

2. (ψ′ (x))2 < ψ (x)ψ′′ (x) for all x

Assumption 3.1 ensures that the equilibrium amount of new investment at t = 1 ispositive. Assumption 3.2 means that ψ (·) is sufficiently convex that new investment doesnot respond too strongly to asset prices.

The entrepreneur’s t = 1 problem, both in the canonical model and in the asymmetricinformation model, becomes:

V (k, q) ≡ maxcE1 ,c

E2 ,s,k

E ,αcE1 + cE2

s.t.

cE1 + αzk + ψ (x) ≤ sq

kE ≤ αk − s+ x

cE2 ≤ AkE

α ∈ [0, 1] ; cEt ≥ 0 t = 1, 2; kE ≥ 0; s ∈ [0, αk] ; x ≥ 0

(46)

The first order condition for x is:

ψ′ (x) = q

The entrepreneur equates the marginal cost of delivering capital into period t = 2 across thetwo ways of obtaining it: new investment at a cost of ψ′ (x) or retaining maintained capital,which has an opportunity cost of q. Assumption 3.1 ensures that the entrepreneur choosesan interior solution.

Replacing this back into the entrepreneur’s problem gives:

V (k, q) =A

q︸︷︷︸rate of return on wealth

(q − z) k︸ ︷︷ ︸value of initial capital

+ qx (q)− ψ (x (q))︸ ︷︷ ︸net present value of new investement

(47)

where x (q) ≡ (ψ′)−1 (q). As in the baseline model, this implies that there will be an interiorlevel of investment at t = 0 if and only if q = Az

A−1. Using the t = 1 budget constraint, sales

of capital will be:

s (q, k) =zk + ψ (x (q))

q(48)

21

Canonical Model

In the canonical model, sales of capital will be purchased by households, so the price ofcapital will be equal to the marginal product of capital in the hands of households. Theequilibrium price will therefore satisfy:

F ′(zk + ψ (x (q))

q

)= q (49)

Lemma 3. If equation (49) has a solution, it is unique. The solution q (k) is a decreasingfunction.

Lemma 3 is an extension of Lemma 1. It means that the fire sale effect in the canonicalmodel extends to the variant where there is a new investment opportunity at t = 1. Settingq = Az

A−1implies that the level of investment at t = 0 must satisfy:

F ′

(zk + ψ

(x(AzA−1

))AzA−1

)=

Az

A− 1(50)

The social planner will solve problem (21), with the difference that q (k) will be given bythe solution to (49), the entrepreneur’s value function is (47) and the amount of capital soldby entrepreneurs is given by (48). Using q = F ′ (s (q, k)) = ψ′ (x (q)) = Az

A−1, expression (22)

reduces to:dW (k)

dk=

1

q2

[q3x′ (q) + [ψ (x (q)) + zk] (A− q)

]q′ (k) < 0

This establishes the following result:

Proposition 3. In the canonical model with reinvestment, the social planner can obtain aPareto improvement by lowering investment relative to the competitive equilibrium.

Introducing a new-investment margin does not change the overinvestment result. It is stilltrue that in equilibrium the ratio of marginal utilities across entrepreneurs and householdsis A

q> 1 and the net present value of investment is zero, so shifting the terms of trade in

favor of entrepreneurs results in a Pareto improvement.

Asymmetric Information Model

Equilibrium in the asset market at t = 1, given a level of k, is determined the same way asin the model without reinvestment. The only difference is that sales of real capital are given

22

by equation (48) instead of equation (15). As in the baseline, there is an equilibrium withq = 0 and no investment and possibly interior equilibria with q = Az

A−1or corner equilibria

with q > AzA−1

and k = n.

Lemma 4.

1. Taking k as given, the system of equations (35), (36), (48) has a unique solution q (k),θ∗ (k), s (k) for any k ≤ n.

2. q′ (k) < 0 if and only if condition (38) holds.

Lemma 4 is an extension of Lemma 2. It means that the presence of fire sale effects inthe variant of the asymmetric information model with reinvestment at t = 1 depends onthe same forces as in the baseline. The uniqueness result relies on Assumption 3.2, whichlimits how strongly sales of real capital react to asset prices. Without it, it is possible toconstruct examples where higher asset prices induce higher sales of real capital to financenew investment, which improves the pool of assets on sale and justifies the higher prices, astandard source of multiple equilibria in variants of the Akerlof (1970) model.

The social planner will solve problem (40), with the only difference that V (k, q (k)) isdefined by equation (47) and q (k), s (k) and θ∗ (k) are defined as the solution to the systemof equations (35), (36) and (48). At an interior point where the nonnegativity constraint oncF0 (k) and cH0 (θ, k) don’t bind, the same steps that lead to equation (45) result in:

dW (k)

dk= A− 1− z +

(A− q (k)

ψ′′ (x (q (k)))

)q′ (k) (51)

Evaluating this expression at an interior equilibrium where q = AzA−1

and rearranging:

dW (k)

dk

∣∣∣∣k=k∗

=A− 1− z

(A− 1)ψ′′ (x (q (k)))[(A− 1)ψ′′ (x (q (k))) + Aq′ (k)]

This implies the following result:

Proposition 4. In the asymmetric information model with reinvestment:

1. If there are no fire-sale effects (q′ (k) ≥ 0), the social planner can obtain a Paretoimprovement by raising investment relative to the competitive equilibrium.

2. If there are fire-sale effects (q′ (k) < 0), the social planner can obtain a Pareto im-provement by raising investment relative to the competitive equilibrium if the cost of

23

t = 1 investment is sufficiently convex relative to the magnitude of fire-sale effects:

A

A− 1

|q′ (k)|ψ′′ (x (q (k)))

< 1 (52)

Otherwise, the social planner can obtain a Pareto improvement by lowering investmentrelative to the competitive equilibrium.

As in the baseline asymmetric information model, the planner recognizes that entrepreneurshold back on investment because they anticipate having to cross subsidize lemons; the plan-ner wants to counteract this implicit tax and encourage higher investment at t = 0, knowingthat the social net present value of t = 0 investment is A− 1− z > 0.

On the other hand, in equilibrium there is also underinvestment at t = 1. Entrepreneurschoose investment at t = 1 according to the first order condition ψ′ (x) = q, while the efficientlevel of investment (for instance, if entrepreneurs were able to borrow) would be ψ′ (x) = A.The net present value of marginal investments at t = 1 is A−ψ′ (x) = A− q > 0. . If raisingthe amount of investment at t = 0 results in a change in asset prices of q′ (k), this results inx′ (q) q′ (k) = q′(k)

ψ′′(x)additional units of investment, with net present value

(A−qψ′′(x)

)q′ (k).

When condition (38) holds so that the model has fire sale effects, this is a negativenumber: higher investment at t = 0 pushes down investment at t = 1 by lowering assetprices. Effectively, the planner faces a tradeoff between underinvestment at t = 0 andunderinvestment at t = 1. Depending on how these two compare, the planner may wantto raise or lower t = 0 investment relative to the competitive equilibrium. Condition (52)determines the direction of the net effect. The planner wants to lower t = 0 investmentwhen (i) A

A−1is high; at an interior equilibrium where q = Az

A−1, AA−1

= A−qA−1−z is the ratio of

the net present value of t = 1 investment to t = 0, so high AA−1

means t = 1 investment isrelatively valuable; (ii) ψ′′ (x) is low, so the marginal cost of investment is relatively flat andt = 1 investment responds strongly to asset prices;9 and (iii) q′ (k) is very negative, meaningstrong marginal fire sale effects.

Instead, if condition (38) does not hold, then there is no tradeoff. By raising t = 0

investment, the planner raises asset prices and raises t = 1 investment. In this case, theplanner unambiguously wants to raise investment relative to the competitive equilibrium.

Note that the effect on t = 1 investment is not, in the terminology of Dávila and Korinek(2017), a distributive externality. The reason why the planner wants to raise asset prices is

9ψ′′ (x) is the inverse of the coefficient of a regression of investment on Tobin’s q, so in principle it’sstraightforward to measure it empirically.

24

not as a way to redistribute towards entrepreneurs at t = 1 and indirectly replace the missingcredit market. Holding q constant, entrepreneurs use their marginal unit of wealth to retainexisting capital, not to build new capital. As in the baseline asymmetric information model,whether or not entrepreneurs sell capital at t = 1 is neutral from the social planner’s pointof view, and therefore there is no social value in increasing entrepreneurs’ wealth at t = 1.Instead, the reason why higher asset prices at t = 1 matter is that they determine thediscount factor that entrepreneurs apply to new investment. Entrepreneurs obtain A

qgoods

at t = 2 by dedicating one good at t = 1 towards retaining existing capital, and thereforerequire the same return from new investment. By raising asset prices the social planner lowersthe required return on new investment, and therefore mitigates t = 1 underinvestment. Oneway to interpret the difference between this effect and a distributive externality is to imaginethat there was a way for the social planner to (unexpectedly) intervene at t = 1 and transferwealth from households to entrepreneurs, while keeping q constant. In the canonical model,this would increase efficiency because entrepreneurs could afford to dedicate more capital toits best use. Here instead keeping q constant would imply no change in t = 1 investmentand therefore no efficiency gain.

4 Investment by Fake Entrepreneurs

In this section I consider a a variant of the asymmetric information model where lemons arecreated deliberately with the intent of being sold, a possibility has been studied recently byCaramp (2016), Neuhann (2017) and Fukui (2018). Concretely, suppose that the amount oflemons is the result of investment by fake entrepreneurs. They can convert ϕ (λ) goods att = 0 into λ lemons (uniformly distributed in [0, 1]) at t = 1, which they can then attemptto sell in exchange for goods.

Assumption 4.

1. ϕ (·) is increasing and convex, with ϕ′ (0) = 0

2. ϕ′(nF)> A

Assumption 4 says that the marginal cost of the first lemon is zero, whereas if the fakeentrepreneurs were to use all their wealth to produce lemons, the marginal cost would exceedthe maximum price they could possibly get for them. This ensures that whenever there ispositive investment in real capital, there is an interior solution for how many lemons areproduced.

25

The fake entrepreneur’s problem becomes

maxcF0 ,c

F1 ,s

F (i)cF0 + cF1

s.t.

cF0 + ϕ (λ) ≤ nF

cF1 ≤ q

1ˆ

0

sF (i)µ (i) di

sF (i) ≤ λ

Using the fact that the fake entrepreneur will always attempt to sell its lemons (i.esF (i) = λ), the first order condition for investment in lemons is:

ϕ′ (λ) = q

1ˆ

0

µ (i) di = ρ (53)

Given a level of investment in real capital k, equilibrium in the t = 1 asset market requiresthat s, q, θ∗, ρ, λ satisfy: the optimal selling condition for (15); the indifference and marketclearing conditions (35) and (36); the rationing formula (39) and the first order conditionfor fake investment (53). As in the baseline model, there always exists an equilibrium withq = 0 and, depending on parameters, there can be equilibria with q = Az

z−1and k ∈ [0, n]

and/or an equilibrium with q > AzA−1

and k = n.

Lemma 5.

1. The system of equations (15), (35), (36), (39), (53) has a unique solution q (k), θ∗ (k),s (k), ρ (k), λ (k) for any k ≤ n.

2. There exist cutoffs eλ and eq such that

(a) λ′ (k) < 0 if and only if e1 (θ∗ (k)) < eλ and

(b) q′ (k) < 0 if and only if e1 (θ∗ (k)) < eq

3. For ϕ′′ (λ (k)) sufficiently large, then eq > eλ, so q′ (k) < 0 is a necessary condition forλ′ (k) < 0

Lemma 5 describes how asset prices and investment by fake entrepreneurs depend onreal investment k. If lemons did not face rationing in equilibrium (i.e. if µ (i) = 1 for all

26

i), or if µ (i) was an exogenous constant, then the first order condition (53) would imply adirect relationship between asset prices q and revenue per unit from selling lemons ρ, andtherefore investment by fake entrepreneurs λ. However, the degree of rationing faced bylemons responds endogenously to the level of investment, so q and ρ need not move in thesame direction. Part 2 of Lemma 5 says that the pattern from the baseline model, establishedby Lemma 2, applies to both q and ρ. If there is a low level of wealth near the cutoff level ofexpertise, higher k will lead to falls in both asset prices and fake investment. Even thoughthe cutoffs might be different, the basic force that leads to fire sale effects (a large fall in themarginal level of expertise among asset buyers) also leads to lower fake investment. Part3 says that if there is enough curvature in ϕ so that fake investment does not respond toostrongly to ρ, then there exist parameter combinations such that higher k leads to lowerq but higher ρ but not the other way around, so fire sale effects are a necessary but notsufficient condition for fake investment to be decreasing in the level of real investment.

The planner solves problem (40) with the only difference that the exogenous constant λis replaced by the function λ (k) and constraint (41) is replaced by

cE0 +

ˆcH0 (θ) dθ + cF0 + k + ϕ (λ (k)) ≤ n+ nF + e0

in order to take into account that creating λ (k) lemons requires investing ϕ (λ (k)) resources.The implicit assumption is that the planner can control the level of investment by realentrepreneurs but cannot control fake entrepreneurs’ investment, which will be given by(53). The planner just assigns a total cF0 + ϕ (λ (k)) goods at t = 0 to fake entrepreneurs,understanding that how they will divide them between consumption and fake investmentdepends on the level of real investment and the resulting asset market equilibrium.

Following the same steps that result in expression (45) and using (53), the marginal socialvalue of investment is given by:

dW (k)

dk= A− 1− z − ϕ′ (λ (k))λ′ (k)

= A− 1− z − ϕ′ (λ (k))

ϕ′′ (λ (k))ρ′ (k)

This implies the following result:

Proposition 5. In the asymmetric information model with endogenous creation of lemons:

1. If real investment lowers the average revenue from selling lemons (ρ′ (k) ≤ 0), the

27

social planner can obtain a Pareto improvement by raising investment relative to thecompetitive equilibrium.

2. If real investment raises the average revenue from selling lemons (ρ′ (k) > 0), thesocial planner can obtain a Pareto improvement by raising investment relative to thecompetitive equilibrium if the cost of producing lemons is sufficiently convex:

1

A− 1− zϕ′ (λ (k))

ϕ′′ (λ (k))ρ′ (k) < 1

Otherwise, the social planner can obtain a Pareto improvement by lowering investmentrelative to the competitive equilibrium.

The social planner cares about the positive net present value of investment A − 1 − z

and the deadweight cost of fake investment. Whenever ρ′ (k) > 0 (which, using (53), impliesλ′ (k) > 0) this involves a tradeoff. Higher real investment increases the average revenue perlemon ρ and therefore induces wasteful investment by fake entrepreneurs. If this effect islarge enough (which will be the case if the marginal cost of producing lemons is relativelyflat, so quantities respond strongly), then it’s possible that the social planner may wantto reduce investment relative to the competitive equilibrium. Instead, when λ′ (k) < 0,there is no tradeoff. Higher investment has a positive net present value and also discourageswasteful fake investment, so the planner unambiguously wants to raise investment relativeto the competitive equilibrium.

Part 3 of Lemma 5 implies that if there are no fire sale effects (q′ (k) > 0), this guaranteesthat higher real investment encourages higher fake investment, so the social planner faces ameaningful tradeoff. Instead, if there are fire sale effects (q′ (k) < 0) then it’s possible thatthe planner faces no tradeoff. Since fake entrepreneurs rely on the asset market, a worseningof asset market conditions for selling lemons is actually helpful in deterring fake investment.If µ were an exogenous constant, this would immediately imply that fire sale effects aredesirable: allowing for an endogenous response of fake investment strengthens the case forraising investment relative to the competitive equilibrium. Lemma 5 gives conditions underwhich the argument can be made precise while taking into account that q and ρ are notexactly proportional.

28

5 Conclusion

The main lesson from the above analysis is that the normative implications of fire sales arenot robust across different possible microfoundations. One should therefore be cautious inextracting conclusive policy implications from the observation that investment booms leadto collapses in asset prices and tightening financial constraints, even if the empirical factitself could be firmly established.

In addition, it is worth devoting attention to try to determine what model best appliesin practice, which might be context-specific. A priori, one would expect the canonical modelto be a good fit for applications to investment in highly specific real assets such as airplanes,where alternative uses run into sharply diminishing marginal returns. Instead, the asymmet-ric information model may be a better fit for investment in complex financial instrumentssuch as tranches of mortgage-backed CDOs, which are hard for non-experts to evaluate. Em-pirically, distinguishing between the models is challenging because many of the predictionssimilar: in both models, higher initial investment leads to falling asset prices and to increasedprofits for the average arbitrageur who buys at the fire sale price. An open challenge is todevelop precise empirical tests that can tell the models apart.

Of course, these are not the only two possible explanations of fire sales. Other prominenttheories include models with differences of opinion (Fostel and Geanakoplos 2008, Geanako-plos 2009), cash-in-the-market pricing (Allen and Gale 1994, 1998, Acharya and Yorulmazer2008) or signal extraction in the presence of noise traders (Grossman and Stiglitz 1980, Calvo1999). I expect the non-robustness of the normative implications of fire sales to to extendto this broader class of models a well.

Appendix

Proof of Lemma 1

Let Γ (q, k) ≡ F ′(zkq

)− q. Differentiating with respect to q:

∂Γ (q, k)

∂q= −F ′′

(zk

q

)zkq−2 − 1

29

If q∗ is a solution, then

∂Γ (q, k)

∂q

∣∣∣∣q=q∗

= −F ′′(zkq∗

)zkq∗

F ′(zkq∗

) − 1 < 0

where the inequality follows from Assumption 1.4. Since Γ is decreasing in q at any solution,there is at most one solution.

Using the implicit function theorem:

q′ (k) =F ′′(zkq

)zq

F ′′(zkq

)zkq−2 + 1

=F ′′(zkq

)zq

F ′′(zkq

)zkqF ′(zkq

)+ 1

< 0

where the inequality follows from the concavity of F and Assumption1.4.

Proof of Lemma 2

1. Replacing (15) into (35) and solving for q:

q (θ∗) =−zk +

√(zk)2 + 4λ (1− θ∗) zkA

2λ (1− θ∗)(54)

Replacing (15) and (54) in (36) and rearranging:

Φ (θ∗) ≡1ˆ

θ∗

1

zk + q (θ∗)λ (1− θ)e1 (θ) dθ − 1 = 0 (55)

(θ∗, q, s) is a solution to the system of equations if and only if Φ (θ∗) = 0, q = q (θ∗)

and s = zkq. To establish existence, note that since q (θ∗) ≤ A, and k < n, it follows

that:

Φ (θ∗) >

1ˆ

θ∗

1

zk + Aλe1 (θ) dθ − 1

30

so Assumption 2 ensures that for sufficiently low θ∗, Φ (θ∗) > 0. Furthermore, for θ∗

sufficiently close to 1, Φ (θ∗) < 0 and since Φ (θ∗) is a continuous function, a solutionto (55) exists. To establish uniqueness, take the derivative of Φ (θ∗):

Φ′ (θ∗) = − 1

zk + q (θ∗)λ (1− θ∗)e1 (θ∗)− q′ (θ∗)

1ˆ

θ∗

λ (1− θ)[zk + q (θ∗)λ (1− θ)]2

e1 (θ) dθ

Taking the the derivative of (54):

q′ (θ∗) =2λzk

(zk + 2λ (1− θ∗)A−

((zk)2 + 4λ (1− θ∗) zkA

)0.5)

4λ2 (1− θ∗)2 ((zk)2 + 4λ (1− θ∗) zkA)0.5 > 0

This is positive because:

4λ2 (1− θ∗)2A2 > 0

⇒ 4λ2 (1− θ∗)2A2 + (zk)2 + 4λ (1− θ∗)Azk > (zk)2 + 4λ (1− θ∗)Azk

⇒ (zk + 2λ (1− θ∗)A)2 > (zk)2 + 4λ (1− θ∗) zkA

⇒ zk + 2λ (1− θ∗)A >((zk)2 + 4λ (1− θ∗) zkA

)0.5

The fact that q′ (θ∗) > 0 implies that Φ′ (θ∗) < 0. Since Φ (θ∗) is monotonicallydecreasing, there is a unique solution to equation (55).

2. Rewrite (35), (36), (15) asΓ (q, θ∗, s; k) = 0

where

Γ (q, θ∗, s; k) =

qs+ qλ (1− θ∗)− sA´ 1

θ∗1

s+λ(1−θ)e1 (θ) dθ − qsq − zk

and let ∇ be the gradient of Γ. Taking derivatives:

∇ =

s+ λ (1− θ∗) −qλ q − A

−1 − 1s+λ(1−θ∗)e1 (θ∗) −

´ 1

θ∗1

[s+λ(1−θ)]2 e1 (θ) dθ

s 0 q

(56)

31

By the implicit function theorem q′ (k)

θ∗′ (k)

s′ (k)

= −∇−1

∂Γ1

∂k∂Γ2

∂k∂Γ3

∂k

= −∇−1

0

0

−z

and therefore

q′ (k) =1

|∇|

qλ 1ˆ

θ∗

1

[s+ λ (1− θ)]2e1 (θ) dθ − A− q

s+ λ (1− θ∗)e1 (θ∗)

z

where

|∇| = qλ

1ˆ

θ∗

s

[s+ λ (1− θ)]2e1 (θ) dθ − q

− e1 (θ∗)

[(A− q) s

s+ λ (1− θ∗)+ q

]

< qλ

1ˆ

θ∗

s

[s+ λ (1− θ)]2e1 (θ) dθ − q

< qλ

1ˆ

θ∗

1

s+ λ (1− θ)e1 (θ) dθ − q

= 0 (57)

where the last step uses (36). Therefore q′ (k) < 0 if and only if

qλ

1ˆ

θ∗

1

[s+ λ (1− θ)]2e1 (θ) dθ − A− q

s+ λ (1− θ∗)e1 (θ∗) > 0

Rearranging gives the result.

Proof of Lemma 3

Let Γ (q, k) ≡ F ′(zk+ψ(x(q))

q

)− q. Differentiating with respect to q:

∂Γ (q, k)

∂q= F ′′

(zk + ψ (x (q))

q

)(ψ′ (x (q))x′ (q) q − zk + ψ (x (q))

q2

)− 1

32

If q∗ is a solution, then

∂Γ (q, k)

∂q

∣∣∣∣q=q∗

= −F ′′(zk+ψ(x(q))

q

)zk+ψ(x(q))

q

F ′(zk+ψ(x(q))

q

) − 1 + F ′′(zk + ψ (x (q))

q

)ψ′ (x (q))x′ (q)

q< 0

where the inequality follows from Assumption 1.4, the concavity of F and the convexity ofψ. Since Γ is decreasing in q at any solution, there is at most one solution.

Using the implicit function theorem:

q′ (k) =F ′′(zk+ψ(x(q))

q

)zq

∂Γ(q,k)∂q

< 0

where the inequality follows from the concavity of F and the first part of the result.

Proof of Lemma 4

1. Replacing (48) into (35), q (θ∗) solves:

Ω (q, θ∗) =zk + ψ (x (q))

q− q

A− qλ (1− θ∗) = 0

Note that

∂Ω (q, θ∗)

∂q=ψ′ (x (q))x′ (q) q − zk − ψ (x (q))

q (θ∗)2 − A

(A− q)2λ (1− θ∗)

<ψ′ (x (q))x′ (q) q − ψ (x (q))

q2

=[ψ′ (x (q))]2 1

ψ′′x(q)− ψ (x (q))

q2< 0

where the last step follows from using the definition x (q) ≡ (ψ′)−1 (q) and Assumption3.2. Also, limq→0 Ω (q, θ∗) > +∞ and limq→A Ω (q, θ∗) > −∞. This implies that foreach θ∗ ∈ [0, 1] there is a unique q (θ∗) that satisfies (48) and (35). Also note that

∂Ω (q (θ∗) , θ∗)

∂θ∗=

q (θ∗)

A− q (θ∗)λ > 0

33

so, using the implicit function theorem:

q′ (θ∗) =−∂Ω(q(θ∗),θ∗)

∂θ∗

∂(q(θ∗),θ∗)∂q

> 0

Replacing q (θ∗) into (36) gives (55), so the same steps that lead to the proof of part1 of Lemma 2 apply.

2. Rewrite the system of equations as

Γ (q, θ∗, s; k) = 0

where

Γ (q, θ∗, s; k) =

qs+ qλ (1− θ∗)− sA´ 1

θ∗1

s+λ(1−θ)e1 (θ) dθ − qsq − zk − ψ (x (q))

and let ∇ be the gradient of Γ. Taking derivatives:

∇ =

s+ λ (1− θ∗) −qλ q − A

−1 − 1s+λ(1−θ∗)e1 (θ∗) −

´ 1

θ∗1

[s+λ(1−θ)]2 e1 (θ) dθ

s− ψ′ (x (q))x′ (q) 0 q

By the implicit function theorem q′ (k)

θ∗′ (k)

s′ (k)

= −∇−1

∂Γ1

∂k∂Γ2

∂k∂Γ3

∂k

= −∇−1

0

0

−z

(58)

and therefore

q′ (k) =1

|∇|

qλ 1ˆ

θ∗

1

[s+ λ (1− θ)]2e1 (θ) dθ − A− q

s+ λ (1− θ∗)e1 (θ∗)

z (59)

where

|∇| = qλ

1ˆ

θ∗

s− ψ′ (x (q))x′ (q)

[s+ λ (1− θ)]2e1 (θ) dθ − q

− e1 (θ∗)

[(A− q) s− ψ

′ (x (q))x′ (q)

s+ λ (1− θ∗)+ q

]

34

Assumption 3.2 and (48) imply that

s− ψ′ (x (q))x′ (q) > 0

and therefore

|∇| < qλ

1ˆ

θ∗

s− ψ′ (x (q))x′ (q)

[s+ λ (1− θ)]2e1 (θ) dθ − q

< qλ

1ˆ

θ∗

s

[s+ λ (1− θ)]2e1 (θ) dθ − q

< qλ

1ˆ

θ∗

1

s+ λ (1− θ)e1 (θ) dθ − q

= 0

so the last step of the proof of part 2 of Lemma 2 applies.

Proof of Lemma 5

1. By Lemma 2, the system (15), (35), (36) has a unique solution q (λ, k) , θ∗ (λ, k) , s (λ, k)

for any λ, k. Using (39), this implies that for any λ,k there is a unique

ρ (λ, k) =

1ˆ

θ∗(λ,k)

1− θs (λ, k) + λ (1− θ)

e1 (θ) dθ (60)

that satisfies (15), (35), (36) and (39). Taking the derivative:

∂ρ

∂λ= − 1− θ∗

s+ λ (1− θ∗)e1 (θ∗)

∂θ∗

∂λ−

1ˆ

θ∗

(1− θ)(∂s∂λ

+ 1− θ)

[s+ λ (1− θ)]2e1 (θ) dθ

By the implicit function theorem:∂q∂λ∂θ∗

∂λ∂s∂λ

= −∇−1

q (1− θ∗)

−´ 1

θ∗1−θ

[s+λ(1−θ)]2 e1 (θ) dθ

0

(61)

35

where ∇ is defined by (56). Therefore:

∂θ∗

∂λ= − 1

|∇|

−s 1ˆ

θ∗

1

[s+ λ (1− θ)]2e1 (θ) dθ + q

q (1− θ∗)− (qλ (1− θ∗) +As)

1ˆ

θ∗

1− θ[s+ λ (1− θ)]2

e1 (θ) dθ

∂s

∂λ= − 1

|∇|

s

s+ λ (1− θ∗)e1 (θ∗) q (1− θ∗) + qλs

1ˆ

θ∗

1− θ[s+ λ (1− θ)]2

e1 (θ) dθ

Equation (57) says that |∇| < 0 and therefore that ∂s

∂λ> 0. This implies:

∂ρ

∂λ≤ − 1− θ∗

s+ λ (1− θ∗)e1 (θ∗)

∂θ∗

∂λ−

1ˆ

θ∗

(1− θ)2

[s+ λ (1− θ)]2e1 (θ) dθ

=1

|∇|

(−s´ 1θ∗

1[s+λ(1−θ)]2 e1 (θ) dθ + q

)q (1− θ∗)

− (qλ (1− θ∗) +As)´ 1θ∗

1−θ[s+λ(1−θ)]2 e1 (θ) dθ

1− θ∗

s+ λ (1− θ∗)e1 (θ∗)−

1ˆ

θ∗

(1− θ)2

[s+ λ (1− θ)]2e1 (θ) dθ

If−s 1ˆ

θ∗

1

[s+ λ (1− θ)]2e1 (θ) dθ + q

q (1− θ∗)− (qλ (1− θ∗) +As)

1ˆ

θ∗

1− θ[s+ λ (1− θ)]2

e1 (θ) dθ > 0

then |∇| < 0 implies ∂ρ∂λ< 0. Otherwise, condition (57) implies that

|∇| < e1 (θ∗)

[(A− q) s

s+ λ (1− θ∗)+ q

]and therefore

∂ρ

∂λ≤

(−s´ 1θ∗

1[s+λ(1−θ)]2 e1 (θ) dθ + q

)q (1− θ∗)− (qλ (1− θ∗) +As)

´ 1θ∗

1−θ[s+λ(1−θ)]2 e1 (θ) dθ

e1 (θ∗)[(A− q) s

s+λ(1−θ∗) + q] 1− θ∗

s+ λ (1− θ∗)e1 (θ∗)

−1ˆ

θ∗

(1− θ)2

[s+ λ (1− θ)]2e1 (θ) dθ

= − (1− θ∗)2 sq[As+ qλ (1− θ∗)]

1ˆ

θ∗

1

[s+ λ (1− θ)]2e1 (θ) dθ +

q2 (1− θ∗)2

[As+ qλ (1− θ∗)]− (1− θ∗)

1ˆ

θ∗

1− θ[s+ λ (1− θ)]2

e1 (θ) dθ

−1ˆ

θ∗

(1− θ)2

[s+ λ (1− θ)]2e1 (θ) dθ (62)

36

Using (35) and (36) and rearranging, this simplifies to:

∂ρ

∂λ≤ −

(1− θ∗)

1ˆ

θ∗

(1− θ)[s+ λ (1− θ)]2

s+ λ (1− θ∗)s+ 2λ (1− θ∗)

e1 (θ) dθ

+

1ˆ

θ∗

(1− θ)2

[s+ λ (1− θ)]2e1 (θ) dθ

< 0

(63)

Since ϕis convex, (53) defines an increasing relationship between ρ and λ. Togetherwith (63), this implies that there can be at most one value of λ that satisfies (15), (35),(36), (39) and (53). Assumption then implies that a solution exists.

2.

(a) DefineΛ (ρ) ≡ (ϕ′)

−1(ρ)

Convexity of ϕ implies that Λ is an increasing function. ρ (k) must solve:

ρ (k) = ρ (Λ (ρ (k)) , k)

so

ρ′ (k) =∂ρ(λ,k)∂k

1− ∂ρ∂λ

Λ′ (ρ)

By part 1, ∂ρ∂λ

< 0. Therefore ρ′ (k) < 0 if and only if ∂ρ(λ,k)∂k

< 0. Taking thederivative of (60):

∂ρ

∂k= − 1− θ∗

s+ λ (1− θ∗)e1 (θ∗)

∂θ∗

∂k−

1ˆ

θ∗

(1− θ)[s+ λ (1− θ)]2

∂s

∂ke1 (θ) dθ

Using (58):

∂θ∗

∂k=

1

|∇|

A− q + (s+ λ (1− θ∗))

1ˆ

θ∗

1

[s+ λ (1− θ)]2e1 (θ) dθ

z∂s

∂k= − 1

|∇|[e1 (θ∗) + qλ] z

and therefore:

∂ρ

∂k=

z

|∇|

− 1−θ∗s(k)+λ(1−θ∗)e1 (θ∗)

[A− q + (s+ λ (1− θ∗))

(´ 1θ∗

1[s+λ(1−θ)]2 e1 (θ) dθ

)]+ [e1 (θ∗) + qλ]

´ 1θ∗(k)

(1−θ)[s(k)+λ(1−θ)]2 e1 (θ) dθ

(64)

37

Taking the limit:

lime1(θ∗)→0

∂ρ

∂k=

z

|∇|

qλ 1ˆ

θ∗(k)

(1− θ)[s (k) + λ (1− θ)]2

e1 (θ) dθ

< 0

so by continuity there exists eλ such that ∂ρ∂k< 0 if e1 (θ∗ (k)) < eλ. Since λ (k) =

Λ (ρ (k)) and Λ is increasing, the result follows.

(b) q (k) must solveq (k) = q (k,Λ (ρ (k)))

so

q′ (k) =∂q

∂k+

∂q∂λ

Λ′ (ρ)

1− ∂ρ∂λ

Λ′ (ρ)

∂ρ

∂k(65)

Lemma 2 implies that for sufficiently low e1 (θ∗) , then ∂q∂k. Furthermore: using

(61):

∂q

∂λ=

1

|∇|q2

1

s+ λ (1− θ∗)e1 (θ∗) (1− θ∗) + λ

1ˆ

θ∗

1− θ[s+ λ (1− θ)]2

e1 (θ) dθ

< 0 (66)

Therefore if ∂ρ∂k

> 0 then q′ (k) < 0 follows immediately. Instead if ∂ρ∂k

< 0, theright hand side of (65) is increasing in Λ′ (ρ), so it suffices to show the result forΛ′ (ρ)→∞. Therefore:

q′ (k) ≤ ∂q

∂k−

∂q∂λ∂ρ∂λ

∂ρ

∂k(67)

Replacing (59), (66), (62) and (64) and (57) in (67) and taking the limit andrearranging, then:

lime1(θ∗)→0

q′ (k) ≤ lime1(θ∗)→0

∂q

∂k−

∂q∂λ∂ρ∂λ

∂ρ

∂k

=

(lim

e1(θ∗)→0∆

) (´ 1θ∗(k)

(1−θ)[s(k)+λ(1−θ)]2 e1 (θ) dθ

)2−´ 1θ∗

1[s+λ(1−θ)]2 e1 (θ) dθ

´ 1θ∗

(1−θ)2

[s+λ(1−θ)]2 e1 (θ) dθ

≤ 0

where∆ =

1∂ρ∂λ

z

|∇|λ

sA

s+ λ (1− θ∗)> 0

38

and the inequality follows from the Cauchy-Schwarz inequality. Therefore bycontinuity there exists eq such that q′ (k) < 0 if and only if e1 (θ∗ (k)) < eq.

3. Since Λ′ (ρ) = 1ϕ′′(Λ(ρ))

, then for sufficiently large ϕ′′ we have Λ′ (ρ)→ 0, which impliesq′ (k) → ∂q

∂k. Therefore it is sufficient to prove that λ′ (k) < 0 implies ∂q

∂k< 0. Note

that:

(A− q) 1

s+ λ (1− θ∗)

1ˆ

θ∗

θ∗ − θ[s+ λ (1− θ)]2

e1 (θ) dθ

< 0 <

1ˆ

θ∗

1

[s+ λ (1− θ)]2e1 (θ) dθ

1ˆ

θ∗

θ − θ∗

[s+ λ (1− θ)]2e1 (θ) dθ

Rearranging implies that:

qλ´ 1θ∗

1−θ[s+λ(1−θ)]2 e1 (θ) dθ

1−θ∗s+λ(1−θ∗) (A− q) +

´ 1θ∗

θ−θ∗[s+λ(1−θ)]2 e1 (θ) dθ

<qλ´ 1θ∗

1[s+λ(1−θ)]2 e1 (θ) dθ

(A− q) 1s+λ(1−θ∗)

(68)

Assume λ′ (k) < 0. By the argument in part 2a, this requires ∂ρ∂k< 0, so using (64):

e1 (θ∗) <qλ´ 1

θ∗1−θ

[s+λ(1−θ)]2 e1 (θ) dθ

1−θ∗s+λ(1−θ∗) (A− q) +

´ 1

θ∗θ−θ∗

[s+λ(1−θ)]2 e1 (θ) dθ

Together with (68), this implies

e1 (θ∗) <qλ´ 1

θ∗1

[s+λ(1−θ)]2 e1 (θ) dθ

(A− q) 1s+λ(1−θ∗)

and therefore, using (59), ∂q∂k< 0.

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