Takeovers with Resistant Boards, Political Pressure and

Risk-Arbitrageurs

Guillem Ordonez-Calafı

University of Warwick∗John Thanassoulis

University of Warwick†

January 30, 2016

Abstract

Managerial resistance plays an important role in the market for corporate control.

Such resistance is often reinforced by government pressure. Yet the ability of a Board

to resist a takeover offer is tempered by share sales to arbitrageurs during the offer

period. Using a global games approach we document how changes in market fun-

damentals (bid premium, external pressure, and shareholder voting rights) affect the

main takeover outcomes: probability of takeover success, the market price during the

offer period, and the volume of shareholders who choose not to wait for the Board’s

recommendation but rather sell early to risk-arbitrageurs.

JEL: C72, G34, G38.

Keywords: Takeovers, takeover resistance, shareholder coordination, risk arbitrage,

political pressure, global games.

∗Department of Economics, University of Warwick. g.ordonez-calafi@warwick.ac.uk†Warwick Business School, University of Warwick; Oxford-Man Institute, University of Oxford, Associate

Member; and Nuffield College, University of Oxford, Associate Member.

1 Introduction

In 2009 Kraft Foods, a US company, launched a hostile bid for Cadbury, the UK-listed

chocolatier. Cadbury’s managerial board declared the offer “unattractive”. Moreover, the

British government and some unions publicly stood against the takeover, with the business

secretary warning Kraft that it could meet “huge opposition” from the British Government.

Despite the evident political distaste for the deal, Kraft eventually won the Cadbury’s board’s

blessing. The agreement was acknowledged to be unpopular amongst many Cadbury’s share-

holders, but the board’s approval led to 72% of the stock being tendered – sealing the deal.

Sir Roger Carr, Cadbury’s Chairman, admitted that the increase in the proportion of the

company held by short-term investors during the bidding period (from 5% to 31%) led him

to close the deal:1

“In the final analysis of the deal, it was the shift in the shareholder register

that lost the battle for Cadbury.” Roger Carr, former Chairman of Cadbury

(emphasis added).2

Managerial resistance is acknowledged to play an important role in the market for corpo-

rate control. The ability of a Board to resist a takeover offer is tempered however by share

sales during the offer period – as noted in the quote above. After a hostile takeover offer is

made, the trading volume of the target company’s stock rises, typically dramatically. This

is due in large part to the activity of risk-arbitrageurs.3 The arbitrage community often

comes to control 30-40% of the stock and therefore becomes a crucial element in determining

whether a Board can credibly recommend against a takeover as it is perceived to be in the

interests of arbitrageurs to tender their stock (Hillier, Grinblatt and Titman (2012)).

The strength of managerial resistance is often reinforced by external pressure exerted by

the political system, labour, the media and other stakeholders whose interests are aligned

with those of the incumbent managers (Hellwig 2000). Governments in particular use both

formal and informal instruments to oppose takeovers for the sake of their interpretation

of the public interest.4 In numerous countries the takeover legislation is considered to be

1Press coverage includes: Wiggins, J. (2010). The inside story of the Cadbury takeover. Financial Times,[online]. Available at: http://www.ft.com/cms/s/2/1e5450d2-2be5-11df-8033-00144feabdc0.html [Accessed17 Jan. 2016]; or Moeller, S. (2012). Case Study: Kraft’s takeover of Cadbury. Financial Times, [online].Available at: http://www.ft.com/cms/s/0/1cb06d30-332f-11e1-a51e-00144feabdc0.html#axzz3xVfEY4Dk[Accessed 17 Jan. 2016]

2‘UK takeover threshold should be raised, says ex-Cadbury chairman Roger Carr,’ Daily Telegraph, 10February 2010.

3For example see the numerous HBS cases cited in Cornelli and Li (2002).4The motive “public interest” provides flexibility to governments for intervening in takeovers. In the

UK, the competition authorities are required to assess whether a merger should be prohibited on the basis

1

protectionist as it provides legal tools for governments to block takeovers. Furthermore,

the government is in a position to use moral persuasion to stop an acquisition. The threat

for acquirers is to have to deal with a hostile government on many regulatory issues if

the acquisition goes through (see Dinc and Erel (2013) for a classification of government

instruments to oppose takeovers).

This paper formally analyses the takeover process from a novel perspective allowing for

strategic Boards who have a reluctance to sell, and whose reluctance is in part a function

of external pressure and shareholder sales during the offer period. The shareholders are

themselves strategic and evaluate the trade-off between selling during the takeover process

to lock in gains, against the benefits of holding on in the hope the pressure created by the

sales of other shareholders will be sufficient to counter the expected external pressure on the

Board.

The fundamentals of our analysis will be the bid premium, the expected Board resistance

to takeovers, and the influence of new owners of the stock – many of whom will be arbitrageurs

during a takeover process. We study how changes in these parameters affect the main

outcomes of our analysis: the probability a takeover succeeds, the market price during

the offer period, and the volume of shareholders who choose not to wait for the Board’s

recommendation but rather sell early to risk-arbitrageurs.

The relationship between these fundamental and outcome variables is unclear without

further analysis. For example, if the bid premium rises it seems natural that the probability

of a successful deal will rise, and so too will the market price of the stock in the offer

period. It is not however clear whether the volume of shareholders selling during the offer

period should rise or fall; both the benefits of selling early and of waiting to the Board’s

decision have risen and so which dominates is not clear. Similarly, if a government should

increase the pressure on a Board to resist a sale then one can anticipate that the probability

of the deal and so the interim share price will decline. The former may therefore make

shareholders more likely to sell early, though the latter force works against this. However,

a third possibility exists: if enough shareholders sell early then it may be possible for the

force of the new shareholders (mostly risk-arbitrageurs) to overturn the effectiveness of the

government’s intervention and increase the probability of the deal being consummated. We

will offer answers to such questions.

Our analyses are in some cases relevant for current policy debates concerning shareholder

voting rights and the merits of political interference in the market for corporate control.

of whether it can be expected to lead to a substantial lessening of competition. Furthermore, since theEnterprise Act 2002, the Secretary of State is allowed to intervene in mergers where they give rise to certainpublic concerns other than competition. These are: national security, media quality, plurality & standards,and financial stability.

2

There is a long tradition of scholarship arguing that shareholders are too short-term and

can distort managerial decision making (see Thanassoulis and Somekh (2016) and references

therein). In response to these concerns there have been prominent calls for the voting

power of new shareholders to be curtailed. This would particularly affect the ability of risk-

arbitrageurs to influence takeovers. In the U.S. the Alpen Institute propose that shareholders

should be able to vote only after a minimum holding period. In the EU the European

Commission are looking into increasing the voting weight of shareholders who are long-term

holders of the stock.5 In the UK the takeover panel also considered disenfranchising new

shareholders following the Cadbury takeover. As these proposals, designed to encourage long

term shareholder ownership, have the effect of diminishing the power of risk-arbitrageurs, it

is conceivable that they will in fact encourage more shareholders to sell early during an offer

period to maintain selling pressure on the Board; exacerbating not correcting the alleged

problem. We will show that this can indeed be the case and offer a characterization of when

it is.

This paper formally studies the coordination problem of target shareholders deciding

whether to sell their stock before the end of the offer period and the Board’s decision. A

Bidder makes an acquisition offer to the target company. For the takeover to succeed, i.e.

for enough shareholders to tender, it requires the acceptance of the target Board. Thus, the

takeover premium needs to outweigh the Board’s resistance. However, the resistance is alle-

viated by the effect of the risk-arbitrageurs. During the offer period (interim), arbitrageurs

take long positions in the target stock in the hope that the takeover will go through. Hence,

they put pressure on the Board to reach an agreement with the Bidder. A target shareholder

has to decide whether to hold her stock or to sell it at the interim. If enough shareholders

sell, the internal pressure of risk arbitrageurs yields the takeover success. Conditional on a

takeover’s success, a shareholder prefers to hold her shares so as to gather the full takeover

premium. In contrast, when many shareholders hold the takeover fails due to the low pres-

sure of arbitrageurs relative to the Board’s resistance. Then, the stock price returns to its

original value, leaving those shareholders who held with no gain. Therefore, selling becomes

a “public good” to which no one wishes to contribute, but everyone hopes others will.

In equilibrium both the interim stock price and the order flow are determined endoge-

nously. This generates multiple equilibria and impedes discrimination between takeover

outcomes. To deal with multiplicity, we introduce the information structure of a global

game. This global games analysis is itself novel as shareholders’ actions are strategic sub-

stitutes, not complements, which arises because of the public good nature of the early sale

decision. The Board’s type, which represents its resistance to the takeover, is not known with

5See Brussels aims to reward investor loyalty, Financial Times, Jan 23, 2013.

3

certainty by the shareholders. Nonetheless, the shareholders share a common prior about its

distribution. Moreover, each shareholder receives an independent private signal about the

Board’s type. Hence, current shareholders have an information advantage over arbitrageurs.

Finally, the interim stock price at which trade between shareholders and arbitrageurs takes

place is set by market makers with zero expected profits. This setting pins down a unique

equilibrium as a function of the fundamental variables, where all agents are rational and

trading arises due to informational asymmetries.

We characterize the equilibrium and conduct comparative statics to study the effects

of the model fundamentals on the main outcomes. Our goal is to better understand the

consequences of government pressure and policy making in a takeover setting. Our model

is extended to evaluate the effectiveness of different policy measures against takeovers. We

show that marginal increases in political pressure have the greatest impact on the deal success

probability when the counterfactual takeover probability of success and failure are equally

likely. Furthermore, we show that in jurisdictions which have strong respect for shareholder

rights – including new shareholders – the marginal impact of increases in political pressure

on the Board are most pronounced. Finally we extend the model to consider a strategic

bidder and so study how typical bid levels themselves are influenced by the political and

shareholder regime existing in a given jurisdiction.

1.1 Related Literature

This paper contributes to the literature analysing shareholders’ coordination issues in takeover

offers. The seminal work of Grossman and Hart (1980) described a free-riding incentive in

shareholders’ tendering decision. Shareholders would rather not sell and gain from holding

shares in an improved company, conditional on enough other shareholders selling to en-

sure sufficient incentive for the acquirer to monitor the target firm. Since then, numerous

studies characterized mechanisms mitigating the coordination problem so as to explain why

takeovers succeed (see Tirole 2006, and references therein). In line with recent evidence,

we propose a different coordination problem: During the offer period, shareholders decide

whether to sell their shares to arbitrageurs that make the takeover more likely, or whether

to wait to sell to the acquirer at the end of the offer period if the deal is successful.

Our work allows for managerial resistance to a value-enhancing takeover in this setting.

The economic literature mainly focused on the agency problems between ownership and

control to explain takeover defenses (see Tirole 2006 for a systematic approach). While not

neglecting this approach, we also put emphasis on the pressure against takeovers exerted

by the broader stakeholders. This is in line with Hellwig (2002), who reviews the alliance

4

between incumbent managers and the political system (see, e.g., Jensen (1991) and Roe

(1994)). In a similar spirit, Dinc and Erel (2013) find evidence of widespread economic

nationalism by governments who oppose takeovers of domestic firms by foreign acquirers.

The effects of risk arbitrage on takeovers has been previously analysed. In an early study,

Larcker and Lys (1987) find that takeovers in which arbitrageurs bought shares have a higher

probability of success. Relatedly, numerous studies find that risk arbitrage generates excess

returns (see e.g. Baker and Savasoglu 2002). Nonetheless, while there is consensus on the

existence of excess returns, less is known about their sources. Most notably, Mitchell and

Pulvino (2001) offer evidence that some of the excess returns reflect a premium paid to

risk-arbitrageurs for providing liquidity to the market.

An alternative explanation for the excess returns is that risk arbitrage makes a takeover

more likely. Since arbitrageurs are keener to tender, they are typically perceived as favouring

the acquirer (see Hillier, Grinblatt and Titman 2012). Cornelli and Li (2002) suggest that

risk-arbitrageurs have an information advantage in making inferences from the order flow

because they know they will tender with certainty. This leads them to accumulate more

stock and in turn, to increase the probability of a takeover. Our model is close Cornelli and

Li (2002) on the positive impact of arbitrage in the probability of a takeover. Nonetheless,

it differs on the mechanism. In particular, we assume that arbitrageurs put pressure on the

Board to accept the offer and that existing shareholders are strategic in deciding whether to

sell early or sell late.

We use global games techniques and take an approach similar to Bebchuk and Goldstein

(2011). Nonetheless, our setting differs from most literature in global games in that agents’

actions are strategic substitutes. This has not been as thoroughly studied as the case of

strategic complements. Hence, while the properties of equilibria are analysed in Morris

and Shin (2005) and Harrison (2005) among others, we find little application of this case.

Last but not least, we use the information asymmetries of a global game to motivate trade.

This relates our work to other frictions proposed in the literature, such as Kyle (1985) who

introduces a proportion of noise traders. In our model all agents are rational and have the

same reservation value, which makes information asymmetries the unique rationale for trade.

This paper is structured as follows. Section 2 presents the benchmark model and describes

the information structure of the global game. In Section 3, we characterize the equilibrium

and the main outcomes. We also study the insights of the information structure which is

compatible with equilibrium uniqueness. Section 4 describes the main results and presents

the comparative statics. Section 5 examines the effectiveness of political pressure. In section

6, we study the offer of a strategic bidder. Finally, Section 7 concludes. The appendices

contain two extensions. Appendix A studies the robustness of the results to the setting in

5

which some target shareholders are also holders of the bidder’s stock. Appendix B studies

a perturbed model in which the information structure is altered to give arbitrageurs an

informational advantage over existing shareholders. The third and final appendix collects

all the technical proofs.

2 The Model

Consider a target company which has a market value V. The target company is owned by a

continuum of shareholders of mass one. At t = 0, the target receives an offer P ≥ V from a

company that we call the Bidder. In the core model this price P is exogenous. This captures,

for instance, the highest bid from an auction process and so represents the maximum value

which can be extracted from the target’s assets. We consider the case of a single bidding

firm and so strategically chosen P in Section 6.

We assume that the target’s Board is influential enough that their advice will deter-

mine whether the acquisition can be completed. Typically, perhaps due to the dispersion

of shareholders in publicly traded firms, the managerial team is expected to use its infor-

mation advantage and advise shareholders as to whether tendering their shares is in their

long-term interests. As a consequence, the Board’s verdict is usually deterministic for the

takeover outcome. More systematically, Baker and Savasoglu (2002) in their study of 1,901

US takeover offers between 1981 and 1996 report that in approximately 80% of cases, the

outcome was in line with the Board’s advice. We model the Board’s decision is being made

at t = 2. Both shareholders and external investors understand that if the Board accepts, the

share value will move to P . Instead, if the offer is rejected, the value will return to V . In the

interim period t = 1, before the Board decides whether to accept the offer, shares are traded

at a market price M . This price is endogenously set by market makers with zero expected

profits. We denote by ρ the proportion of shareholders selling at t = 1.

We assume that the Board causes the offer to be accepted at t = 2 if

P ≥ V + θ − κρ. (1)

The parameter θ ∈ R represents the Board’s resistance to accept the offer. This can capture

political pressure or managerial rents. The Board’s type θ is private information. It is

drawn from a normal distribution with mean y and precision τθ. The parameter y measures

the expected Board resistance to a takeover. The resistance could be due to the perks of

control or to political pressure. If the realised Board’s resistance, θ, is high then the Board

will only be willing to sell if the offer price P exceeds the stand alone value of the firm, V ,

6

by a sufficiently large margin. Both θ and y are permitted to take negative values.

The variable ρ ∈ [0, 1] is determined by equilibrium play and measures the proportion of

stock acquired by risk-arbitrageurs at t = 1, which we assume is proportional to the trading

volume or order flow. As shareholders sell out, ρ increases so the pressure on the Board

to accept the offer grows. This effect is scaled by the parameter κ ≥ 0, which represents

the effectiveness of risk-arbitrageurs on influencing the takeover’s outcome. Hence, higher ρ

increases the range of values of θ at which the Board accepts the offer.

2.1 Shareholders’ game

To explain this game succinctly, suppose for the moment that at the interim t = 1 everyone

observes the Board’s type θ. Then, each shareholder decides whether to hold her share or

to sell it for a price M , which is simultaneously set by market makers. Selling to outside

investors gives a payoff M . Instead, holding the share has a payoff of V if the Board rejects

the offer at t = 2, and a payoff P > V if the takeover goes through. There are two key

features driving the equilibria in this game. First, whether the Board accepts the takeover

bid may depend on the number of shareholders selling. Second, the interim stock price M is

set by market makers with no profits. We characterize the symmetric mixed strategy Nash

Equilibria so as to provide a neat exposition of the main mechanisms in this game.

Let h(θ) ≡ θ−(P−V )κ

represent the proportion of shareholders selling that makes the Board

indifferent between accepting and rejecting the offer. Hence, the takeover fails if ρ < h(θ)

and succeeds otherwise. Then, shareholders’ payoff can be represented as follows:

ρ ≥ h(θ) ρ < h(θ)

Sell M M

Hold P V

where both the stock price M and the order flow ρ are endogenous. There are a continuum

of equilibria to this game that can be of two types. First, market makers set M = P ,

shareholders sell with probability h(θ) or higher and the takeover succeeds. Second, market

makers set M = V , shareholders sell with probability lower than h(θ) and the takeover fails.

Notably, in all equilibria shareholders are indifferent between selling and holding their shares

and risk-arbitrageurs make zero profits.

Finally, note that shareholders face a coordination problem when θ ∈(θ, θ]

where

θ ≡ P − V and θ ≡ P − V + κ. (2)

7

The Board always accepts the offer (and M = P ) if θ ≤ θ and rejects it (and M = V )

when θ > θ, no matter shareholders’ selling decisions - formally, note that h(θ) ∈ [0, 1] when

θ ∈[θ, θ]. Therefore, multiplicity of equilibria arises for realizations θ ∈

(θ, θ]. Then, the

decision of the Board depends on the strategic interaction of shareholders and the simple

full information setting does not allow us to make predictions about the takeover’s outcome.

Furthermore, in contrast to the results here, evidence shows that the stock price at the

interim assigns positive probabilities to both a takeover success and a failure: V < M < P .

To analyse takeovers more fully we introduce the information structure of a global game so

as to obtain equilibrium uniqueness and to provide more plausible predictions about both

the interim price and the order flow.

2.2 Global game

Our goal is to conduct comparative statics to better understand the effects of policy making

in takeovers under the presence of risk-arbitrageurs. To this end, we address the problem of

multiplicity of equilibria by introducing the information structure of a global game. Recall

that the Board’s type θ is a normally distributed random variable with mean y and precision

τθ and its realisation is only observable to the Board. Thus τθ represents the precision of

the public information about the Board’s resistance to a takeover (and the pressures they

face). In Section 5 we will analyse the efficacy of political pressure on the Board which we

will interpret as a shift in the Board’s expected type: y. Market makers have zero expected

profits. Therefore, M equals the expected value of a share conditional on the common prior.

Shareholders obtain additional information about the Board’s type through a private noisy

signal. This may come, for instance, from private knowledge of Board members, access

to investor calls, or from exposure to discussions and views of company employees. More

specifically, each shareholder i receives an iid signal xi = θ + εi where εi is distributed

according to a normal of mean 0 and precision τε. A shareholder sells her shares when she

receives a sufficiently large signal, which indicates that the board is likely to reject the offer.6

For clarity we bring together the entire timeline of the model: At t = 0 the target company

receives an offer P from the Bidder that everyone observes and θ is privately realised. During

the interim t = 1, market makers set a price M which is a function of the probability that

the takeover succeeds. Furthermore, each shareholder receives a private signal xi, updates

her beliefs about θ and then decides whether to sell or to hold her share. Finally, at t = 2

6Market makers do not receive these signals. Thus, shareholders are at least as well informed as risk-arbitrageurs as to the type of the Board. Appendix 1 shifts the information structure to represent thesituation where arbitrageurs are weakly better informed. We show that the main results are robust to thenew assumption.

8

the Board decides whether to accept the offer after observing the proportion of shareholders

who have sold during the interim period, ρ. We ignore temporal discounting between periods

for the sake of simplicity. Figure 1 depicts the sequence of events:

The Bidder makesan offer P and θ isprivately realised

t = 0

Market makers set M and share-holders receive private signals ofθ and either sell or hold

t = 1

The Board eitheraccepts or rejectsthe Bidder’s offer

t = 2

Figure 1: Time line

3 Equilibrium Analysis

We solve the model using global-games techniques. Before presenting our results, it is con-

venient to define a term that captures the effect of information in numerous expressions of

our analysis:

∆ ≡ τθ√τε

[√1 +

τετθ− 1

](3)

Recall that τθ and τε represent the precision of public information and private signals respec-

tively. The impact of the relative precision of public and private information in equilibrium

outcomes is one of the most studied issues in global games. However, our setting differs

from most applications in that agents’ actions are strategic substitutes rather than strategic

complements. We present the study of information in our model in subsection 3.2.

We derive a critical Board type θ∗ such that the takeover succeeds if an only if θ < θ∗.

We also determine a critical signal x∗ such that shareholders sell their shares when they

receive a signal x > x∗. Proposition 1 states the basic equilibrium result.

Proposition 1 Suppose the influence of new shareholders is not too great, κ <√

2π∆

. There

exists a unique Bayesian Nash Equilibrium in which all shareholders sell their shares to the

market if they observe a signal above x∗ and hold them otherwise. The takeover succeeds

if, and only if, the Board’s takeover resistance is below the threshold θ∗. The thresholds are

characterized implicitly by the following equations:

θ∗ = P − V + κ · Φ (∆ (θ∗ − y)) (4)

x∗ = θ∗ − ∆√τε

(θ∗ − y) (5)

where Φ(·) is the cumulative distribution function for the standard normal.

9

Proof. See Technical Appendix

The fundamental variables of the model {P − V, κ, y, τε, τθ} determine the critical Board’s

type θ∗ and in turn our main outcomes, namely the probability that the takeover succeeds,

the interim stock price and the order flow. First, note that the decision of the Board depends

on the value of its type θ with respect to the threshold θ∗. Hence, for a given prior distribution

θ ∼ N (y, 1/τθ), the equilibrium threshold θ∗ determines the probability of a takeover, which

we denote as β. The characterization follows immediately:

β = Pr[θ ≤ θ∗] = Φ (√τθ (θ∗ − y)) (6)

Now recall that the interim stock price M is set by market makers with no profits in

expectation. As their valuation is based on public information, M equals the expected value

of a share conditional on the common prior:

M = βP + (1− β)V (7)

Finally, we derive the proportion of shareholders selling during the interim period t = 1

or equivalently, the order flow. Assuming a continuum of shareholders implies that they do

not consider the possibility of being ”pivotal,” i.e., affecting the takeover’s outcome through

their selling decision. As a consequence, shareholders take M as given by their expectations

of others’ actions. The next Corollary characterizes the order flow:

Corollary 2 For a given realization of the Board’s type θ, the order flow is

ρ = Φ (∆ (θ∗ − y)−√τε (θ∗ − θ)) (8)

Proof. See Technical Appendix.

The shareholders selling received a high signal (x > x∗) and thus they believe there is a

high probability that the Board will reject the offer. Hence, such a shareholder sells because

the value offered by the market is higher than her expected value of a share.

3.1 Uniqueness

In Proposition 1, κ <√

2π/∆ is an equilibrium existence and uniqueness condition that

limits the influence of new shareholders on the Board. In other words, it restraints the

externality of players’ actions. This result is in line with previous studies analysing global

games where actions are strategic substitutes (see, for instance, Morris and Shin 2005).

The condition can also be interpreted in terms of information precision: ∆ <√

2π/κ. We

10

require the combination of public and private signals to be noisy enough.7 To see this, it is

convenient to plot ∆ as a function of both τθ and τε:

0 2 4 6 8 100

2

4

6

8

10

τε

τθ

uniqueness

multiplicity

κ = 3

↓ κ↑ κ

Figure 2: The solid line delimits the regions of uniqueness (∆ <√

2Πκ

) and multiplicity

(∆ >√

2Πκ

) when κ = 3. The dashed lines depict the same regions for both higher and lowervalues of κ (κ = 3.25 and κ = 2.75 respectively)

Figure 2 shows that public and private information are substitutes on delivering unique-

ness. Hence if, for example, private signals are highly precise (large τε), we require a rela-

tively uninformative common prior (small τθ). The condition differs from most applications

of global games, where actions are strategic complements. There, uniqueness is guaranteed

for relatively precise private signals (Morris and Shin, 2003), so the colored area corresponds

to the region below an upward sloping line. By contrast, here if agents can predict with high

accuracy what signals others received, then the uniqueness of the pure strategy equilibrium

breaks down. The reason is that they wish to mis-coordinate, selling (holding) if enough

other shareholders hold (sell).

3.2 Information Structure

To better understand the effects of the information structure imposed by global games, we

present the intuition provided by the model at the limiting values of the private information

precision, τε. First, the next corollary identifies the limits of the term capturing the effects

of information precision:

7Limitations on the uniqueness result are studied in Angeletos, Hellwig and Pavan (2006, 2007) andAngeletos and Werning (2006).

11

Corollary 3 The limits of ∆ as a function of τε are as follows:

limτε→∞

∆ =√τθ and lim

τε→0∆ = 0 (9)

Proof. See Technical Appendix.

As private signals become infinitely precise, i.e. as τε approaches infinity, shareholders’

beliefs on θ converge to the true value. Hence, while there is an information advantage of

shareholders with respect to market makers, the heterogeneity of beliefs among shareholders

vanishes. As a consequence, they all take the same action. Furthermore, since their signals

reveal the true value of θ, their decision threshold converges to the critical Board’s type x∗ →θ∗ - see the expression in (5) with τε → ∞. Therefore, as revealed by the characterization

of the order flow (8), all shareholders hold (ρ = 0) when θ ≤ θ∗ and sell (ρ = 1) if θ > θ∗.

In contrast, when private signals are completely uninformative, i.e. as τε approaches zero,

all information asymmetries disappear. Hence, shareholders and market makers share the

same belief about the realisation of θ. In equation (4) it is possible to see that information

precision does not affect the critical Board type, which equals the mean of the interval where

the strategic interaction of shareholders determines the takeover outcome: θ∗ =(θ + θ

)/2 =

P−V +κ/2. As with fully revealing signals, here all shareholders have the same information.

Furthermore, now market makers have the same information, so they set a price that makes

shareholders indifferent about their selling decision. Thus, both selling and holding become

dominant and the model predicts that exactly half of the shareholders will take either action

(ρ = 1/2), which follows from (8).

4 Comparative Statics

We conduct comparative statics to study the reaction of the main outcomes in our model to

changes in the fundamentals: namely the Bidder’s offer, the influence of new shareholders,

and the Board’s expected type. However, it is first convenient to describe the effect of these

variables on the strategic game. The following lemma describes the reaction of the critical

threshold θ∗ to changes in the fundamentals.

Lemma 4 The threshold θ∗ is positively related to both the Bid premium and the extent of

new shareholders’ pressure, and negatively related to the Board’s expected type, i.e. ∂θ∗

∂(P−V )>

0, ∂θ∗

∂κ> 0 and ∂θ∗

∂y< 0.

Proof. See Technical Appendix.

12

Lemma 4 establishes that a higher bid premium P −V increases the critical Board’s type

θ∗. Furthermore, note from expression (2) that it also has an impact on the range of values

where shareholders face a coordination problem,(θ, θ]. In particular, it shifts the interval up

without affecting its size. We find that the bid premium has a larger effect on the decision

threshold θ∗ than on the boundaries of the strategic game,{θ, θ}

. As a consequence, a

higher offer increases the range of realizations of θ at which there is a takeover driven by the

presence of risk arbitrage, θ∗−θ. The influence of arbitrageurs on the Board’s decision κ also

has a positive impact on the critical type θ∗. Furthermore, it is possible to see in (2) that

raising κ also increases the range of values where incumbent shareholders face a coordination

problem. While new shareholder pressure κ has no effect on the lower bound θ, it raises the

upper bound θ, thereby increasing the region where an acquisition can take place due to the

pressure of risk-arbitrageurs. Finally, the expected Board’s type y is negatively related to

the decision threshold θ∗ but has no impact on the strategic interval(θ, θ].

The effect of fundamentals on the critical type θ∗ (Lemma 4) only captures partially

their impact on the main outcomes. Importantly, fundamentals affect the outcomes both

directly, as well as indirectly - through θ∗. Hence, we must account for the interaction of

both effects to study the overall response. The results of this analysis are summarized by

the next proposition.

Proposition 5 Responses of the main outcomes to marginal increases of the model funda-

mentals are as follows

Fundamental Effect on outcomes

Pr. Takeover β Interim Price M Order Flow ρ

Bid premium: P − V + + −New-shareholder pressure: κ + + −Expected Board resistance: y − − ±

where dρdy

=sign

√τεκϕ(∆ (θ∗ − y)) − 1 with ϕ(·) denoting the probability distribution function

for the standard normal.

Proof. See Technical Appendix.

Takeover Premium. In expression (6) it is clear that the bidder’s offer only affects the

probability of a takeover indirectly, i.e. through the critical Board type θ∗. Furthermore,

Lemma 4 shows that the decision threshold θ∗ is positively related to the takeover premium.

Hence, a higher premium increases the range of realisations of θ such that θ < θ∗ and thus

raises the probability of a takeover. The bid premium also has a positive effect on the interim

13

price. It is possible to see in (7) that the effect operates through two channels working in the

same direction. First, a positive direct impact ∂M∂(P−V )

> 0 occurs because a higher premium

increases the target’s value if the takeover succeeds. Second, there exists a positive indirect

effect ∂M∂β

∂β∂θ∗

dθ∗

d(P−V )> 0 caused by the increase in the probability that the takeover takes

place already established above. Hence, both mechanisms contribute to raising the price

of shares. Finally, equation (8) shows that the order flow is only affected indirectly by the

bid premium. In particular, a higher offer raises θ∗, which in turn reduces the proportion of

shareholders selling - note√τε−∆ > 0. Thus, even though the interim stock value increases,

less shareholders are willing to sell. The reason is that shareholders are better informed than

outside investors about θ. As a consequence, the price adjustment by market makers does

not compensate the increase in shareholders’ expected value. Notably, the better informed

shareholders are with respect to market makers, the bigger is the effect:∂(√τε−∆)

∂τε> 0.

Arbitrage Pressure. The parameter κ accounts for the influence of risk-arbitrageurs

on the Board’s decision. The probability of a takeover is higher when the Board is more

sensitive to the pressure exerted by these investors. The Bidder benefits from their influence

as it pushes up the decision threshold of the Board, which in turn becomes more likely

to accept the offer. The interim stock value is also positively related to the extent of the

new-shareholder pressure κ. When the latter rises, the probability of a takeover increases

and therefore so does the expected value of shares. Despite this, the order flow is negatively

affected by the influence of risk-arbitrageurs κ. The mechanism is similar to the one described

for the takeover premium. To see this, note in expression (8) that κ only affects the order

flow indirectly through the decision threshold θ∗. Moreover, according to Lemma 4, κ is

positively related to θ∗. Hence, higher influence of arbitrageurs increases θ∗ and makes the

takeover more likely. Finally, since shareholders are better informed than market makers,

this has a negative impact on the order flow.

Takeover Resistance. The parameter y represents the expected Board resistance to

takeovers. It captures the expectation of market makers as well as the common prior shared

by all shareholders. In equation (6) it is clear that y has a direct negative effect on the prob-

ability of a takeover. However, an indirect effect operating in the same direction exacerbates

the overall impact. More specifically, Lemma 4 shows that a bigger y also diminishes the

critical type θ∗, lowering the range of values θ at which the transaction takes place. Hence,

more resistance to deal leads to both higher expected realizations of θ (direct effect) and a

lower range of values θ at which the Board accepts the offer (indirect effect). As can be seen

from (7), the negative effect is transmitted to the interim stock price. Thus, increases in y

make the takeover less likely, which in turn lowers the expected value of shares.

14

Despite the previous results, the Board’s expected type y has ambiguous effects on the

order flow. Equation (8) shows that the proportion of shareholders selling is negatively

related to the Board’s reluctance to accept in a direct manner ∂ρ∂y< 0. Increasing y lowers

the probability of a takeover and in turn the interim stock price. Thus, shareholders have less

incentive to sell, which affects the order flow negatively. However, an indirect effect works in

the opposite direction. A higher y reduces the critical type (∂θ∗

∂y< 0) which in turn induces

more shareholders to sell ( ∂ρ∂θ∗

< 0), thereby increasing the the order flow: ∂ρ∂θ∗

∂θ∗

∂y> 0.

In the Proof of Proposition 5 we show that the dominating effect is determined by the

relation between the expected Board’s type y and the threshold θ∗. If the two values are

considerably different, the probability that the takeover takes place is either high (y << θ∗)

or low (θ∗ << y). Then, the marginal effect of y on the decision threshold ∂θ∗

∂yis small and

as a consequence, the direct effect dominates, leading to a negative impact of y on the order

flow. In contrast, when the values of y and θ∗ are close, the critical type θ∗ becomes more

sensitive to changes in the common prior. As a result, the indirect effect is bigger and drives

the sign of the overall response.

4.1 Alternative Settings and Robustness

To illustrate the main drivers of our analysis most simply, we have proposed a setting that

omits some relevant characteristics of the market for corporate control. Hence, the reader

might be concerned about the effects that accounting for these features have on the main

insights of the model. This section considers two variations of the benchmark model.

Target Shareholders With Bidder’s Stock. In numerous cases target shareholders

own stock of the bidding firm. If the takeover succeeds, these shareholders receive an ad-

ditional payoff (either positive or negative) that accounts for the effect of the takeover on

the bidder’s stock. Notably, this is independent of whether they personally sold their target

shares or not at the interim. Appendix I considers this setting. We show that despite the

change in the payoff function, the strategic interaction between shareholders is not affected.

As a consequence, results in both Proposition 1 and Proposition 5 remain unchanged.

Informed Arbitrageurs. We assume that shareholders are better informed about the

value of their assets than other investors. As a result, risk-arbitrageurs make zero profits

in expectation whereas shareholders have positive expected returns. However, risk arbitrage

commonly invokes images of informed funds that appear to make substantial profits (see

Mitchell and Pulvino 2001 and references therein). Appendix II reverses the information

structure. It characterizes a setting where risk-arbitrageurs are better informed than existing

shareholders and thus, they have positive expected profits. In particular, it is assumed

15

that outside investors receive private signals about the Board’s type whereas shareholders’

selling decisions are only based on the common prior. Notably, this is a game of strategic

complementarities for arbitrageurs, whose expected payoff from acquiring stock increases

with the stock owned by other arbitrageurs. Despite the new information structure, both

the probability of a takeover and the interim stock value react to changes in the fundamentals

as in Proposition 5. In contrast, the relation between order flow and fundamentals is changed.

Increases in the expected resistance of the Board to sell lower the probability of the Board

agreeing to sell and lower the interim stock price. The potential benefits to arbitrageurs grow

if enough arbitrageurs buy shares. The better information of the arbitrageurs implies that the

reduction in the stock price over-states the reduction in the probability of sale allowing the

expected profits of the arbitrageurs to grow. Thus with this reversed information structure

a greater expected reluctance of the Board to sell is associated with a greater proportion of

risk-arbitrageurs buying, conditional on any given Board type.

5 Political Pressure and Shareholder Voting Rights

As noted in the introduction, the pressure the British government was willing to exert did

not manage to block Kraft’s takeover of Cadbury. In contrast, analysts argue that it was

important in the failure of Pfizer’s attempt to acquire AstraZeneca in 2014. The case, which

was characterized by prominent political grandstanding, even lead politicians to publicly

interrogate Pfizer executives in a manner that, according to some observers, bordered on

vilification.8 More recently, after the rumors of a potential offer of ExxonMobil for BP, the

British oil and gas company, the government said it would oppose any potential takeover -

even if it involved Royal Dutch Shell, the Anglo-Dutch oil major - because it wants Britain

to have two big global oil companies.9

The previous examples, though drawn from the UK, highlight a common phenomenon in

the market for corporate control across the world: the frequent opposition of governments

and other lobbying groups to takeovers. Scholars have sought to clarify why governments

should wish to oppose a free market in corporate control. For example Roe (1994) identifies a

systematic alignment of the politicians’ interests with those of corporate management. In the

same spirit, Hellwig (2000) argues that the political system can be seen as a stakeholder in

8Press coverage includes: Bogdanor, A. (2014). The regulators were right to force Pfizer’shand. Financial Times, [online]. Available at: http://www.ft.com/cms/s/0/a476fdf0-e5bc-11e3-a7f5-00144feabdc0.html#axzz3xbkC23LC [Accessed 17 Jan. 2016]

9See Parker, G. and Christopher, A. (2015). UK government warns BP over potentialtakeover. Financial Times, [online]. Available at: http://www.ft.com/cms/s/2/06a3207e-e901-11e4-87fe-00144feab7de.html#axzz3xbkC23LC [Accessed 17 Jan. 2016]

16

its own right. First, because there is an immediate financial interest ranging from corporate

taxation to campaign contributions. Second, because politicians represent other stakeholder

concerns. In contrast, Jensen (1991) regards the treatment of corporate control by the

political system as a reaction to populist rhetoric without understanding of the systemic

implications.

Here we do not take a stand on why government should wish to influence a Board’s

decision. Instead our focus is on understanding when political pressure is likely to be most

effective. Thus this section considers the case of a government that seeks to alter y by

exerting political pressure. Nonetheless, actions aiming to block a takeover usually carry

some costs, e.g. lower reputation or erosion of diplomatic relations. As a consequence,

political pressure against a takeover is only worthwhile if it has a sufficiently large impact.

We evaluate the effectiveness of such pressure by analysing its impact on the probability of

a takeover. Hence our analysis studies when governments are most likely to be willing to

incur the costs of increasing the pressure exerted on target Boards.

Suppose that after a bidding offer is made at t = 0, there exists a time t = 12

where the

government can increase y at some cost before the Board type θ is privately drawn. Thus a

government can encourage a Board to reject takeover bid, but they cannot compel them to

do so. From t = 1 the game continues as in the benchmark setting. The next result relates

the effectiveness of political pressure with the takeover premium:

Proposition 6 ∂β∂y≤ 0 is quasiconvex in P and reaches a minimum when β = 1

2. Therefore,

political pressure is most effective when the premium makes the takeover’s fail and success

equally likely.

Proof. See Technical Appendix.

The proposition indicates that increases in political pressure have the greatest impact on

the takeover’s outcome when the probability of a success is otherwise close to a half. Hence,

a government is most likely to find the cost of increasing political pressure in opposition

to a bid worthwhile if the bidding offer is competitive, but not overly so that the success

of the deal hangs in the balance. In contrast, extreme takeover premiums make the Board

likely to either accept or reject the offer. Then, it is unlikely that a marginal increase in

government pressure can change the takeover’s outcome thus raising the bar to political

intervention. A graphical intuition is provided by Figure 3, where we plot ∂β∂y

as a function

of P . The dashed line represents an arbitrary marginal cost of increasing y and hence, it

delimits the corresponding range of takeover premiums for which the government is willing to

exert political pressure. Finally, it is possible to see that political pressure reaches a highest

impact when the bidder’s offer is such that β = 12.

17

0

-1

-2

V = 10 12

∂β∂y

P

example marginal costof political pressure

region of pressure

β = 12

Figure 3: Effectiveness of marginal increases in political pressure y as a function of thetakeover bid P . When ∂β

∂yovercomes the marginal cost, the government would be willing to

increase pressure on the Board. The dot indicates P = V + y − κ2, which satisfies β = 1

2.

Parameter values are V = 10, y = 2, τε = 4, τθ = 5 and κ = 1.5

Governments are also repeatedly urged to take advantage of more formal instruments

to oppose takeovers. In the UK, for example, the case of Kraft and Cadbury reopened the

debate about the voting rights of short term shareholders. After the takeover, Mr. Carr, then

Cadbury’s CEO, argued that he was forced to recommend shareholders to tender due to the

dramatic increase in the proportion of stock held by opportunist investors. The unpopularity

of the case led regulatory bodies to consider the implementation of disfranchisement rules,

which aim that only the shareholders that are registered at the start of the offer period

are eligible to ”vote” on the takeover proposal. A significant goal was ”to ensure that the

outcome of takeover bids is determined by the core shareholder base, not by short term

speculative investors who may acquire shares in order to facilitate the takeover.”10 Once

again the UK is not unique in this respect. As noted in the introduction, in the both the US

and EU there have been prominent calls for shareholders of longer standing to be privileged

while newer shareholders, including arbitrageurs, it is argued should be disadvantaged in

their ability to influence the Board. From a political analysis (e.g. Hellwig 2000), this might

be viewed as an other measure to protect the incumbent corporate management.

10See: The Takeover Panel (2010). Review of certain aspects of the regulation of takeover bids. Consulta-tion paper issued by the Code Committee. London: 1 June 2010; and Department for Business & InnovationSkills (BIS), (2014). Practical and legal issues related to limiting the rights of short-term shareholders duringtakeover bids. Note of BIS roundtable, October 2014.

18

We have already established in Proposition 5 that reductions in the influence of new

shareholders (and so of arbitrageurs) will reduce the probability of a takeover for any set

of fundamentals (∂β/∂κ > 0) . However it will also increase the order flow – so more exist-

ing shareholders will sell to try and compensate for the reduced pressure from the Board

(∂ρ/∂κ < 0) . Thus favouring long-term owners will not lead to greater long-term ownership

in a takeover setting.

Next, we examine the effectiveness of political pressure against a takeover, i.e. the in-

centives to marginally increase y, as a function of the influence allowed for risk-arbitrageurs,

κ. The parameter κ accounts for the weight that the interests of those shareholders that

acquired stock at the interim have in the Board’s verdict. Thus, the goal of disfranchisement

rules can be represented in our setting with a reduction of κ. Suppose that before a takeover

offer is made, there is a time t = −1 at which the government can set κ. Then, the game

continues as in the benchmark model. The following partial result is available:

Proposition 7 ∂β∂y≤ 0 is monotonically decreasing in κ when public information is suffi-

ciently noisy, i.e. small τθ. Hence, political pressure is most effective when risk-arbitrageurs

have voting rights.

Proof. See Technical Appendix.

The proposition shows that in jurisdictions which have strong respect for shareholder

rights – including new shareholders (κ large) – the marginal impact of increases in pressure

on the Board are most pronounced: |∂β/∂y| is large in magnitude. If new shareholders have

strong influence then deal success is sensitive to the actions of the existing shareholders. A

small impact on the proportion of the shareholders selling can therefore achieve a substantial

reduction in the deal probability. Thus the incentive for the government to intervene rises.

6 Strategic Bidder

Thus far we considered an exogenous takeover premium, and we suggest that this is helpful in

capturing a contested takeover in which the bid price is forced above the level an individual

bidder would bid in the absence of competing buyers. Nonetheless, numerous takeover offers

are characterized by only one bidder. For instance, in the Kraft-Cadbury case, Hershey and

the group Ferrero contemplated teaming up to try and trump Kraft’s offer, although they

never came forward with a formal bid of their own.

This section studies the case of the takeover premium being strategically set by a Bidder

with no competitors. Hence, the Bidder makes an offer maximizing its expected profit, which

is the product of the surplus from the takeover and the probability that the takeover takes

19

place. Suppose that the Bidder’s value of the target company is W > V . The game proceeds

as in the benchmark setting with the only difference that, at t = 0, the offer P solves the

following bidder optimisation problem:

maxP∈[V,W ]

{Π = (W − P ) β(θ∗ (P ))} (10)

= maxP∈[V,W ]

{Π = (W − P ) Φ (√τθ [θ∗ (P )− y])}

where θ∗ (P ) is implicitly characterized by equation (4). We assume P ∈ [V,W ] for analytical

simplicity and for realism. Nonetheless, in principle an offer P < V can be optimal if the

Board have y < 0. This would be the case after political pressure urging a sale for example.

We omit those cases where the target Board is keen to be replaced. The takeover offer

determines both the Bidder’s surplus if the takeover succeeds and the probability that this

occurs. Furthermore, in Section 3 it was shown that P is directly related to the Board’s

decision threshold (∂θ∗

∂P) and as a result, increasing P raises the probability of a takeover

( ∂β∂P

). Therefore, increasing the takeover premium lowers the Bidder’s surplus but makes

the takeover more likely. The following Lemma characterizes the solution to the Bidder’s

problem:

Proposition 8 Suppose the Bidder’s value for the target company is sufficiently large, i.e.

W − V > y − κ2

+√π/2τθ. Then, Π in (10) is quasiconcave and has an interior maximum

that at the optimal bid P ∗ is implicitly characterized by:

P ∗ = W −[1− κ∆ϕ (∆ [θ∗ (P ∗)− y])] Φ

(√τθ [θ∗ (P ∗)− y]

)√τθϕ

(√τθ [θ∗ (P ∗)− y]

) (11)

Furthermore, at the optimal bid the probability of a successful deal, β(θ∗ (P ∗)) > 12.

Proof. See Technical Appendix.

For analytic solutions we restrict to the case in which the surplus available is large

enough. This allows us to show that the Bidder’s profit function is quasiconcave. Notably,

the Proposition implies that the optimal bid always satisfies θ∗ (P ∗)− y > 0 and therefore,

the probability that the takeover occurs is above one half.

We are interested in the effects of takeover resistance when the bidder is strategic. The

following proposition relates the parameter y with both the bidding offer and the probability

of a takeover:

Proposition 9 With a strategic Bidder, higher expected Board takeover resistance y in-

creases the takeover premium P , but reduces the probability that the takeover goes through β,

i.e. ∂P∂y≥ 0 and ∂β

∂y≤ 0.

20

Proof. See Technical Appendix.

The Proposition indicates the Bidder responds to increases in the expected takeover

resistance with bigger premiums. Nonetheless, the positive effect of a higher bid on the

probability of a takeover (Proposition 5) is insufficient to outweigh the negative impact

of greater resistance. Hence, the takeover becomes less likely. The result suggests that

corporate management and governments achieve two goals through their campaigns against

takeovers: first, they make the acquisition less likely; but second, they increase the surplus of

target shareholders if the takeover finally takes place. An additional feature of takeovers with

strategic bidders is the ambiguous response of interim stock prices to takeover resistance.

V = 10

11

12

13

14

W = 15

0 2 4 6 8y

premium

spread

P ∗

M

Figure 4: Optimal takeover bid P ∗ and the corresponding interim stock price M as a functionof takeover resistance y. The takeover premium is P − V whereas the spread from whichrisk-arbitrageurs aim to make profit is P − M . Parameter values are W = 15, V = 10,τε = 4, τθ = 5 and κ = .3.

Corollary 10 With a strategic Bidder, higher resistance y has ambiguous effects on the

interim stock price, M .

Figure 4 plots both the takeover offer P ∗ and the interim stock price M as a function of

the Board’s type y. It is possible to see that the takeover premium increases monotonically

in response to higher expected Board resistance. The concavity of P ∗ implies that when

expected Board resistance is relatively low, a marginal increase in the takeover premium has

a large impact on the probability that the takeover goes through, which allows the interim

price M to track the bid P . This effect diminishes as y increases. It is possible to see

that the interim stock price presents an inverted-U shape relationship with expected Board

takeover resistance. Intuitively, when the probability of a takeover is high (small y), an

increase of the takeover premium in response to additional resistance compensates for the

21

lower probability of a takeover. As a consequence, the stock price rises. This is in contrast

to when the takeover is less likely (large y). Then, a small premium rise is overcome by the

lower probability of a takeover and thus, the stock price falls.

7 Concluding Remarks

Takeover offers are often characterized by managerial resistance and frequently also political

pressure against the takeover. This paper formally studies a novel coordination problem

of the target shareholders. Once an offer is made, before the tendering decision there is a

speculative period where the takeover outcome is uncertain. Shareholders have to decide

whether to hold their stock in the hope that the takeover goes through or to sell it to

risk-arbitrageurs that in turn, increase the probability that the takeover is consummated.

As a consequence, each shareholder wants to sell her stock when other shareholders hold

it, but to hold it when other shareholders sell. Hence, selling becomes a public good to

which no one wants to contribute. In equilibrium, selling decisions and takeover success are

determined by the bid premium, the external pressure against the takeover and the influence

of risk-arbitrageurs.

It is shown that, even though political pressure against a takeover reduces the probability

that the takeover succeeds and reduces the stock price during the speculative period, it may

contribute to an increase in the volume of stock brought to market in the interim period and

purchased by arbitrageurs. Whether it does so depends on the expected Board resistance

and its relationship to the critical level needed to block the takeover. Moreover, marginal

increases in the pressure on Boards is most effective, and so most likely to exceed its marginal

cost, when takeover success and failure are otherwise equally likely. Additionally, we show

that limiting the voting rights of risk-arbitrageurs (disfranchisement rules) makes marginal

increases in political pressure against a given takeover less effective. Finally, the takeover

offer of a strategic bidder is examined. Results suggest that bidders increment takeover

premiums in response to additional external resistance. Hence, political pressure not only

reduces the probability that the takeover succeeds, but it also contributes to an increase in

the surplus of target shareholders if it occurs.

22

A Appendix I: Target Shareholders With Bidder’s Stock

Suppose that a proportion γ of the target shareholders own stock of the Bidder. These

shareholders obtain an additional payoff B ∈ R if the takeover succeeds. Denote common

shareholders those that own stock only in the target company and bidding shareholders those

who also own stock of the Bidder. Then, the order flow reads ρ = γρB + (1− γ) ρC and the

shareholders’ payoff is:

ρ ≥ h(θ) ρ < h(θ)

Common Bidding Common/Bidding

Sell M M +B M

Hold P P +B V

We derive the equilibrium as in Proposition 1. However, we initially assume that the two

groups of shareholders (common and bidding) have different decision thresholds: x∗C and x∗B.

The failure threshold reads

θ∗ = P − V + κ

(1− γ) [1− Φ (√τε [x∗C − θ∗])]︸ ︷︷ ︸ρC

+ γ[1− Φ (√τε [x∗B − θ∗])]︸ ︷︷ ︸ρB

(12)

Furthermore, the signal thresholds for the common shareholders satisfies

V Pr[θ > θ∗|x∗C ] + P Pr[θ ≤ θ∗|x∗C ] = M

Pr[θ ≤ θ∗|x∗C ] =M − VP − V

(13)

whereas for bidding shareholders it must hold

V Pr[θ > θ∗|x∗B] + (P +B) Pr[θ ≤ θ∗|x∗B] = M Pr[θ > θ∗|x∗B] + (M +B) Pr[θ ≤ θ∗|x∗B]

Pr[θ ≤ θ∗|x∗B] =M − VP − V

(14)

Expressions in (13) and (14) imply that x∗C = x∗B = x∗. Moreover, in equation (12) it

is possible to see that ρC = ρB = ρ. Hence, the two types of shareholders have the same

decision threshold and in turn, a proportional order flow. As a result, the fact that some

shareholders own stock of the bidding company has no effect in equilibrium. The derivation

of the thresholds θ∗ and x∗ follows as in the proof of Proposition 1.

23

B Appendix II: Informed Investors

The common prior about the Board’s type is θ ∼ N(y, 1/τθ). While shareholders do not

have additional information, there is a continuum of risk-arbitrageurs of mass one where

each arbitrageur i receives an iid signal xi = θ+ xi with εi ∼ N(0, 1/τε) and updates beliefs

accordingly. The interim stock price M is set as in (7) by market makers with zero expected

profits. Arbitrageurs acquire stock at the interim if their value is higher than M , i.e. if they

receive a sufficiently low signal of the Board’s type. The expected payoff of acquiring stock

is increasing on the number of arbitrageurs that follow the same strategy because these exert

higher pressure on the Board and thus, make the takeover more likely. Hence, arbitrageurs’

actions are strategic complements. Their payoffs are as follows:

ρ ≥ h(θ) ρ < h(θ)

Don’t buy 0 0

Buy P −M V −M

where V −M ≤ 0 by construction.

The equilibrium analysis is analogous to that of Section 4. As a result, the next propo-

sition follows:

Proposition 11 There exists a unique Bayesian Nash Equilibrium in which all investors

buy shares if and only if they observe a signal below x∗. The takeover succeeds if, and only if,

the takeover resistance is below the threshold θ∗. The thresholds are characterized implicitly

by the following equations:

θ∗ = P − V + κ [1− Φ (∆ (θ∗ − y))] (15)

x∗ = θ∗ − ∆√τε

(θ∗ − y) (16)

Proof. Analogous to the proof of Proposition 1. Note that the expressions for the probability

of a takeover (6) and the interim stock price (7) remain unchanged. Instead, the order flow

now reads ρ = Φ(√

τε (θ∗ − θ)−∆ (θ∗ − y)).

Comparing the threshold θ∗ with the expression in (4) it is possible to appreciate that

reversing the information structure switches the mass of strategic decisions influencing the

takeover, i.e. 1 − Φ (∆ (θ∗ − y)). In particular, the proportion of shareholders selling now

corresponds to the probability that an arbitrageur receives a signal x < x∗. This is in

contrast to the benchmark model, where the order flow equals the probability that x > x∗.

As a consequence, while the effect of the fundamental variables on both the probability of

24

a takeover and the interim stock price has the same sign than in the original setting, their

effect on the order flow is different:

Proposition 12 Responses of the main outcomes to marginal increases of the model funda-

mentals are as follows

Fundamental Effect on outcomes

Pr. Takeover β Interim Price M Order Flow ρ

P − V + + +

κ + + +

y − − +

Proof: An analysis analogous to the proof of Lemma 4 yields ∂θ∗

∂(P−V )> 0, ∂θ∗

∂κ> 0 and

∂θ∗

∂y∈ [0, 1). Then results are obtained with an analysis analogous to the proof of Proposition

5.

A higher premium incentivizes both arbitrageurs to acquire stock and shareholders to

hold it, thereby raising the interim price M . Nonetheless, due to the information advantage

of arbitrageurs, the former effect is bigger and hence, there is a higher order flow. An

analogous intuition applies to the influence of arbitrageurs on the Board’s decision κ, which

increases the probability that the takeover goes through. Finally, takeover resistance has a

positive impact on the order flow. Intuitively, since arbitrageurs beliefs are more accurate

than those of shareholders, the decrease of the stock price M outweighs the decrease of the

stock value for arbitrageurs. As a result, arbitrageurs have higher incentives to acquire stock.

25

C Technical Appendix

Proof of Proposition 1. A shareholder receiving a signal xi updates beliefs so that θ|xiis normally distributed with mean τθy+τεxi

τθ+τεand precision τθ + τε. Two equations allow us to

find an equilibrium threshold represented by a fixed point {θ∗, x∗} such that shareholders

sell their shares when they receive a signal x ≥ x∗ and the takeover succeeds if θ ≤ θ∗.

The signal threshold computes the signal x∗ that at a given critical θ∗ makes a shareholder

indifferent between selling and holding:

V Pr[θ > θ∗|x∗] + P Pr[θ ≤ θ∗|x∗] = M

Pr[θ ≤ θ∗|x∗] =M − VP − V

Introducing the belief updating we obtain

Φ

(√τθ + τε

[θ∗ − τθy + τεx

∗

τθ + τε

])=M − VP − V

which we can rearrange as

θ∗ − x∗ = −τθτε

(θ∗ − y) +

√τθ + τετε

Φ−1

(M − VP − V

)(17)

The failure threshold defines the θ∗ that at a given critical x∗, the Board declines if and

only if θ > θ∗.

θ∗ = P − V + κPr[x ≥ x∗|θ∗]︸ ︷︷ ︸ρ = %SHs selling

which, after considering the information structure, reads

θ∗ = P − V + κ [1− Φ (√τε [x∗ − θ∗])] (18)

By plugging (17) into (18) we obtain an equilibrium expression in terms of θ∗:

θ∗ = P − V + κ

[1− Φ

(τθ√τε

[θ∗ − y −

√τθ + τετθ

Φ−1

(M − VP − V

)])](19)

Now, recall that outside investors do not receive private signals and so for them θ ∼N (y, 1/τθ). Hence, the probability of the Board selling is equal to the probability that θ lies

below the critical θ∗, Φ(√

τθ (θ∗ − y)). Therefore, the value achieved by shareholders when

26

they sell to outside investors is given by the Market Maker’s zero profit condition as

M = PΦ (√τθ (θ∗ − y)) + V [1− Φ (

√τθ (θ∗ − y))] (20)

= (P − V ) Φ (√τθ (θ∗ − y)) + V

Substituting this into the equilibrium threshold we obtain the expression in (4), where we

already use the definition ∆ and the fact that Φ (x) = 1− Φ (−x).

For uniqueness, note that if P > V , the Right Hand Side (RHS) in expression (4) is

positive when θ∗ = 0. Hence, uniqueness is guaranteed if the slope of the RHS is strictly

lower than one. Note that

d

dθ∗[RHS (4)] = κ ·∆ · ϕ (∆ (θ∗ − y)) (21)

and that a normal pdf reaches a maximum at 1/√

2π.

For the signal threshold, use equation (17) to define

x∗ = θ∗(

1 +τθτε

)− τθτεy −√τθ + τετε

Φ−1

(M − VP − V

)= θ∗ − ∆

√τε

(θ∗ − y)

Where in the second line we used (20). �

Proof of Corollary 2: The proportion of shareholders selling their shares is

ρ = Pr[x > x∗|θ] = Pr[θ + εi > x∗|θ] = 1− Φ (√τε [x∗ − θ]) (22)

Plug the expression for x∗ in (5) into that of ρ and obtain the equation in (8). �

Proof of Corollary 3: First, note that ∆ =√τθ√

1 + τθτε− τθ√

τε, where it is clear

that limτε→∞∆ =√τθ. Second, use L’Hopital’s rule and let f(τε) = τθ

[√1 + τε

τθ− 1]

and

g(τε) =√τε. Then f ′(τε)

g′(τε)=√

τθτετθ+τε

and clearly limτε→0f ′(τε)g′(τε)

= 0. �

Proof of Lemma 4

- (a) Bid Premium. Define F (θ∗, P − V ) = θ∗−RHS (4) and use the Implicit Function

Theorem (IFT) so that ∂θ∗

∂(P−V )= −∂F (θ∗,P−V )/∂(P−V )

∂F (θ∗,P−V )/∂θ∗. Then,

∂θ∗

∂ (P − V )=

1

1− κ∆ϕ(∆ [θ∗ − y])> 0 (23)

27

where denominator is positive under uniqueness, i.e. when ∆ <√

2π/κ, as ϕ(·) takes

maximum value at 1/√

2π. Furthermore, note that ∂θ∗

∂(P−V )> ∂θ

∂(P−V )= ∂θ

∂(P−V )= 1.

- (b) Internal Pressure. Use the IFT so that dθ∗

dκ= − ∂F (θ∗,κ)/∂κ

∂F (θ∗,κ)/∂θ∗. Then,

dθ∗

dκ=

Φ(∆ [θ∗ − y])

1− κ∆ϕ(∆ [θ∗ − y])> 0 (24)

Note from (2) that ∂θ∂κ

= 0 and ∂θ∂κ> 0.

- (c) Board’s Expected Type. Use the IFT so that ∂θ∗

∂y= − ∂F (θ∗,y)/∂y

∂F (θ∗,y)/∂θ∗. Then,

∂θ∗

∂y= − κ∆ϕ(∆ [θ∗ − y])

1− κ∆ϕ(∆ [θ∗ − y])< 0 (25)

Notice also that ∂θ∂y

= ∂θ∂κ

= 0. �

Proof of Proposition 5

- (a) Bid Premium. Note in (6) that dβd(P−V )

= ∂β∂θ∗

∂θ∗

∂(P−V )where ∂β

∂θ∗> 0 and ∂θ∗

∂(P−V )> 0

- see (23). Furthermore, dMd(P−V )

= ∂M∂(P−V )

+ ∂M∂β

∂β∂θ∗

∂θ∗

∂(P−V )with ∂M

∂(P−V )> 0 and ∂M

∂β> 0.

Finally, dρd(P−V )

= ∂ρ∂θ∗

∂θ∗

∂(P−V )where ∂ρ

∂θ∗< 0 because

√τε −∆ > 0.

- (b) Internal Pressure. First, notice in (6) that dβdκ

= ∂β∂θ∗

∂θ∗

∂κwhere ∂β

∂θ∗> 0 and ∂θ∗

∂κ> 0 -

see (24). Moreover, dMdκ

= ∂M∂β

∂β∂θ∗

∂θ∗

∂κ> 0 since ∂M

∂β> 0. Last, dρ

dκ= ∂ρ

∂θ∗∂θ∗

∂κ< 0 because

∂ρ∂θ∗

< 0.

- (c) Board’s Expected Type. The effect on β is dβdy

= ∂β∂y

+ ∂β∂θ∗

∂θ∗

∂y< 0 because ∂β

∂y< 0;

∂β∂θ∗

> 0 and ∂θ∗

∂y< 0. Furthermore, dM

dy= ∂M

∂β∂β∂θ∗

∂θ∗

∂y< 0 because ∂M

∂β> 0. Finally, note

in (8) that

dρ

dy=sign

∆

(∂θ∗

∂y− 1

)−√τε∂θ∗

∂y

=√τε

(κ∆ϕ(∆ [θ∗ − y])

1− κ∆ϕ(∆ [θ∗ − y])

)−∆

(κ∆ϕ(∆ [θ∗ − y])

1− κ∆ϕ(∆ [θ∗ − y])+ 1

)= [√τεκϕ(∆ [θ∗ − y])− 1]

(∆

1− κ∆ϕ(∆ [θ∗ − y])

)=sign

√τεκϕ(∆ [θ∗ − y])− 1

Hence, if |θ∗ − y| is small (large) then ϕ(∆ [θ∗ − y]) is large (small) and dρdy> (<)0.

However, note that ϕ(·) ≤ 1√2π

. As a consequence, if κ <√

2πτε

it always holds dρdy< 0.

28

Furthermore√

2πτε<√

2π∆

, where the term in the RHS is the condition for uniqueness.

Therefore, we conclude that if κ ∈[0,√

2π/τε

], then dρ

dy< 0. Instead, when κ ∈[√

2π/τε,√

2π/∆]

the sign of dρdy

depends on the value of |θ∗ − y|. �

Proof of Proposition 6: The relation between θ∗ and y is crucial to characterize the

effect of fundamentals. In expression (4) it is possible to see that θ∗ = y ←→ y = θ+θ2

=

P − V + κ2. Hence, ∂θ∗

∂y< 0 from Lemma 4 implies that θ∗ > y ←→ y < P − V + κ

2.

Furthermore, notice in (6) that β > 12←→ y < P − V + κ

2. We use these results as a

reference in our analysis.

Note that∂2β

∂P∂y=

∂

∂y

( √τθϕ(√τθ (θ∗ − y)

1− κ∆ϕ(∆ (θ∗ − y))

)Therefore ∂2β

∂P∂y> 0 ↔ y < P − V + κ

2or equivalently, ∂2β

∂P∂y> 0 ↔ P − V > y − κ

2. Hence,

∂β∂y

is quasiconvex in P − V with a minimum at P − V = y − κ2.�

Proof of Proposition 7:Notice that

∂2β

∂κ∂y=

∂

∂κ

(−√τθϕ(√τθ (θ∗ − y)

1− κ∆ϕ(∆ (θ∗ − y))

)=sign−[(

∂θ∗

∂κ

)τθϕ

′(√τθ (θ∗ − y))

][1− κ∆ϕ(∆ (θ∗ − y))]

− [√τθϕ(√τθ (θ∗ − y))]

[∆ϕ(∆ (θ∗ − y)) +

(∂θ∗

∂κ

)κ∆2ϕ′(∆ (θ∗ − y))

]where ∂θ∗

∂κ> 0 by Lemma 4. Furthermore, note both ϕ′(

√τθ (θ∗ − y)) > 0 and ϕ′(∆ (θ∗ − y)) >

0 if and only if θ∗ < y. As a result, ∂2β∂κ∂y

< 0 for θ∗ ≤ y or equivalently, for κ ≤2 [y − (P − V )].

Consider κ ∈(2 [y − (P − V )] ,

√2π/∆

)so that θ∗ > y. Using that ϕ′ (x) = −xϕ (x) we

have

∂2β

∂κ∂y=sign

√τθϕ(√τθ (θ∗ − y))

(θ∗ − y)

(∂θ∗

∂κ

)τ

3/2θ [1− κ∆ϕ(∆ (θ∗ − y))]

+ (θ∗ − y)(∂θ∗

∂κ

)κ∆3ϕ (∆ (θ∗ − y))

−∆ϕ(∆ (θ∗ − y))

Now using the fact that

(∂θ∗

∂κ

)[1− κ∆ϕ(∆ (θ∗ − y))] = Φ (∆ (θ∗ − y)) we can write

∂2β

∂κ∂y=sign

√τθϕ(√τθ (θ∗ − y))∆ϕ (∆ (θ∗ − y))

{(θ∗ − y)

[τ

3/2θ

∆

Φ (∆ (θ∗ − y))

ϕ (∆ (θ∗ − y))+

(∂θ∗

∂κ

)κ∆2

]− 1

}

29

As θ∗ < P − V + κ we can guarantee that the brace is negative if

(θ∗ − y)

[τ

3/2θ

∆

Φ (∆ (θ∗ − y))

ϕ (∆ (θ∗ − y))+

Φ (∆ (θ∗ − y))

[1− κ∆ϕ(∆ (θ∗ − y))]κ∆2

]< 1

We know that this is true if τθ is small enough as limτθ→0τθ/∆ = 0 and limτθ→0

∆ = 0.�

Proof of Proposition 8. As we are maximising a continuous function over a compact

set, a solution exists. We have that the optimal bid P ∗ < W as [dΠ/dP ]P=W = −β (θ∗ (P )) <

0. Now note that

∂Π

∂P= −Φ (

√τθ [θ∗ (P )− y]) + (W − P )

√τθ∂θ∗ (P )

∂Pϕ (√τθ [θ∗ (P )− y]) (26)

=

[(W − P )

√τθ

1− κ∆ϕ (∆ [θ∗ (P )− y])−

Φ(√

τθ [θ∗ (P )− y])

ϕ(√

τθ [θ∗ (P )− y])]ϕ (

√τθ [θ∗ (P )− y])

We can guarantee that a solution satisfies P ∗ ≥ V + y− κ2, if ∂Π

∂P> 0 for P < V + y− κ

2.

Next, we show that this holds for P < W −√π/2τθ. Use the following claims:

• Claim 13 θ∗ (P ) < y and β(θ∗ (P )) < 12

if and only if P < V + y − κ2.

Proof of Claim 13 : Note that ∂θ∗(P,y)∂y

< 0 and θ∗ (P, y) = y when P = y + V − κ2

Claim 14 Φ(x)ϕ(x)

is increasing in x. Hence, maxx

{Φ(x)ϕ(x)

}|x≤0=

√π2.

Proof of Claim 14 : The result is immediate for x ≥ 0. Consider x < 0. The result

follows by differentiation if ϕ2 (x) − Φ (x)ϕ′ (x) > 0. As ϕ′ (x) = −xϕ (x) we wish to

show ϕ (x) + xΦ (x) > 0. Set y = −x > 0, and using symmetry we require ϕ (y) −y (1− Φ (y)) > 0 for y > 0. This follows if (1− Φ (y)) /ϕ (y) < 1/y. This inequality is

confirmed in Gordon (1941). Furthermore, note that Φ (0) /ϕ (0) =√π/2.

The combination of Claim 13 and Claim 14 implies that ∂Π∂P

> 0 when P < V + y − κ2

if

(W − P )√τθ

1− κ∆ϕ (∆ [θ∗ (P )− y])−√π

2> 0 (27)

Therefore, a sufficient condition for (27) is P < W −√π/2τθ. By setting ∂Π

∂P= 0 we obtain

the expression in (11), where it is possible to see that P ∗ < W −√π/2τθ.

Now the concavity of Π for P ≥ V + y − κ2

is sufficient to guarantee a unique interior

solution. Notice that

∂2Π

∂P 2= −2

∂β(θ∗(P ))

∂P+ (W − P )

∂2β(θ∗(P ))

∂P 2(28)

30

where ∂β(θ∗(P ))∂P

> 0 and

∂2β(θ∗(P ))

∂P 2=

[√τθ∂θ∗

∂P

]2

ϕ′ (√τθ [θ∗ (P )− y]) + ϕ (

√τθ [θ∗ (P )− y])

√τθ

(∂2θ∗

∂P 2

)(29)

Given P ≥ V+y−κ2, it must be that θ∗ (P )−y > 0 by Claim 2 and therefore ϕ′

(√τθ [θ∗ (P )− y]

)<

0. Moreover, algebra confirms that ∂2θ∗

∂P 2 ≤ 0.

Finally, notice that we require V+y−κ2< W−

√π/2τθ and that P ∗ ∈

[V + y − κ

2,W −

√π/2τθ

].

�

Proof of Proposition 9.

9.A) Proof dPdy≥ 0

The optimal bid P ∗ satisfies

β (θ∗ (P )) = (W − P )

[∂β (θ∗ (P ))

∂θ∗∂θ∗ (P )

∂P

]= (W − P )

(√τθϕ

(√τθ [θ∗ (P )− y]

)1− κ∆ϕ(∆ [θ∗ (P )− y])

)(30)

Differentiate both sides with respect to y so as to get dLHSdy

= dRHSdy

. Note that β (θ∗ (P (y) , y) , y).

HencedLHS

dy=

∂β

∂θ∗

(∂θ∗

∂P

dP

dy+∂θ∗

∂y

)+∂β

∂y(31)

and

dRHS

dy= −dP

dy

(√τθϕ

(√τθ (θ∗ − y)

)1− κ∆ϕ(∆ (θ∗ − y))

)+

(W − P )√τθ

[1− κ∆ϕ(∆ (θ∗ − y))]2·

·

(∂θ∗

∂PdPdy

+ ∂θ∗

∂y− 1)√

τθϕ′ (√τθ (θ∗ − y)

)[1− κ∆ϕ(∆ (θ∗ − y))]

+ϕ(√

τθ (θ∗ − y)) (

∂θ∗

∂PdPdy

+ ∂θ∗

∂y− 1)κ∆2ϕ′(∆ (θ∗ − y))

Rewriting we have

dRHS

dy= −dP

dy

∂β

∂θ∗∂θ∗

∂P+

(∂θ∗

∂P

dP

dy+∂θ∗

∂y− 1

)·

·(W − P )

√τθ

[1− κ∆ϕ(∆ (θ∗ − y))]2

{ √τθϕ

′ (√τθ (θ∗ − y))

[1− κ∆ϕ(∆ (θ∗ − y))]

+ϕ(√

τθ (θ∗ − y))κ∆2ϕ′(∆ (θ∗ − y))

}︸ ︷︷ ︸

Ψ

31

where Ψ < 0←→ θ∗ > y. Rearranging we get

dRHS

dy= −dP

dy

∂β

∂θ∗∂θ∗

∂P+

(∂θ∗

∂P

dP

dy+∂θ∗

∂y− 1

)Ψ (32)

=dP

dy

(∂θ∗

∂PΨ− ∂β

∂θ∗∂θ∗

∂P

)+ Ψ

(∂θ∗

∂y− 1

)Finally, dLHS

dy= dRHS

dyyields

dP

dy

∂β

∂θ∗∂θ∗

∂P+∂β

∂θ∗∂θ∗

∂y+∂β

∂y=

dP

dy

(∂θ∗

∂PΨ− ∂β

∂θ∗∂θ∗

∂P

)+ Ψ

(∂θ∗

∂y− 1

)(33)

dP

dy

2∂β

∂θ∗︸︷︷︸+

∂θ∗

∂P︸︷︷︸+

− ∂θ∗

∂P︸︷︷︸+

Ψ︸︷︷︸−

︸ ︷︷ ︸

+

= Ψ︸︷︷︸−

∂θ∗

∂y︸︷︷︸−

− 1

− ∂β

∂θ∗︸︷︷︸+

∂θ∗

∂y︸︷︷︸−

− ∂β

∂y︸︷︷︸−︸ ︷︷ ︸

+

9.B) Proof dβdy≤ 0

Note that β (θ∗ (P (y) , y) , y). Hence, differentiating both sides of the expression above

we havedβ

dy=

∂β

∂θ∗

(∂θ∗

∂P

∂P

∂y+∂θ∗

∂y

)+∂β

∂y

We can plug in all corresponding expressions in the RHS with the exception of ∂P∂y

. Then,

dβ

dy=√τθϕ (

√τθ (θ∗ − y))

(∂P∂y

)− κ∆ϕ(∆ [θ∗ − y])

1− κ∆ϕ(∆ [θ∗ − y])

−√τθϕ (√τθ (θ∗ − y))

=

√τθϕ

(√τθ (θ∗ − y)

)1− κ∆ϕ(∆ [θ∗ − y])

[(∂P

∂y

)− 1

]

So dβdy

=sign

(∂P∂y

)− 1.

Now recall

P = W −[1− κ∆ϕ (∆ (θ∗ − y))] Φ

(√τθ (θ∗ − y)

)√τθϕ

(√τθ (θ∗ − y)

)and let

F = P −W +[1− κ∆ϕ (∆ (θ∗ − y))] Φ

(√τθ (θ∗ − y)

)√τθϕ

(√τθ (θ∗ − y)

)so we can use the IFT ∂P

∂y= − ∂F/∂y

∂F/∂P. For this we fix P , so we consider P rather than P (y)

and θ∗ (y) rather than θ∗ (P (y), y).

32

First, compute ∂F∂y

:

∂F

∂y=

(A+B)√τθϕ

(√τθ (θ∗ − y)

)+ C

τθ[ϕ(√

τθ (θ∗ − y))]2 < 0

where

A ≡ −(∂θ∗

∂y− 1

)κ∆2ϕ′ (∆ (θ∗ − y)) Φ (

√τθ (θ∗ − y)) < 0

B ≡ [1− κ∆ϕ (∆ (θ∗ − y))]

(∂θ∗

∂y− 1

)√τθϕ (

√τθ (θ∗ − y)) < 0

C ≡ − [1− κ∆ϕ (∆ (θ∗ − y))] Φ (√τθ (θ∗ − y))

(∂θ∗

∂y− 1

)τθϕ

′ (√τθ (θ∗ − y)) < 0

Second, compute ∂F∂P

:

∂F

∂P= 1 +

− (A+B)√τθϕ

(√τθ (θ∗ − y)

)− C

τθ[ϕ(√

τθ (θ∗ − y))]2 > 0

using the fact that ∂θ∗

∂y− 1 = −∂θ∗

∂P.

Now

∂P

∂y= −

(A+B)√τθϕ

(√τθ (θ∗ − y)

)+ C

τθ[ϕ(√

τθ (θ∗ − y))]2 − (A+B)

√τθϕ

(√τθ (θ∗ − y)

)− C

∈ [0, 1)

�

33

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