Journal of Mathematical Economics 45 (2009) 361–375

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Journal of Mathematical Economics

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Short-selling restrictions, takeovers and the wealth of long-runshareholders

Marcelo Pinheiro ∗

Cornerstone Research, 1919 Pennsylvania Avenue NW, Suite 600, Washington, DC 20006,United States

a r t i c l e i n f o

Article history:Received 31 January 2008Received in revised form 16 March 2009Accepted 16 March 2009Available online 27 March 2009

Keywords:Short-sellingTakeovers

a b s t r a c t

In this paper we consider a situation in which a firm may be able to influence the investors’ability to short-sell its stock. We analyze the effect short-selling restrictions have on themarket price and the subsequent effect generated on the market for corporate control.More precisely, we argue that short-selling restrictions may lead to exclusion of pessimisticbeliefs and may therefore inflate prices. Thus, if a company is poorly managed and has astock with strong short-selling restrictions, a profitable takeover will not emerge becauseof the high stock price. The raider may not have the incentives to acquire the company as itsprice will be above its fundamental value, conditional on takeover, even accounting for thepotential benefits of takeover. We then argue that such effects are detrimental to long-runshareholders and that a value-maximizing strategy is to have a stock with no short-sellingrestrictions.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

In this paper we are interested in the relationship between short-selling restrictions, overvalued equity, managerialdecision-making, takeovers and the welfare of long-run shareholders.1

We first consider a simple model in which a firm may be able to influence the investors’ ability to short-sell its stock, byinfluencing the supply and demand (e.g., initially choosing with whom to place convertible securities—placing with hedgefunds would increase demand, as they hedge their positions, and placing with other institutions would increase supplyand decrease demand). We analyze the effect short-selling restrictions have on the market price and the subsequent effectgenerated on the market for corporate control. More precisely, we argue that short-selling restrictions may lead to exclusionof pessimistic beliefs and may therefore inflate prices (as in Miller, 1977). Thus, if a company is in a poor situation (be it becauseof the economy or because the manager is doing something wrong) and has a stock with strong short-selling restrictions, aprofitable takeover will not emerge because of the high stock price. The raider may not have the incentives to acquire thecompany as its price will be above its fundamental value, conditional on takeover, even accounting for the potential benefitsof takeover. We then argue that such effects are detrimental to long-run shareholders and that a value-maximizing strategyis to have a stock with no short-selling restrictions.

In another, more elaborate, model we allow for additional effects through direct managerial decision-making. I.e., weenhance the initial model by allowing the manager to choose an effort level. So, now we make a clear distinction between

∗ Tel.: +1 609 2 73 1069.E-mail address: marcelo@alumni.princeton.edu.

1 The ideas in this paper are related to, among others, the following articles: Bebchuk (2003); Cohen et al. (2007); Diamond and Verrecchia (1987); Duarteet al. (2006); Ferreira and Laux (forthcoming); Harrison and Kreps (1978); Jensen (2004); Lamont (2004) and Scheinkman and Xiong (2003).

0304-4068/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.jmateco.2009.03.004

362 M. Pinheiro / Journal of Mathematical Economics 45 (2009) 361–375

the situation that a firm is poorly managed and the economy is doing well vis-à-vis a situation where the manager is doingwell but the economy is doing poorly and hence the company is suffering for reasons outside management’s control. Themanager’s effort choice is costly and depends on his compensation package. Effort affects the long-run expected value of thefirm, and therefore may also affect prices. We then analyze the model trying to understand how effort choice, takeovers andshort-selling constraints interact to generate overvalued equity in the short-run, less takeovers and lower long-run valuesof the firm.

As a final note the reader should be aware that, even though we write and analyze the model in terms of takeovers, thereis nothing special about this. One could think about an alternative interpretation where instead of a takeover what happensis that the board fires the manager when it appears profitable to do so. I.e., the board would replace the manager wheneverthe expected value generated by his replacement is higher than what the market currently assess as this manager’s addedvalue, which is the same condition under which takeovers may happen.

2. Simple model

We start with a simple model where we abstract from managerial decision-making. We do this just to stress the effectsthat short-selling constraints may have on prices, takeovers and the welfare of long-run shareholders.

2.1. Company

Imagine a situation where a firm can be either well managed and have positive prospects or poorly managed and havenegative prospects. What we have in mind is a situation where both the manager’s ability as well as the stochastic nature ofthe economy may influence firm’s value. For simplicity we model this by assuming that with probability p the firm is overallgood – the manager is doing a good job and the economy is strong – and with complementary probability (1 − p) the firm isbad but can be improved—manager is performing poorly and the economy is weak.

When the firm is bad, firm value can be enhanced by replacing management, but not to the point of turning it into a goodcompany, since the economy is not strong. If the company is good replacing the manager does not enhance or destroy value.This idea translates into the following assumptions. If the company is good, then its expected value is equal to �G . Notice thatthis is independent of who is managing the firm. If the company is bad then its expected value is equal to �M if managementis replaced and �B otherwise. Furthermore, we assume that �G > �M > �B.

We model the replacement of the manager as a takeover. More precisely, we assume that with probability q a raider existsand with complementary probability no such agent is present. If a raider exists than with probability t he decides to takeoverthe firm and with probability 1 − t he does not. Notice that t is an endogenous variable that will depend on market priceat the time of takeover. Clearly, takeover only takes place if the price is below the expected value of the firm conditionalon takeover, i.e., �M . The situation of this firm is summarized in the picture below. Finally, we assume that in the long-runthere cannot be a separation between price and fundamental value. Therefore, we assume that, at some point, all investorslearn the true state of the economy and incorporate such information on their beliefs and, hence, prices converge to a firm’sexpected fundamental value (Fig. 1).

The situation described above is the same for all firms in the economy. The only difference across firms is the cost toinvestors of short-selling its stock. We take a very simplified approach by assuming that a company’s stock either can befreely short-sold or not at all. So, one company has a stock traded in a completely perfect market , while the other company’sstock is traded in a market where investors are short-sale constrained.

2.2. Short-selling restrictions, investors and market prices

We model the easiness of short-sales as a choice variable for the company, so the board may take actions that lead thefirm to have a stock that is freely traded or one that is traded with short-sale restrictions.

Independently of the type of market we have two types of investors (in addition to the raider): informedtraders/arbitrageurs and irrational/overconfident traders. The informed traders know whether the company is good or badand understand that a raider may takeover the company and replace management, therefore enhancing value. We alsoassume that these traders, as a group, have unlimited wealth. Therefore, the informed traders believe that the company’sexpected value is �G if the state is good and (1 − q)�B + q(1 − t)�B + qt�M if the state is bad; where the last expressionincorporates the fact that if the company is bad a takeover might take place. The irrational traders are uninformed and do notanticipate takeovers, and we assume that they always assign � as the company’s expected value, with �G > � > �M > �B.

We also assume that these irrational traders are small relative to the informed traders. The idea here is that these tradersare not big enough to generate inefficiency if the informed traders are able to trade freely. So, whenever informed traderstrade freely price equals fundamental values, i.e., the beliefs of these traders

We are now ready to analyze market prices and the decision to takeover under perfect markets and under short-sellingrestrictions.

M. Pinheiro / Journal of Mathematical Economics 45 (2009) 361–375 363

Fig. 1. Timeline of the model.

2.2.1. Perfect marketsWith no short-selling constraints we have that the company’s stock price is either PSS

G or PSSB , where superscripts SS

represent the situation where short selling is allowed, and subscripts stand for the good, G, or the bad, B, state. With theassumptions discussed above we have that

PSSG = �G;

PSSB = (1 − q)�B + q(1 − t)�B + qt�M.

Notice that if the informed trader conjectures that a takeover will take place whenever there is a wealthy raider, i.e., t = 1,then

PSSB = (1 − q)�B + q�M.

With this conjecture then PSSB < �M and therefore it is optimal for the raider to takeover the firm and the informed traders

conjecture is verified in equilibrium.So, when the state is bad and a raider exists he takes over the firm enhancing its value.

2.2.2. Short-selling restrictionsIf there are short-selling restrictions then equilibrium prices will differ slightly from the ones described above.In the good state everything is as before, since informed traders are trading and hence

PSSRG = PSS

G = �G,

where the superscript SSR stands for short-selling restrictions. However, in the bad state prices will diverge from the com-pany’s expected fundamental value since informed traders cannot arbitrage away such inefficiency. So, PSSR

B = �. With theseprices informed traders would like to short-sell the stock, since

E[Value] = (1 − q)�B + q(1 − t)�B + qt�M < PSSRB ,

but they are prohibited from doing so.In these circumstances there is no takeover, even if the firm is in the bad state, since PSSR

B = � > �M . So, the market forcorporate control is not able to function properly and an opportunity to enhance firm (and hence shareholder) value is lost.

2.3. Shareholders’ welfare

We just saw that short-selling restrictions may render the market for corporate control inept to deal with poor manage-ment. That is, if a firm has a stock that is traded under short-selling restrictions then a profitable takeover may never happen.The main idea behind this result is that under short-selling constraints the beliefs of the more pessimistic traders may not

364 M. Pinheiro / Journal of Mathematical Economics 45 (2009) 361–375

be reflected in prices Therefore, even though a manager should be replaced through a hostile takeover he will not, since thistakeover may not be profitable due to inflated prices that do not reflect fundamental values. Ultimately, we expect long-runshareholders to pay the price for this inefficiency.

In this simplified model it is easy to see that the ex-ante expected long-run value of the firm is equal to

VSS = p�G + (1 − p)(q�M + (1 − q)�B)

if short-selling is allowed and

VSSR = p�G + (1 − p)�B

if there are short-selling restrictions in place. Clearly VSS > VSSR and long-run shareholders are, ex-ante, better off with noshort-selling restrictions.

3. More elaborate model

In this section we modify the previous model to directly allow for managerial decision-making. We discuss a more generalversion of this second model in Appendix

The timing and structure of the model are similar to the one previously discussed with the exception that we now allowthe manager to exert effort. In the present framework the expected firm’s value depends on the overall state of the economyand on the manager’s effort choice. The state of the economy may be either good or bad and in either case the managerchooses an effort level e ∈ [e, e] that affects the firm’s expected final value. For now, we take the effort choice as given anddescribe the model in more detail, afterwards we specify the manager’s objective function and his choice. Finally, we alsoallow for two firm types: with or without short-selling constraints. We derive the equilibrium for each firm type and comparethe two situations and then allow for an endogenous choice of constraints (Fig. 2).

The picture briefly describes the timing of the model. We allow prices to be functions of the effort choice, since effortaffects expected values (in the absence of takeovers). So, PG(e) is the price that obtains if the economy is good and themanager exerts effort e. As a matter of fact, prices depend on expected values and these depend on effort, so we could havewritten PG(�G(e)) but chose not to in order to save on notation.

We make the following assumptions concerning the relationship between effort and expected values.

1.∂�j(e)

∂e≥ 0 and

∂2�j(e)

∂e2 ≤ 0 for j = G, B, �G(e) > �B(e), ∀e and �MG > �MB;

2. ∃e1 ≤ e2 such that �G(e1) = �MG and �B(e2) = �MB;3. �G(e) ≥ �MB.

Fig. 2. New model.

M. Pinheiro / Journal of Mathematical Economics 45 (2009) 361–375 365

The first set of assumptions just describes how effort affects expected values. The more effort the manager exerts thehigher the expected value of the firm, but expected values grow at a decreasing rate with effort, and the better the economythe higher the expected values of the firm, for a given effort level. The second assumption guarantees that there are effortlevels such that takeovers are not profitable if the manager’s choice is above them, and it is easier to avoid takeovers if theeconomy is good. And, the last assumption just says that if a trader believes that the economy is good, then, no matter whatthe manager does, if a takeover can only generate �MB then it is not profitable (the importance of this assumption will beclear later).

We also assume that the effort choice is observable and that, as before, there are 2 types of traders: rational and irra-tional/myopic/optimist. The rational trader fully understands the model and knows whether the economy is good or bad,however the irrational trader always thinks the economy is good. Therefore traders’ beliefs are

�I(e) = qtI�MG + (1 − qtI)�G(e),always; and

�R(e) = qtR�Mj + (1 − qtR)�j(e),withj = GorB,

where superscript I stands for irrational and R for rational. (Notice that we allow the traders to hold different beliefs concerningthe decision of the raider, e.g., the rational trader, given his beliefs, may anticipate a takeover while the irrational trader maysee the takeover as unprofitable.) And there are also long-run shareholders that do not trade and are only interested inmaximizing expected long-run firm value. In the long-run the value of the firm converges to the true value, i.e., the effect ofthe irrational traders disappear and long-run shareholders expect to get the objective long-run value of the firm. So, short-rundeviations of prices from long-run expected values only affect long-run shareholders in as much as they affect manager’sand raiders’ decisions.

We are now ready to analyze the model. We start with the case of short-selling constraints and then move to the casewhere markets are perfect.

3.1. Short-selling constraints

If the stock of this company is subject to short-selling constraints we know that its price only reflects the most optimisticbelief. We characterize this equilibrium below.

In case the economy is good, both traders hold the same beliefs, hence, prices are such that:

• If e < e1, then �G(e) < �MG , so, if traders conjecture that a takeover occurs, the price equals

PG(e) = q�MG + (1 − q)�G(e) < �MG,

and the conjecture is verified.• And, if e ≥ e1, then �G(e) ≥ �MG , traders conjecture t = 0 and

PG(e) = �G(e) > �MG,

and again the conjecture is verified.

In case the economy is bad, traders hold different beliefs, with �I(e) > �R(e), ∀e whenever tI = tR. So, whenever thetraders agree on the takeover decision, prices are given by the irrational traders beliefs. Otherwise, two cases may happen:

1. tI = 0 and tR = 1;2. tI = 1 and tR = 0.

Case 1 can only happen if e1 < e < e2. In this case then:

�I(e) = �G(e) > q�MB + (1 − q)�B(e) = �R(e),

where the inequality follows from the assumption that �G(e) ≥ �MB, ∀e and �G(e) > �B(e). So, under Case 1 we also havethat prices reflect only the beliefs of the irrational trader. For Case 2 to happen we need e < e1 (for tI = 1 to be justified), butthen if tR = 0 we have

�R(e) = �B(e) < q�MG + (1 − q)�G(e) = �I(e),

since �B(e) < �G(e) < �MG, ∀e < e1 and again prices only reflect the beliefs of the irrational trader. We summarize thisdiscussion in the proposition below.

366 M. Pinheiro / Journal of Mathematical Economics 45 (2009) 361–375

Proposition 1. If the state of the economy is bad, then prices reflect only the beliefs of the irrational traders, since �I(e) >�R(e), ∀e for any (tI, tR) pair. Therefore, prices are given by

PB(e) = �I(e) ={

�G(e),if e > e1;

q�MG + (1 − q)�G(e),if e ≤ e1.

And, it the economy is good we have PG(e) = PB(e), ∀e.

Proof. We need only verify that traders’ beliefs are consistent. If e > e1 then PB(e) > �MG > �MB and tI = tR = 0 is consistentand �I(e) = �G(e) > �B(e) = �R(e). If e ≤ e1 then tI = 1 is consistent with the irrational trader’s beliefs. But, in equilibriumthe irrational trader is wrong and t = 0 since PB(e) > �MB (because �MG > �MB and by assumption 3 above �G(e) > �MB, ∀e)so tR = 0 is consistent and �R(e) = �B(e) < q�MG + (1 − q)�G(e) = �I(e). �

The results discussed above clearly imply that takeovers never occur when we have short-selling constraints. If theeconomy is good, then takeovers occur if e ≤ e1 and a raider is present. More precisely, whenever e ≤ e1 the irrationalsbelieve that takeovers occur (conditional on the appearance of the raider). However, if the economy is bad these beliefs arenot correct (and are not shared by the rationals) and no takeovers occur. If the economy is good the rationals also believethat a takeover occur, and it indeed does when the raider is present.

We now proceed to analyze prices when there are no short-selling constraints.

3.2. Frictionless markets

By the same reasoning used in the original model, we know that, in case the stock is freely traded, prices are always “right”,reflecting only the rational trader’s beliefs. In frictionless markets there is no departure of prices from long-run values, evenif there is disagreement among traders. For prices to diverge from long-run values we need both imperfect markets andheterogeneous beliefs. Therefore, we have the following proposition.

Proposition 2. (Equilibrium prices) If the economy is good we have

PG(e) ={

�G(e),if e > e1;

q�MG + (1 − q)�G(e),if e ≤ e1.

If the economy is bad

PB(e) ={

�B(e),if e > e2;

q�MB + (1 − q)�B(e),if e ≤ e2.

Proof. We just need to verify the beliefs about takeover. If the economy is good (bad), and given the definition of e1 (e2),the rational trader anticipates that takeover only happens if e ≤ e1(e2). And, it is indeed the case that if e ≤ e1, PG(e) < �MG

and if e ≤ e2, PB(e) < �MB, so takeover is optimal under these circumstances and the beliefs are verified in equilibrium. �

3.3. Implications

Given our previous analysis we see that if effort is exogenously determined (or if the manager chooses the same levelindependent of the presence of short-selling constraints) takeovers are more likely under frictionless markets, since in thiscase takeovers can occur both under good or bad states of the world, while with constraints takeovers can only occur underthe good state of the world. Since takeovers are value enhancing, i.e., a necessary condition for takeovers is that it increasesthe expected value of the firm, we see that, in a partial equilibrium framework, short-selling constraints cannot be optimalfrom the point of view of long-run shareholders. This result is analog to the one discussed before.

The problem with this argument is that effort is a choice variable of the manager and he may act differently dependingon the presence or absence of short-selling constraints. To fully understand these effects we must allow for an objectivefunction for the manager and model his choices. We do this next.

3.4. Manager’s choice

Assume that the manager’s payoff takes the following form: w + f (P) + C − g(e) in case of no takeover and w − g(e)otherwise. Where we think of w as a fixed wage, C as a private benefit of continuation, f (P) as his compensation package/bonusthat only vests after the period where takeovers may happen and g(e) is the cost of effort. Furthermore, we assume that thefunctions satisfy: f ′ > 0, f ′′ ≥ 0 and g′ > 0, g′′ ≥ 0. By this formulation, it should be clear that the effort choice is made beforethe takeover period.

At this point we make some simplifying assumptions in order to completely characterize the actions of the agents andanalyze the properties of the equilibrium. Without these assumptions the model is not very tractable. However, we do presentsome general results in Appendix; results that hold with much milder assumptions than the ones we now make.

M. Pinheiro / Journal of Mathematical Economics 45 (2009) 361–375 367

We further assume that f (P) = P, g(e) = (e2/2), and q = 1, so that raiders always exist. Furthermore, �G(e) = ke + G and�B(e) = ke + B. Where we also impose the condition that G − B > �MG − �MB > 0, i.e., for any given effort choice, the gainsthat arise due to a better economy are higher under the incumbent management than under the raider’s management.And, finally, let e = 0 and �MG > G > �MB. These two last assumptions together imply that, if prices reflect beliefs that theeconomy is good (P = G), with minimum effort (zero) takeovers are profitable only if the economy is indeed good. I.e., if theeconomy is bad but the prices are given by the irrational trader’s beliefs, then takeovers do not happen. But, if the economyis good and the manager shirks then takeovers will happen.

Following our previous definitions and using the assumptions above we have that e1 = (�MG − G/k) and e2 = (�MB − B/k)so

e2 − e1 = �MB − B − �MG + G

k= (G − B) − (�MG − �MB)

k> 0

as required.Since markets and decisions function in the same way if the economy is good, there is no reason to concentrate our

analysis there. Therefore, we concentrate on the bad economy results (mentioning the good economy when needed) andcompare the cases with and without short-selling restrictions. Later on, we analyze the whole game using the fact that mostof the decisions when the economy is good resemble the decisions when the economy is bad and the firm’s stock facesshort-selling restrictions.

With the assumptions above in place and using the results derived previously we know that if there are short-sellingrestrictions then the firm’s stock price when the economy is bad is given by

PB,SS(e) = �I(e) ={

ke + G,if e > e1;

�MG ,if e ≤ e1,

and the manager’s problem is

maxePaySS(e) = PB,SS(e) + w + C − e2

2.

So,

eSS = ∂PB(e)∂e

={

k,if e > e1;

0,otherwise,

where superscript SS stands for the presence of short-selling constraints. If k < e1 even if he chooses k he cannot make theirrational traders believe that there is no takeover, then e∗ = 0. If k > e1 he might be able to convince the irrational trader thathe avoids takeover. So e∗ = 0 and e∗ = k are possible and he chooses the level that gives him the most payoff. If he chooses khe gets

PaySS(k) = k2 + G + C + w − k2

2= G + C + w + k2

2,

and if he chooses zero he gets (recall that the irrational trader thinks there is takeover, but none happen in equilibrium)

PaySS(0) = �MG + C + w − k2

2.

Clearly,

PaySS(k) − PaySS(0) = G + k2 − �MG > 0,

since G + k2 = �G(k) > �MG , if k > e1, so choosing k is always optimal.If, on the other hand, there are no short-selling constraints, then

PB,NoSS(e) = �R(e) ={

ke + B,if e > e2;

�MB,if e ≤ e2,

and the manager’s problem is

maxePayNoSS(e) = PB,NoSS(e) + w + C − e2

2.

So,

eNoSS = ∂PB(e)∂e

={

k,if e > e2;

0,otherwise,

368 M. Pinheiro / Journal of Mathematical Economics 45 (2009) 361–375

where superscript NoSS stands for the absence of short-selling constraints. If k < e2, then e∗ = 0. If k > e2, he chooses k andavoids takeovers.

The following proposition summarizes this discussion.

Proposition 3. (Effort choice) If the economy is in the bad state then we have

eSS ={

k,if k > e1

0,otherwise

and

eNoSS ={

k,if k > e2;

0,otherwise.

In the good state the behavior of the manager is like the one in the bad state with short-selling constraints and takeovers neverhappen with short-selling constraints in the bad state, but may happen in the good state.

3.5. Analyzing the effects on markets and managerial decisions

In this section we analyze the effects that short-selling restrictions have on markets and managerial decisions. Moreprecisely, what is the effect on effort choice, probability of takeovers, prices and long-run welfare of shareholders.

3.5.1. Effects on takeoversLet us start by analyzing the effects that short-selling restrictions may have on the probability that a takeover occurs in

equilibrium.To start the analysis notice that the previous proposition has the following as a corollary.

Corollary 1. (Effort choice and takeovers) Three situations are possible:

1. k > e2 > e1, in which case takeovers never happen and the manager always exerts the same effort (eSS = eNoSS = k),2. e2 > k > e1 in which case takeovers only happen if markets are frictionless and the economy is bad and he exerts more effort

if there are short-selling constraints (eSS = k and eNoSS = 0),3. e2 > e1 > k, in which case takeovers only happen if markets are frictionless and the economy is bad and he shirks (eSS =eNoSS =

0).

The corollary above shows that takeovers are more likely under frictionless markets. There are instances where takeoverhappens when markets are frictionless, but does not if short selling is prohibited.

In order to do a more general and interesting analysis let us now assume that the marginal benefit of effort is random.More precisely, what we have in mind is a situation where, ex-ante, k is random. The idea being that at the very beginningof times we cannot know how much the expected final value of the firm depends on effort. So, we do not know what is themarginal benefit of effort, i.e., how much the effort choice affects the expected long-run value of the firm. Or, alternatively,different firms may have different k s and we may see it as random if we are looking at a cross-section of firms.

For the rest of this study assume that k is random and let F be its c.d.f. As it will become clear as we go through the resultsand discussions, no specific assumptions have to be made concerning this distribution, other than what is mathematicallyrequired of a cumulative distribution function.

Under this additional assumption we can proof a proposition that strengthens and clarifies the ideas discussed above.

Proposition 4. (Probability of takeover) Let �NoSS be the ex-ante probability of takeover if markets are frictionless (similarlyfor �SS) and �J,m be the probability of takeover in state J = G, B with m = NoSS, SS. Then,

�NoSS,G = �SS,G ≡ �G, �SS,B = 0

since when the economy is good everything is the same independent of the assumption concerning short-selling and there is notakeover when the economy is bad and there are short-selling restrictions. So,

�NoSS = p�G + (1 − p)�NoSS,B,

�SS = p�G + (1 − p)�SS,B,

where �NoSS,B = F(R2) > 0 = �SS,B, F(R1) = �G . Takeovers are more likely under frictionless markets.

Proof. We know that if there are short-selling constraints then takeovers never happen. And if the economy is good thenthey happen if k < e1, i.e., if

k <�MG − G

k⇔ k <

√�MG − G ≡ R1.

M. Pinheiro / Journal of Mathematical Economics 45 (2009) 361–375 369

So, takeovers happen with probability F(R1) in good times. If markets are frictionless and the economy is bad takeovershappen if k < e2, i.e., if

k <�MB − B

k⇔ k <

√�MB − B ≡ R2 > R1,

so F(R1) < F(R2) and takeovers are more likely under frictionless markets in the bad state than under the good state, and0 < F(R2), so takeovers are more likely than with constraints. The expressions can also be written as

�NoSS = pF(√

�MG − G) + (1 − p)F(√

�MB − B),

�SS = pF(√

�MG − G) + (1 − p)0,

so clearly �NoSS − �SS = (1 − p)F(√

�MB − B) > 0. �

There are two effects that drive the probability of takeover down when we have short-selling restrictions. Both of them arelinked to the fact (discussed below) that prices are inflated under short-selling restrictions. First, prices are higher becausewith short-selling constraints only the beliefs of the optimists are incorporated. With higher prices we have less takeovers,since the value-added of a takeover is inversely related to prices. But, then, since prices are higher, avoiding takeovers iseasier so the manager exerts effort more often to convince the irrational traders that he is avoiding takeovers which leads toa second “positive” effect on prices and an even lower probability of takeover.

3.5.2. Effects on shareholder welfareWe argued before that there is a reduced probability of takeovers, if takeovers only happen when profitable then we

might expect that long-run shareholders may suffer. The following proposition makes this idea more formal.

Proposition 5. (Long-run shareholders’ welfare) The ex-ante long-run expected value of the firm is maximized with frictionlessmarkets. So, long-run shareholders are better of without short-selling restrictions.

Proof. Notice that the ex-ante expected value of the firm is

LVSS = p[�G�MG + (1 − �G)�SSG ] + (1 − p)�SS

B ;

LVNoSS = p[�G�MG + (1 − �G)�NoSSG ] + (1 − p)[�NoSS,B�MB + (1 − �NoSS,B)�NoSS

B ],

where

�SSG = E(�G(k)|k > R1) = E(k2 + G|k > R1) = �NoSS

G

�SSB = E(�B(k)) = E(k2 + B)

�NoSSB = E(�B(k)|k > R2) = E(k2 + B|k > R2)

Alternatively,

LVSS = p[�G�MG + (1 − �G)�SSG ] + (1 − p)

[F(R2)�B(k0) + (F(R2) − F(R1))�B(k1) + (1 − F(R2))�B(k2)

];

LVNoSS = p[�G�MG + (1 − �G)�SSG ] + (1 − p)

[F(R1)�MB + (F(R2) − F(R1))�MB + (1 − F(R2))�B(k2)

],

where with a strong abuse of notation we let �B(k0) = E(�B(k)|R1 > k), �B(k1) = E(�B(k)|R1 < k < R2) and �B(k2) =E(�B(k)|k > R2).

So,

LVSS − LVNoSS = (1 − p)[F(R1)�B(k0) + (F(R2) − F(R1))�B(k1) + (1 − F(R2))�B(k2)] − [F(R1)�MB + (F(R2) − F(R1))�MB

+ (1 − F(R2))�B(k2)] = (1 − p)F(R1)(�B(k0) − �MB) + [F(R2) − F(R1)](�B(k1) − �MB) < 0

where the inequality follows since R2 > R1 and

�B(k0) < �B(k1) = E(�B(k)|R1 < k < R2) < �MB

because

E(�B(k)|R1 < k < R2) = E(k2 + B|R1 < k < R2) < E((R2)2 + B|R1 < k < R2) = E((√

�MB − B)2 + B|R1 < k < R2)

= E(�MB − B + B|R1 < k < R2) = �MB

Therefore, LVSS < LVNoSS and the long-run shareholders are better off without short-selling restrictions (i.e., in a frictionlessmarket). �

370 M. Pinheiro / Journal of Mathematical Economics 45 (2009) 361–375

With short-selling constraints long-run value is smaller because there are less takeovers, and takeovers only occur ifprofitable in terms of long-run value. Hence, profitable takeovers are avoided more often if there are short-selling constraints.So, long-run shareholders suffer.

3.5.3. Effects on pricesFinally, the idea that short-selling restrictions may lead to overvalued companies is not new and has been discussed

extensively in the literature. But, to the best of the authors’ knowledge, this is the only model that allows for effort choiceand takeovers as well as short-selling constraints. So, let us discuss the potential intertwined effects of considering all theseissues together. We do that in the following proposition.

Proposition 6. (Overvaluation) The ex-ante expected value of the company’s stock price at the time of trading is higher if thereare short-selling restrictions.

Proof. With notation similar to the one developed above we have that

PSS = p[�G�MG + (1 − �G)�SSG ] + (1 − p)[�SS,B�MG + (1 − �SS,B)�SS

G ];

PNoSS = p[�G�MG + (1 − �G)�SSG ] + (1 − p)[�NoSS,B�MB + (1 − �NoSS,B)�NoSS

B ],

Alternatively,

PSS = p[�G�MG + (1 − �G)�SSG ] + (1 − p)[F(R1)�MG + (F(R2) − F(R1))�G(k1) + (1 − F(R2))�G(k2)];

PNoSS = p[�G�MG + (1 − �G)�SSG ] + (1 − p)[F(R1)�MB + (F(R2) − F(R1))B + (1 − F(R2))�B(k2)],

So,

PSS − PNoSS = (1 − p){[F(R1)�MG + (F(R2) − F(R1))�G(k1) + (1 − F(R2))�G(k2)]

− [F(R1)�MB + (F(R2) − F(R1))B + (1 − F(R2))�B(k2)]}= (1 − p)F(R1)(�MG − �MB) + [F(R2) − F(R1)](�G(k1) − �MB)

+ (1 − F(R2))[�G(k2) − �B(k2)] > 0,

where the inequality follows because every term inside curly brackets is positive:

1. The first term, F(R1)(�MG − �MB), is positive because �MG > �MB;2. The second because

�G(k1) = E(�G(k)|R1 < k < R2) = E(k2 + G|R1 < k < R2) > �MB;

3. And, finally, the third term is positive since

�G(k2) − �B(k2) = E(�G(k)|k > R2) − E(�B(k)|k > R2) = G − B > 0.

Therefore, PSS > PNoSS and we see that the ex-ante expected stock price of this company is higher in the presence ofshort-selling constraints than under frictionless markets. �

The result of the proposition is due to two effects that interact with each other. Prices are higher because with short-sellingconstraints only the beliefs of the optimists are reflected on the prices. Since prices are higher, convincing the irrationalsthat takeovers are avoided is easier and the manager exerts effort more often, since when he convinces the irrationals thattakeovers are avoided prices increase, what is better for him. This excess effort leads to even more inflated prices.

3.5.4. Effects on managerial decisions and welfareFinally one last result speaks to the expected effort choice of the manager and his welfare. How do managers respond to

the overpricing? Do they exert more or less effort? (More importantly, why?) Is the extra effort (if any) good or bad? And,finally, are managers better off with or without short-selling restrictions?

The following two proposition deals with these questions.

Proposition 7. (Managerial effort) The ex-ante expected value of the manager’s effort choice is higher under short-sellingrestrictions than with frictionless markets.

M. Pinheiro / Journal of Mathematical Economics 45 (2009) 361–375 371

Proof.

eSS = p[�G0 + (1 − �G)eSSG ] + (1 − p)[�SS,B0 + (1 − �SS,B)eSS

B ];

eNoSS = p[�G0 + (1 − �G)eNoSSG ] + (1 − p)[�NoSS,B0 + (1 − �NoSS,B)eNoSS

B ],

eSSG = E[k|k > e1] = E[k|k > R1] = eSS

B = eNoSSG

and

eNoSSB = E[k|k > e2] = E[k|k > R2]

Therefore,

eSS = p(1 − �G)eSSG + (1 − p)[(1 − �SS,B)eSS

G ];

eNoSS = p(1 − �G)eSSG + (1 − p)(1 − �NoSS,B)eNoSS

B ,

Alternatively,

eSS = p(1 − �G)eSSG + (1 − p)[(F(R2) − F(R1))e1 + (1 − F(R2))e2];

eNoSS = p(1 − �G)eSSG + (1 − p)[(F(R2) − F(R1))0 + (1 − F(R2))e2],

where we let e1 = E(k|R1 < k < R2) and e2 = E(k|k > R2). So,

eSS − eNoSS = (1 − p)(F(R2) − F(R1))(e1 − 0) > 0,

so, ex-ante, we expect the manager to exert more effort if there are short-selling restrictions than if markets arefrictionless. �

The idea of the proof is that it is easier to convince that takeovers are avoided when markets have imperfections, i.e., whenthere are short-selling constraints prices are inflated, for a given effort. So the manager will exert effort more often; thereare more circumstances under which it is optimal to exert effort and have the irrationals believe that takeovers are avoided(which leads to higher prices). But, as shown before, long-run shareholders suffer, the firm has a lower long-run expectedvalue. So, equity will be overvalued in the short-run and we would observe underperformance in the long-run for firms thathave their stock traded in a market with short-selling constraints.

Proposition 8. (Managerial welfare) Managers are better off working for a company that has a stock subjected to strongshort-selling restrictions.

Proof. First, define the following:

Pay0 = E[�G(k) + w + C − k2

2|R1 > k]

Pay1 = E[�G(k) + w + C − k2

2|R1 < k],

Pay2 = E[�G(k) + w + C − k2

2|R1 < k < R2],

Pay3 = E[�G(k) + w + C − k2

2|R2 < k],

Pay4 = E[�B(k) + w + C − k2

2|R2 < k],

and let �SS and �NoSS be the manager’s ex-ante expected payoff under short-selling restrictions and frictionless markets,respectively. Then we can write

�SS = p[�Gw + (1 − �G)Pay1] + (1 − p)[F(R1)Pay0 + (F(R2) − F(R1))Pay2 + (1 − F(R2))Pay3],

�NoSS = p[�Gw + (1 − �G)Pay1] + (1 − p)[F(R1)w + (F(R2) − F(R1))w + (1 − F(R2))Pay4].

372 M. Pinheiro / Journal of Mathematical Economics 45 (2009) 361–375

And,

�SS − �NoSS = (1 − p)[F(R1)(Pay0 − w) + (F(R2) − F(R1))(Pay2 − w) + (1 − F(R2))(Pay3 − Pay4)],

= (1 − p)[F(R1)(E[�G(k) + C − k2

2|R1 > k]) + (F(R2) − F(R1))E[�G(k) + C − k2

2|R1 < k < R2]

+ (1 − F(R2))E[�G(k) − �B(k)|R2 < k]]

= (1 − p)[F(R1)E[G + C + k2

2|k < R1] + (F(R2) − F(R1))E[G + C + k2

2|R1 < k < R2]

+ (1 − F(R2))E[�G(k) − �B(k)|R2 < k]]

< 0,

where the inequality follows since G + C + (k2/2) > 0 and �G(e) > �B(e), ∀e. Therefore, the manager is better off with short-selling constraints in place. �

This last proposition shows that if the manager could choose the level of short-selling restriction that the company’s stockfaces he would choose to have short-selling restriction prohibited. Intuitively, not only can he enjoy inflated prices (due toshort-selling) but he also enjoys more of the benefits of continuation, since there are less takeovers. And, these advantagesmore than compensate him for the extra effort he exerts in order to avoid the takeovers.

In conclusion we have the following corollary.

Corollary 2. If the level of short-selling restrictions is under the firm’s control then we would observe firms with strong shareholderpower having their stock traded in markets that are close to frictionless, and companies with weak shareholder power would facea lot of short-selling costs/constraints.

Proof. This result follows from the fact that the initial (long-run) shareholders prefer no short-selling restriction while themanager prefers to have short-selling constraints in place. �

3.5.5. Discussion and empirical implicationsPutting all the results together we see that, because prices are inflated with short-selling constraints, the manager exerts

more effort, this, by its turns, inflates prices even further and results in less profitable takeovers, reducing the expected long-run value of the firm and hurting shareholders. Furthermore, the manager himself is better off working for a company whosestock is traded in a market plagued by short-selling restrictions. Even though he exerts more effort, the lower probabilityof takeover, and the associated benefits of continuation (both private and through his compensation package), more thancompensates him.

If we allow short-selling restrictions to be endogenous then we would see firms with restrictions if the managers areallowed to choose and firms trading in frictionless markets if the long-run initial shareholders are the ones choosing. Thisresult can be tested by using a measure of shareholder power, as in Gompers et al. (2003), with the expected result beingthat firms in the so-called democracy portfolio should have less short-selling constraints.2

The results above are also directly related to Jensen’s costs of overvalued equity3: short-selling constraints leads toartificially inflated prices, this may render the market for control ineffective and may by its turn preserve the desire tothe manager to inflate prices, through inefficient exertion of effort. The manager has incentives to exert more effort, thisartificially inflates prices avoiding takeovers when they are profitable for long-run shareholders. So, even though the highereffort choice positively affects value, it does so to a lesser extent than what a takeover might have done. I.e., the managerexerts too much effort making prices too high and avoiding takeovers, the long-run expected value is lower because thetakeover would generate more value than the additional effort of the manager, but prices are higher. So prices are artificiallyinflated while long-run values show underperformance.

Acknowledgments

I would like to thank Jefferson Duarte, David Robinson and Deniz Igan for helpful comments and discussions. The materialdiscussed herein may not reflect the opinions of Cornerstone Research.

Appendix A. General results

In this appendix we study a more general model and show that some of our results are qualitatively the same under thismore general setting.

2 I highlight the implications of the endogenous case since I perceive these to be more interesting and novel. However, the exogenous restriction casecould also be tested. E.g., one could test if constraints on short-sales are associated with differing probabilities of takeover.

3 Jensen (2004).

M. Pinheiro / Journal of Mathematical Economics 45 (2009) 361–375 373

We do away with most of our parameterization and just revert back to the assumption that the manager’s payoff takesthe following form:

w + f (P) + C − g(e)

in case of no takeover and

(1 − q)[f (P) + C] + w − g(e)

otherwise. Where

f ′ > 0, f ′′ ≥ 0

g′ > 0, g′′ ≥ 0.

Now we also assume that f is concave with respect to e (it may be convex with respect to P):

∂2f (P(e))

∂P(e)2

(∂P(e)

∂e

)2

+ ∂f (P(e))∂P(e)

∂2P(e)∂e2

≤ 0.

So we need (∂2P(e∗No)/∂e2) ≤ 0, and

∂2f (P(e))

∂P(e)2

(∂P(e)

∂e

)2

≤∣∣∂f (P(e))

∂P(e)∂2P(e)

∂e2

∣∣.So, prices must be concave in effort, and, e.g., the compensation package must depend heavily on prices. Notice that thismust hold for both PG(e) and PB(e). Finally, we must also additionally assume that

∂f (PG(e))∂PG

∂PG(e)∂e

≥ ∂f (PB(e))PB(e)

∂PB(e)∂e

,

that is, the manager’s pay depends more on effort in good times than in bad times. When the economy is good, there is a lotthe manager can do to improve prices ((∂PG(e)/∂e) is big) and/or if he improves prices his compensation package moves alot ((∂f (PG(e))/∂PG) big). The idea of the latter effect being that you need to give him more incentives, otherwise he wouldlike to coast, since the economy is already good, without incentives, he may decide to shirk, so you give him extra incentivesto work harder. Basically, since it may be harder to make managers work when everything is already going well, then it maybe valuable to reward him more and make sure that he works even if the economy is already doing well.

We can get similar results even in this general model. For sake of brevity we concentrate on the bad economy case.Without short-selling constraints, in the bad state of the economy the manager gets

w + f (PB(e)) + C − g(e)

in case of no takeover and

(1 − q)[f (PB(e)) + C] + w − g(e),

otherwise. So, if

e∗No = argmaxew + f (PB(e)) + C − g(e) > e2

there is no takeover and the solution is e∗No. Conversely, if e∗No < e2 and

e∗∗No = argmaxe(1 − q)[f (PB(e)) + C] + w − g(e) < e2,

then the solution is e∗∗No and there is takeover with probability 1 − q. The payoffs from the compensation package and theprivate benefit of continuation are multiplied by 1 − q because, if he does not work hard enough (e < e2), then a raider existswith probability q and takeovers the firm (so the manager does not get the bonus or the benefit of continuation). Withcomplementary probability there is no raider so he gets his bonus and his private benefit.

Notice that e∗∗No < e∗No as long as f (.) is concave in e. Because

∂f (PB(e∗No))∂e

− ∂g(e∗No)∂e

= 0

so

(1 − q)∂f (PB(e∗No))

∂e− ∂g(e∗No)

∂e< 0

since f is concave in e (since G is convex). Hence, we must have e∗∗No < e∗No. Therefore, if e∗No < e2 then e∗∗No < e∗No < e2

and the solution is e∗∗No. So the problem always has a solution.

374 M. Pinheiro / Journal of Mathematical Economics 45 (2009) 361–375

If there are short-sale constraints, then the manager gets w + f (PG(e)) + C − g(e) in any case since there is no takeover.But recall that

PG(e) ={

�G(e),it irrational trader believes in no-takeover

q�MG + (1 − q)�G(e),it irrational trader believes in takeover

So, if

e∗SS = argmaxew + f (PG(e)) + C − g(e) > e1

there is no takeover in the view of irrationals and the solution is e∗SS . Conversely, if e∗SS < e1 and

e∗∗SS = argmaxef (q�MG + (1 − q)�G(e)) + C + w − g(e) < e1,

then the solution is e∗∗SS and in the mind of irrationals there is takeover with probability 1 − q, in reality there are no takeovers.Again we can show that these are the only two possible solutions since if e∗SS < e1 then e∗∗SS < e∗SS < e1 and the solution ise∗∗SS .

More importantly, since the manager’s pay depends more on effort in good times, i.e., as we assumed

∂f (PG(e))∂e

≥ ∂f (PB(e))∂e

,

then

0 = ∂f (PB(e∗No))∂e

− ∂g(e∗No)∂e

≤ ∂f (PG(e∗No))∂e

− ∂g(e∗No)∂e

,

and e∗SS ≥ e∗No. Similarly,

∂f (PG(e∗∗No))∂e

− ∂g(e∗∗No)∂e

≥ ∂f (PB(e∗∗No))∂e

− ∂g(e∗∗No)∂e

= 0,

and e∗∗SS ≥ e∗∗No.Furthermore, e2 > e1 so we are more likely to see e∗SS than e∗No, i.e., we are more likely to observe the higher effort

if there are short-sales constraints (managers are more likely to convince irrationals that they avoid takeovers). Putting itdifferently, it may be the case that under short-selling constraints the optimal effort is e∗SS (which is the higher effort) andwithout constraints the optimal effort is e∗∗No (≤ e∗∗SS < e∗SS). But, whenever we have e∗No as the optimal then e∗No > e2 > e1

so e∗SS ≥ e∗No > e2 > e1 and we must have e∗SS as the optimal. So, whenever it is optimal to exert more effort (to convinceirrationals that you are avoiding takeovers) in the absence of short-selling constraints it is also optimal to do so with short-selling constraints. However, the converse does not hold: there are cases when it is optimal to exert more effort and convinceirrationals that you are avoiding takeovers when there are short-sale constraints but it is not optimal to do so without saidconstraints (i.e., we observe e∗SS and e∗∗No). Recall that takeovers never happen in the bad state with constraints.

Overall, then, we can say that the manager is likely to exert more effort under short-sale constraints than under frictionlessmarkets. This, by its turn, means that takeovers are less likely under short-sale constraints. Or, more precisely, under rea-sonable parameter values we can obtain a solution to the model where the manager only exerts a lot of effort and convincesirrationals that takeovers are avoided if short-sale constraints are present. Since takeovers are long-run value enhancing wecan argue that the shareholders are worse off with short-sale constraints.

So, it appears that the qualitative results of the model would go through in a more general setting. The fact that effort islikely to be higher and the probability of takeovers lower under short-selling conditions is certainly true, as showed above.The result that long-run shareholders are worse off is also qualitatively true.

Let us analyze the manager’s welfare. Suppose e∗SS > e2 > e∗∗No > e1. So, there is a takeover (probabilistically) withoutshort-selling restrictions but there is no takeover with constraints. Then the manager’s payoff with short-selling restrictions is

w + f (PG(e∗SS)) + C − g(e∗SS)

and without it is

(1 − q)[f (PB(e∗∗No)) + C] + w − g(e∗∗No)

and the difference is

f (PG(e∗SS)) − (1 − q)f (PB(e∗∗No)) + qC + g(e∗∗No) − g(e∗SS)

and

f (PG(e∗SS)) − (1 − q)f (PB(e∗∗No)) + qC + g(e∗∗No) − g(e∗SS) > f (PG(e∗SS)) − f (PB(e∗∗No)) + g(e∗∗No) − g(e∗SS).

We also have

M. Pinheiro / Journal of Mathematical Economics 45 (2009) 361–375 375

f (PG(e∗SS)) − f (PB(e∗∗No))e∗SS − e∗∗No

− g(e∗SS) − g(e∗∗No)e∗SS − e∗∗No

>f (PG(e∗SS)) − f (PG(e∗∗No))

e∗SS − e∗∗No− g(e∗SS) − g(e∗∗No)

e∗SS − e∗∗No

>∂f (PG(e∗SS))

∂e− ∂g(e∗SS)

∂e= 0,

where the first inequality follows since so f (PG(.)) > f (PB(.)) and the last because of concavity. So,

f (PG(e∗SS)) − (1 − q)f (PB(e∗∗No)) + qC + g(e∗∗No) − g(e∗SS) > f (PG(e∗SS)) − f (PB(e∗∗No)) + g(e∗∗No) − g(e∗SS) > 0.

And the manager is better off with short-selling constraints.Similarly, if e∗SS ≥ e∗No > e2 > e1 there is no takeovers and the manager’s payoff with short-selling restrictions is

w + f (PG(e∗SS)) + C − g(e∗SS) and without it is w + f (PB(e∗No)) + C − g(e∗No) and the difference is f (PG(e∗SS)) − f (PB(e∗No)) +g(e∗No) − g(e∗SS). But

f (PG(e∗SS)) − f (PB(e∗No))e∗SS − e∗No

− g(e∗SS) − g(e∗No)e∗SS − e∗No

>f (PG(e∗SS)) − f (PG(e∗No))

e∗SS − e∗No− g(e∗SS) − g(e∗No)

e∗SS − e∗No

>∂f (PG(e∗SS))

∂e− ∂g(e∗SS)

∂e= 0,

where the first inequality follows since so f (PG(e∗No)) > f (PB(e∗No)) and the last because of concavity. So,

f (PG(e∗SS)) − f (PB(e∗No))e∗SS − e∗No

− g(e∗SS) − g(e∗No)e∗SS − e∗No

> 0,

i.e., f (PG(e∗SS)) − f (PB(e∗No)) − [g(e∗SS) − g(e∗No)] > 0, the manager is better off with short-selling constraints.Similarly, if e2 > e1 > e∗∗SS ≥ e∗∗No the manager’s payoff with short-selling restrictions is w + f (PG(e∗∗SS)) + C − g(e∗∗SS)

and without it is w + [f (PB(e∗∗No)) + C](1 − q) − g(e∗∗No) and the difference is (1 − q)f (PG(e∗∗SS)) − (1 − q)f (PB(e∗∗No)) +g(e∗∗No) − g(e∗∗SS). But

(1 − q)f (PG(e∗∗SS)) − f (PB(e∗∗No))

e∗∗SS − e∗∗No− g(e∗∗SS) − g(e∗∗No)

e∗∗SS − e∗∗No> (1 − q)

f (PB(e∗∗SS)) − f (PB(e∗∗No))e∗∗SS − e∗∗No

− g(e∗∗SS) − g(e∗∗No)e∗SS − e∗∗No

> (1 − q)∂f (PG(e∗∗SS))

∂e− ∂g(e∗∗SS)

∂e= 0,

where the first inequality follows since so f (PG(e∗No)) > f (PB(e∗No)) and the last because of concavity. So

(1 − q)f (PG(e∗SS)) − f (PB(e∗No))

e∗SS − e∗No− g(e∗SS) − g(e∗No)

e∗SS − e∗No> 0,

i.e.,

(1 − q)[f (PG(e∗SS)) − f (PB(e∗No))] − [g(e∗SS) − g(e∗No)] > 0,

the manager is better off with short-selling constraints. Therefore we have just proved that the manager is always better offwith short-selling restrictions. While before we argued that the shareholders were always better off without them. So themain results of the paper follow even under more general conditions.

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