The Timing of Takeovers in Growing and Declining Markets∗ Robin Mason Department of Economics, University of Southampton and CEPR Helen Weeds Department of Economics, University of Essex and CEPR 22 March 2007

Abstract Industry-level studies have found that takeover activity is positively related to the absolute size of industry-wide shocks. In this paper we develop a dynamic model with competing bidders to analyze the timing of takeovers, explaining this pattern within a single framework. Takeovers may create value either by exploiting synergies or by improving efficiency through consolidation and disinvestment. In our model, we assume that the relative value of each type of acquisition is affected by shocks to an industry variable. With competing bidders of different types, takeovers occur only when shocks are sufficiently large in either direction, with no activity taking place in between. We also analyze the dependence of takeover activity on the degree of uncertainty about industry conditions. We examine implications of our model for efficiency of the market for corporate control. Keywords: Takeovers, Mergers, Acquisitions, Real options, Timing games. JEL classification: D44, D81, G34.

∗

We are grateful to seminar participants in Cambridge, Essex, the Olin School of Business, Southampton, Toronto, and the UK Competition Commission. We thank V. Bhaskar and Juuso V¨ alim¨ aki in particular, for helpful comments. Further comments are very welcome. The latest version of the paper can be found at http://www.soton.ac.uk/∼ram2. Robin Mason gratefully acknowledges financial support from the ESRC Research Fellowship Award R000271265.

1

Introduction

When do takeovers occur? At an aggregate level, the empirical evidence shows that takeover activity occurs in waves, peaking with the business cycle and stock market valuations. But at an industry level, takeovers occur after both positive and negative shocks to industry-specific conditions. In this paper, we develop a theory to explain this industry-level phenomenon. In an equilibrium model of competitive takeovers, we show how takeovers do not occur when industry-specific shocks are moderate, but only when they are sufficiently large. We also analyse how takeover activity depends on the degree of uncertainty about industry conditions. Finally, we examine the implications of our theory for the efficiency of the market for corporate control. Our model has two key components. We assume that there are two types of firms who make takeover offers, distinguished by their different comparative advantages in running the acquired firm. The first type (the ‘growth-acquirer’) exploits synergies, such as economies of scale to lower marginal costs, and expands production. The second type (the ‘decline-acquirer’) looks to consolidate and shed capacity, for example through cutting fixed costs of production. (See Andrade and Stafford (2004) for empirical work on this distinction.) The values that these firms place on a target are driven by a stochastic industry variable, such as the level of demand, that varies randomly over time. We assume that the growth-acquirer has the higher valuation when the state is high (e.g., its economies of scale generate greater profit when demand is large). Conversely, the decline-acquirer has the higher valuation when the state is low (fixed costs savings yield greater returns than scale economies when demand is low).1 The second component is that the takeover market is competitive: we assume that when one firm decides to make an offer, a bidding war results, with the target sold to the highest bidder. This is a plausible assumption: there are often multiple suitors for specific targets; and competition between bidders for a single target reduces bidders’ returns (see 1

An alternative case arises when the bidders are in different countries. In this case, the state variable is the exchange rate between the countries. If firm 1 is in the same country as the target, it has the higher valuation when the exchange rate is strong; when the exchange rate is weak, the foreign firm 2 has the higher valuation. We are grateful to Pauli Murto for suggesting this example.

1

e.g., Bradley, Desai, and Kim (1988) and De, Fedenia, and Triantis (1996)). These two components lead to our main result: there is delay in equilibrium. The growth-acquirer makes a takeover offer only when the state is sufficiently high, while the decline-acquirer makes an offer only when the state is sufficiently low. This equilibrium delay occurs because competition makes payoffs convex. The acquiring firms face a combination of irreversibility—they do not get a second chance to acquire the target if they lose—and uncertainty—valuations are driven by a stochastic state variable. (There are, therefore, ‘real options’ involved in the bidding decision.) But on their own, irreversibility and uncertainty are not sufficient to cause delay—we show that competition is also required for delay to occur. Each acquirer’s payoff function is convex, since if it loses, it receives a payoff of zero. The convexity induced by competition gives the acquiring firms an incentive to delay making their takeover offers. We show that the extent of delay increases with the degree of uncertainty: both firms wait for more extreme values of the state variable before making an offer. In deciding when to make an offer, the bidders balance two factors. The first is the benefit from delay that arises due to real option effects. The second is the cost from being pre-empted i.e., the state variable changing by a large amount so that the rival firm acquires the target. Our result shows that the first factor dominates. Thirdly, we show that there is too much delay in equilibrium, relative to the efficient solution. The efficient solution does involve delay: a firm should take over the target only when its valuation is sufficiently greater than its rival’s. But the extent of delay in the efficient solution is less than in equilibrium. There is too much delay in equilibrium because competition creates negative externalities between bidders. These externalities lower a acquirer’s equilibrium payoffs (relative to the efficient payoffs), both when the firm wins the takeover battle and when it loses. The reduction in payoff on winning leads the firm to make its offer later (relative to the efficient solution); the reduction on losing leads to an earlier offer. We show that the former is more important when the firms choose optimally when to make their offers.2 2

The competition externalities may also affect the shape, as well as the level, of equilibrium payoffs. In our model, this shape effect is not present.

2

Finally, we consider two issues for the owner of the target firm. First, we suppose that the owner can choose the time at which the firm is sold. We show that the owner wants to sell immediately and so suffers from the delay in equilibrium takeover offers. Secondly, we suppose that the owner cannot directly control the timing of takeover, but can choose the target’s exposure to industry shocks (e.g., by its choice of investment projects). We show that the owner will choose the lowest possible exposure to uncertainty in order to reduce takeover delay. Finally, we use numerical analysis to contrast the target’s preferences to the efficient choice of the level of uncertainty. In summary, our results have the following implications for takeover activity and the market for corporate control. First, they imply that even when there is competition between acquirers, the target may not be acquired immediately. Secondly, takeover activity will be highest during ‘booms’ and ‘busts’ i.e., when there extreme shocks to an industry, with lower activity in between. Finally, the market for corporate control is inefficient: takeovers occur too infrequently. The field of empirical study of the pattern of takeovers is large (see, e.g., Weston, Chung, and Hoag (1990) for a review). Of particular relevance for our analysis are the papers that study the implications of industry-level shocks for takeover activity. Mitchell and Mulherin (1996) find that takeover activity in the 1980s clustered disproportionately at the industry level. In addition, they find that takeover activity was greatest in those industries exposed to the greatest shocks. They use two measures of shocks: the absolute value of the difference between a particular industry’s sales growth, and the average sales growth across all 51 industries in their sample; and the same, but using employment as the state variable. They then relate these absolute differences to a number of different measures of takeover activity. They find that sales and employment shocks are positively and significantly related to takeover attempts and actual takeovers. In other words, takeovers occur in industries experiencing large negative and positive shocks. In contrast, industry sales growth has no explanatory power for industry variation in takeover activity. Three other papers support this main finding. Harford (2005) observes that absolute changes to a number of different variables (including profitability, employee growth and

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sales growth) are all abnormally high prior to increases in takeover activity. Andrade and Stafford (2004) find a positive relationship between merger activity and the absolute deviation of sales growth from its long-term mean within an industry. Finally, Bernile, Lyandres, and Zhdanov (2006) focus on the effect of demand shocks specifically on merger activity; they also find a significant ‘U’-shaped relationship between mergers and shocks. Compared with the large empirical literature, there are relatively few papers dealing theoretically with the timing of takeovers. A number of papers have concentrated on the relationship between aggregate shocks and takeovers. For example, Shleifer and Vishny (2003) consider the implications of stock market misvaluation for takeover activity. A smaller number of papers have concentrated on industry-level shocks. Lambrecht (2004) argues (like us) that merger synergies are an increasing function of product market demand. Fixed costs to merging and stochastic demand generates real options. Consequently, firms have an incentive to merge only in periods of economic expansion i.e., takeovers are pro-cyclical. In contrast, Lambrecht and Myers (2006) analyse takeovers in declining markets. In their model, the managers of a declining firm are reluctant to shut their firm down. Takeovers are permitted by shareholders as a means to solve this agency problem. They characterize how the timing of takeovers depends on the identity of the acquirer (for example, whether it is a management buy-out, or a raider). In Morellec and Zhdanov (2005), two firms compete to acquire a target; the authors determine when takeover occurs in equilibrium, and the terms (division of surplus between acquirer and target) of the takeover. In their model, takeovers occur only in growing markets (i.e., when the acquirer’s relative cash flows are sufficiently large); and competition between bidders speeds up the takeover process. Our results are quite different: takeovers occur in growing and declining markets; and competition causes delay. There is one key reason for these differences. In Morellec and Zhdanov (2005), bidders’ valuations of the target are positively (and perfectly) correlated. We, in contrast, assume that valuations are negatively correlated. Of course, both assumptions are reasonable, and could both be included in a model with more than two acquirers. We choose the starkest possible case—two acquirers with negatively correlated valuations—in order to understand the

4

empirical relationship between takeovers and the absolute size of industry shocks. Finally, Bernile, Lyandres, and Zhdanov (2006), like us, look to provide a theoretical explanation of this empirical relationship. (They also provide further empirical verification of it.) In their model, incumbent firms have a static incentive to merge, in order to increase their market power. But by merging, the incumbents may create an added incentive for an outside firm to enter their industry. The entry decision is unaffected by merger at extreme values of industry profitability: at low levels, entry will not occur, and at high levels, it will, regardless of the incumbents’ behaviour. Hence, merger occurs at these extreme values. For moderate levels of industry profitability, an entrant will enter only if there is insufficient competition in the market i.e., only if the incumbents have merged. Hence, at these profitability levels, the incumbents choose not to merge, in order to deter entry. Clearly, this theory is quite different from ours, even though the final results are similar.3 Our review of the theoretical literature has been deliberately selective. The papers that we have discussed share with us the modelling framework employed: firms face irreversibility and uncertainty when making their decisions; uncertainty and shocks are modelled by a continuous time stochastic process. All these papers therefore contribute to the small, but growing number of papers analysing strategic interaction in real options settings. See, for example, Smets (1991), Grenadier (1996), Hoppe (2000), Weeds (2002), Lambrecht and Perraudin (2003), and Mason and Weeds (2004). In the rest of the paper, we first describe the model (section 2). We then characterise equilibrium in what we call competitive cautious trigger strategies; in this equilibrium, the timing of takeover is of primary interest. In section 4, we derive the efficient solution and contrast this to the equilibrium outcome. Section 5 summarises the main results, before section 6 considers the implications for the types of project that are chosen by the target firm. Section 7 discusses the robustness of the results. 3

There is, however, one major difference in the results. Bernile, Lyandres, and Zhdanov (2006) find that the zone in which no takeover activity is observed may actually decrease as uncertainty increases. In contrast, in our model, this zone always increases with uncertainty. This offers the prospect of distinguishing empirically between the two models.

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2

The Model

Time is continuous and the time horizon is infinite with t ∈ [0, +∞). Two risk neutral firms each can bid to acquire a target. The decisions to bid and acquire can be delayed indefinitely. Once the target has been acquired by one bidder, no further actions are considered. This limits the analysis to one ‘cycle’ of acquisition, for simplicity. One justification for this assumption is that post-merger integration prevents subsequent acquisition of the original business unit. We do not assume a sunk cost of making an offer (though this could be added). We normalize the value of the target to the owner to zero. The valuations of the acquirers, denoted si (θ) : R+ → R+ , i ∈ {G, D}, are functions of an exogenous, stochastic variable θ ∈ R+ . The variable θ evolves according to a geometric Brownian motion (GBM) without drift: dθt = σθt dWt

(1)

where σ ∈ [0, +∞) is the instantaneous standard deviation or volatility parameter, and dWt is the increment of a standard Wiener process {Wt }t≥0 , so that dWt ∼ N (0, dt). The continuous-time discount rate is r > 0. The parameters σ and r are common knowledge and constant over time. (The choices of continuous time and this representation of uncertainty are motivated by the analytical tractability of the value functions that result.) We interpret θ as an industry variable, such as the level of demand, which varies stochastically over time. The following assumption is made about the acquirers’ valuations: Assumption 1

sG (θ) = λθ, λ > 0;

sD (θ) = η, η > 0.

The fact that the firms’ valuations are non-negative means that we focus on the case in which it is efficient to sell the target; moreover, with two credible bidders at every value 6

of θ, bidding occurs competitively in equilibrium. A crucial feature for our analysis is that the relative valuation δ(θ) ≡ sG (θ) − sB (θ) is upwards single-crossing in θ: firm G’s valuation of the target is greater than firm D’s when θ is high—above θ ∗ ≡ η/λ. Conversely, firm D has the higher valuation when θ is low—below θ ∗ . These linear functional forms are chosen for their analytical convenience. One informal story to support them is that firm G is a firm that has synergies with the target that allow a lower unit cost of production after acquisition. When the state θ is the level of demand in the industry, firm G can generate an extra profit, of the form λθ (where λ is related to the cost difference). In contrast, firm D is a consolidator who aims to reduce the fixed costs of the target by η. In appendix B, we show how the analysis can be generalized beyond the linear case. We formulate the game form and the strategies of the firms to reflect the structure that we have imposed on the valuations. The game has two stages. The first stage is a timing game in which the acquiring firms decide when to bid (i.e., at what level of the state variable). Once one or more of the firms decides to make an offer, the game then enters a second, bidding stage in which the firms both submit bids (if they wish). To formalize this story, at any time t ≥ 0, let x = 0 if firm G has not made a bid for the target at any time τ ≤ t, and x = 1 otherwise; let y ∈ {0, 1} indicate the same for firm D. The state variable of the game is the triple (θ, x, y). At any time t > 0, past realizations of the process described by equation (1) and past decisions whether to bid constitute the history of the game. The firms are assumed to use stationary Markovian strategies: actions depend on only the current state and the strategy formulation itself does not vary with time.4 A (pure) Markovian strategy for firm i ∈ {G, D} (at time t) has two parts: (i) when xt = yt = 0, a measurable function5 mi (θt ) : R+ → {0, 1} which 4

For further explanation, see Maskin and Tirole (1988) and Fudenberg and Tirole (1991). NonMarkovian equilibria may exist. Since we want to analyse how the resolution of uncertainty affects the take-over game, we concentrate on equilibria in Markovian strategies. This allows us to rule out collusive equilibria with continuation strategies that depend on information that is not payoff relevant. 5 Measurability is with respect to the filtration F of the complete probability space (Ω, F, P ) on which

7

takes the value 0 when the firm has not made a bid, and the value 1 when the firm makes a bid; (ii) when xt + yt > 0, a measurable function bi (θt ) : R+ → R describing the bid submitted by the firm. In the bidding stage, firm i acquires the target if and only if its bid bi exceeds its rival’s b−i and is greater than zero. If this is the case, then firm i receives a payoff si (θ) (which can be interpreted as the present discounted value of a flow payoff sˆi (θ) received in perpetuity after acquisition), while paying its bid to acquire the target. If firm i bids unsuccessfully, then it receives a (flow) payoff of zero. In the event of a non-zero tie (bG = bD > 0), the target is allocated randomly between the firms. Hence the intertemporal payoff of firm i is

Vi (T, bG , bD ; θt ) =

    −r(T −t)  (s (θ ) − b (θ ))e E  i T i T t       0

if bi > max[0, b−i ], if bi < max[0, b−i ], (2)

   1 −r(T −t)  E  (s (θ ) − b (θ ))e if bi = b−i > 0, t i T i T  2      0 if bi = b−i = 0,

where T ≡ min[TG , TD ] and Ti is the (random) first time at which mi = 1; the operator Et denotes expectations conditional on information available at time t. Definition 1 (Markov Perfect Equilibrium, MPE) A pair of strategies (m∗G , b∗G ), (m∗D , b∗D ), with T ∗ ≡ min[TG∗ , TD∗ ] and Ti∗ ≡ inf{t|m∗i (θt ) = 1}, is a Markov Perfect Equilibrium if and only if, for all θt and i ∈ {G, D}, Vi (T ∗ , b∗i , b∗−i ; θt ) ≥ Vi (T, b∗i , b∗−i ; θt ), ∀ mi (θt ), Vi (T ∗ , b∗i , b∗−i ; θt ) ≥ Vi (T ∗ , bi , b∗−i ; θt ), ∀ bi ∗ where T ≡ min[Ti , T−i ], Ti ≡ inf{t|mi (θt ) = 1}.

Note that deviation strategies are not required to be Markovian. the Wiener process {Wt }t≥0 in equation (1) is defined.

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2.1

Takeover with a Single Acquirer

In this model, it is relatively straightforward to determine when a single firm, not faced with a rival bidder, will make its offer. Both firms have a linear valuation function. Consequently, there is no option value to either firm and hence no incentive to delay bidding. (The firms would have an incentive to delay only if their valuation functions were sufficiently convex in θ.) As we shall see, this is in marked contrast to the competitive bidding situation: in the game between the two competing firms, equilibrium always involves delay. This is because the prospect of a rival acquiring the target introduces a convexity into the firm’s payoff function; this convexity leads to delay in equilibrium.

3

Equilibrium

Our main interest is in equilibria with the following two properties: Property 1 (Competitive Cautious Bidding) (i) if one firm bids, then so does the other; (ii) the firms epsilon-outbid each other; (iii) the losing firm bids cautiously. 6 In terms of equilibrium strategies, for i ∈ {G, D},

bi (θ)

  = b−i (θ) +  if b−i < si ,  ≤s i

if b−i ≥ si ,

where  > 0 is arbitrarily small.

Property 2 (Trigger Bids) Firm G bids at the first instant that the state variable θ hits the interval [θG , +∞). Firm D bids at the first instant that the state variable θ hits the interval [0, θD ]. Formally,

mG (θ) =

   0   1

θ < θG , mD (θ) =

   0 θ > θ D ,

  1 θ ≤ θ D .

θ ≥ θG ;

I.e., firm G (D) bids when θ rises (falls) to θG (θD ), where θG > θD . 6

I.e., so as to be indifferent between winning and not winning at the equilibrium bids.

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Definition 2 (Competitive, Cautious, Trigger Equilibrium) Any equilibrium that satisfies properties 1 and 2 is called a CCT equilibrium. Restricting attention to competitive bidding means that we can focus on the timing (rather than the amount) of bids as our primary interest. We concentrate on cautious equilibria to rule out arbitrary outcomes in which the losing bidder effectively uses weakly dominated strategies. (The same equilibrium notion has been used in the equilibrium analysis of e.g., Bergemann and V¨alim¨aki (1996) and Felli and Harris (1996).) We only consider trigger strategies, for two reasons. First, they seem natural ones to consider, given the single-crossing property that we have assumed for the relative valuation function δ(θ). The second reason is analytical tractability. There may be equilibria that involve non-trigger strategies i.e., where the ‘stopping region’ for a firm (e.g., the set [θG , +∞) for firm G) is not an interval. Solving explicitly for such equilibria is not likely to be possible. A CCT equilibrium is characterized, therefore, by the two trigger points θG and θD . We now determine and analyze the CCT trigger points. (In appendix A, we prove the existence of a CCT equilibrium.)

3.1

Properties of a CCT Equilibrium

Detailed derivations of the firms’ equilibrium value functions are contained in appendix A. There we show that the value function VG (θ) of firm G has three components, holding over different ranges of θ:    0    VG (θ) = AG θ −α + BG θ α+1      λθG − bG (θG )

10

θ ≤ θD , θD < θ < θ G , θ = θG .

(3)

AG and BG are constants determined by boundary conditions, discussed below. α is the positive root of a ‘characteristic’ equation (see appendix A): 1 α= 2

−1 +

r

8r 1+ 2 σ

!

≥ 0.

BG θ α+1 is an option term anticipating firm G’s bid for the target; AG θ −α is an option-like term anticipating firm D’s bid. By arbitrage, the critical value θG for firm G must satisfy a value-matching condition; optimality requires a second, smooth-pasting condition to be satisfied. (See Dixit and Pindyck (1994) for an explanation.) This condition requires the components of firm G’s value function to meet smoothly at θG . θD is not chosen optimally by firm G; hence the smooth-pasting optimality condition does not apply here for firm G. Value functions are forward-looking, however, and so a value-matching condition applies at θD . Hence there are three relevant equations for firm G—value-matching and smooth-pasting at θG ; and value-matching at θD : −α α+1 AG θG + B G θG = λθG − η, −α−1 α −αAG θG + (α + 1)BG θG = λ, −α α+1 AG θD + B G θD = 0.

These three equations can be combined to give 

θG θD

2α+1

=

(α + 1)θG − αθ ∗ . −αθG + (α + 1)θ ∗

(4)

Equation (4) gives implicitly the best-response correspondence of firm G, which we denote θG (θD ).

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A similar story holds for firm D. Its value function is    η − bD (θD )     VD (θ) = AD θ −α + BD θ α+1      0

θ = θD , θD < θ < θ G ,

(5)

θG ≤ θ.

Value-matching and smooth-pasting hold at θD , and value-matching at θG : −α α+1 AD θD + B D θG = −(λθD − η), −α−1 α −αAD θD + (α + 1)BD θD = −λθD , −α α+1 AD θG + B D θG = 0.

These three equations can be combined to give 

θG θD

2α+1

=

−αθD + (α + 1)θ ∗ , (α + 1)θD − αθ ∗

(6)

which gives implicitly the best-response correspondence of firm D, which we denote θD (θG ). Note that the firms’ best responses are well-defined and non-empty for all θG , θD ∈ ¯ where (θ, θ), ¯ θ¯ ≡



 α+1 ∗ θ > θ∗, α

θ≡ ¯



 α θ∗ < θ∗ . α+1

A CCT equilibrium exists if there is a solution to the simultaneous equations (4) and (6) with θ¯ > θG ≥ θ ∗ ≥ θD > θ. The next proposition establishes the existence of such a ¯ solution. Proposition 1 (CCT Equilibrium) Given assumption 1, there exists a unique solution to equations (4) and (6) with θ¯ > θG > θ ∗ > θD > θ. ¯

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Proof. The symmetry of equations (4) and (6) implies that

θG =

θ∗ 2 . θD

Let x ≡ θG /θ ∗ . Equation (4) then implies that x4α+2 =

(α + 1)x − α . −αx + α + 1

(7)

This equation defines a polynomial in x. We are interested in roots of this polynomial that satisfy x ≥ 1. The first root is clearly x = 1. There must be (at least) a second root, since the left-hand side of equation (7) is greater than zero and finite for x ∈ [α/(α + 1), (α + 1)/α], while the right-hand side is zero at x = α/(α + 1) and tends to infinity as x → (α + 1)/α. We now show that there is no more than 1 root such that x > 1. The slope of the left-hand side of equation (7) is (4α + 2)x4α+1 ; at an intersection point, this must equal 

 4α + 2 (α + 1)x − α . x −αx + α + 1

The slope of the right-hand side is 2α + 1 . (−αx + α + 1)2 At x = 1, the slope of the left-hand side is clearly greater than the slope of the right-hand side. Hence at the ‘next’ intersection point (i.e., the intersection point with the lowest x > 1), it must be that the slope of the right-hand side is greater than the slope of the left-hand side. I.e., at this intersection point, 2α + 1 > (−αx + α + 1)2



 4α + 2 (α + 1)x − α . x −αx + α + 1

13

θG θˆG

θ∗

θ∗

θˆD

θD

Figure 1: Reaction functions Rearranging this inequality gives −α(α + 1)(4α + 2)x2 + ((α + 1)2 (4α + 2) + α2 − 2α − 1)x − α(α + 1) < 0.

The left-hand side of this expression is a quadratic in x. Given the signs of the coefficients in the quadratic, if the inequality is satisfy for an x∗ > 1, it must be satisfied for all x ≥ x∗ . Hence at any intersection point x > 1, it must be that the slope of the right-hand side of equation (7) is greater than the slope of the left-hand side. There can be, therefore, only one intersection point x > 1.



The CCT equilibrium is illustrated in figure 1. Note from the figure (see also the proof of the proposition) that the reaction functions also intersect at θG = θD = θ ∗ . The following corollary, which follows immediately from the relative slopes of the reaction functions around the point θG = θD = θ ∗ , means that this solution can be ignored in the remainder of the analysis. Corollary 1 The solution θG = θD = θ ∗ to equations (4) and (6) is not stable under the best-response dynamic (when strategies are restricted to be CCT). Proposition 1 and corollary 1 imply that takeovers occur only when shocks to the 14

industry variable θ are sufficiently large: when θ rises above θG or below θB . For moderate shocks (so that θ lies in the interval (θB , θG )), equilibrium requires the firms to delay making offers. Our model can, therefore, explain the empirical findings of e.g., Mitchell and Mulherin (1996), that takeovers occur in industries experiencing large negative and positive shocks. Two features of our model are crucial for generating this result. The first is that the firms’ valuations of the target are negatively correlated, since the relative valuation function λθ − η is upward-sloping in θ. We interpret this feature as modelling two types of firms: one whose plans for the target are based on growth (firm G), the other whose plans are based on decline (firm D). The second feature is that the takeover market is competitive. As we pointed out in section 2.1, with one acquirer, there is no delay—the target is acquired immediately. When there is a rival bidder, however, each firm’s payoff function becomes convex: it is bounded below by 0 when the firm loses, and is equal to the difference in the firms’ valuations when the firm wins. This convexity induces an incentive to delay bidding: rather than bidding immediately, or at the first instant when it has the larger valuation, each firm waits until the difference in valuations is sufficiently large. This feature—that competition makes payoffs convex—is quite general; we therefore expect to see bidder delay in more general environments than the one that we consider in this paper. An important comparative static involves the effect of an increase in uncertainty (i.e., the parameter σ) on the trigger points θG and θD . A standard property of single-firm real option models is that irreversible actions are delayed (i.e., occurs at a higher level of the state variable for firm G, and a lower level for firm D) when uncertainty increases. The reason for this is that delay allows for the possibility that the random process (1) might change; if it goes in an adverse direction (down for firm firm G, up for firm D), then the firm need not act. The greater the variance of the process, the more valuable is the option created by this asymmetric situation, and so the more delay occurs. The situation is complicated in the multiple firm case by the threat of pre-emption. If firm G delays, then it may lose its option altogether should firm D act in the meantime.

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This consideration can be seen in the value function in equation (3), for example. The term BG θ α+1 > 0 is firm G’s valuation of the option to delay due to the single-firm effect; the option-like term AG θ −α < 0 is the decrease in the valuation due to the possibility that firm D, not firm G, acquires the target. There are, therefore, two factors pulling in opposite directions when uncertainty increases; the comparative statics of θG and θD with respect to σ are determined by the balance between these two factors. Proposition 2 shows how these effects balance out. Proposition 2 (Uncertainty and equilibrium) ∂θG /∂σ > 0 and ∂θD /∂σ < 0 for all σ ≥ 0. Proof. The proof of both parts concentrates on the reaction function of firm G, equation (4); the symmetry in equations (4) and (6) means that the equivalent result for firm D follows immediately. Equation (4) can be re-written as θG 1

(Θ(θG )) 2α+1

= θD

where Θ(θG ) ≡

(α + 1)θG − αθ ∗ . −αθG + (α + 1)θ ∗

Consider the denominator of the left-hand side of this reaction function. Taking logs and differentiating with respect to σ gives   1 2ασ ∂ 2α+1 = ln (Θ(θ)) ∂σ (2α + 1)2



1 − ln Θ(θ) + 2



Θ2 (θ) − 1 Θ(θ)



where ασ ≡ ∂α/∂σ. Since ασ < 0, the sign of this expression is determined by the sign of 1 − ln Θ(θ) + 2



Consider the function φ(x) ≡ − ln x +

Θ2 (θ) − 1 Θ(θ)

1 2

first derivative is 1 φ (x) = 2 0



x−1 x




x−1 x

16



.

(8)

(1 + x). φ(1) = 0; and for x > 1, the

2

> 0.

Hence φ(x) > 0 for x > 1. The expression in equation (8) is therefore positive for all values of σ. So the denominator of the left-hand side of the reaction function of firm G is decreasing in σ; and consequently, the reaction function of firm G shifts upwards when σ increases. With the symmetric argument for firm D’s reaction function, the proposition follows.



We show in the proposition that the standard option effect (which acts e.g., to increase the trigger θG as σ increases) outweighs the pre-emption effect (which acts in the opposite direction). This need not be the case. When the random process in equation (1) has a strictly positive drift µ,7 there are cases when the pre-emption effect dominates. We have shown analytically (in a proof available on request) that when µ > 0, in the limit as σ → +∞, ∂θG /∂σ < 0 and ∂θD /∂σ > 0. Numerical analysis suggests that this outcome occurs when the drift parameter is large (close to, but below the interest rate r) and there is a substantial degree of uncertainty. When µ is large, the opportunity cost of holding the option to bid is small, and so the option value is large. When σ is large, the option value is large. Hence both conditions (µ and σ large) ensure that the firms’ options to bid are very valuable. But when σ is large, each firm assesses a high probability that the state variable hits the other firm’s trigger point. The optimal response of each firm to an increase in σ is then to decrease delay (so that θG falls and θD rises), to limit the probability of pre-emption and so preserve its large option value. (See Mason and Weeds (2004) for a more general analysis of when the standard comparative static is reversed because of pre-emption.) In a sense, therefore, the result in proposition 2 is not general: the comparative static with respect to σ is unambiguous only when µ = 0. But the numerical analysis suggests that, even with a large drift, the extent of delay increases with σ for almost all values of the volatility parameter. The unusual outcome, in which delay decreases with uncertainty, occurs only for extreme values of σ. Proposition 2 can be contrasted with the findings in Bernile, Lyandres, and Zhdanov 7 I.e., dθt = µθt dt + σθt dWt where µ ∈ (0, r) is the drift parameter. The restriction that µ < r ensures that there is a positive opportunity cost to holding the option to bid to acquire the good. This means that each firm bids at some finite value of the state variable, rather than holding the option in perpetuity.

17

(2006). Like us, they have an upper and lower trigger so that merger occurs if the state variable rises above (falls below) the upper (lower) trigger. They find, however, that both of their triggers increase with the degree of uncertainty. Consequently, in their model, the zone in which no takeover activity is observed may actually decrease as uncertainty increases. In contrast, in our model, this zone always increases with uncertainty. This offers the prospect of distinguishing empirically between the two models.

4

The Efficient Solution

Efficiency requires that the target be acquired by the firm with the higher valuation. The identity of that firm is stochastic—when θ < θ ∗ , it is firm D, when θ > θ ∗ , it is firm G. The timing of acquisition must also be efficient; we denote the efficient trigger points by θL for the lower threshold and θH for the upper one. Hence the efficient allocation rule takes the form “award the target to firm D immediately if θ ∈ [0, θL ], to firm G immediately if θ ∈ [θH , ∞); otherwise wait”. Familiar derivations give the efficient value function as

   η     W = AE θ −α + BE θ α+1      λθ

θ ≤ θL , θ ∈ (θL , θH ),

(9)

θ ≥ θH .

Value-matching and smooth-pasting conditions apply at the efficient triggers θL and θH . These four equations yield, after elimination of the value function coefficients,

θH =



α+1 α

θL =



α α+1

1  2α+1

1  2α+1

θ∗ > θ∗ ,

(10)

θ∗ < θ∗ .

(11)

(The symmetry of the equations means that θH = θ ∗ 2 /θL .) Since pre-emption is not an issue in the efficient solution, the triggers θH and θL should have the the usual property of delay for situations of irreversible action under uncertainty; that is, θH should be increasing and θL decreasing in σ. The next proposition confirms 18

this conjecture. Proposition 3 (Uncertainty and efficient triggers) (i)

limσ→0 θH /θL = 1;

(ii)

limσ→∞ θH /θL = ∞;

(iii) ∂θH /∂σ > 0, ∂θL /∂σ < 0.

Proof. Equations (10) and (11) imply that θH = θL



α+1 α

2  2α+1

.

Parts (i) and (ii) of the proposition follow immediately from observing that as σ → 0 (∞), α → ∞ (0), respectively. Part (iii) of the proposition will be proved for θH ; an equivalent argument holds for θL . Since α is a decreasing function of σ, ∂ ∂σ



α+1 α



∂ > 0 and ∂σ



1 2α + 1



> 0.

This implies immediately that θH is increasing in σ.



We can compare directly the equilibrium and efficient triggers: Proposition 4 (Inefficiency of equilibrium) θG ≥ θH and θD ≤ θL . Proof. The proof works by showing (through contradiction) that θG /θD ≥ θH /θL ; since θG = 1/θD and θH = 1/θL , this gives the result immediately. To simplify expressions, let θ˜k ≡ θk /θ ∗ for k ∈ {G, D, H, L}. From equations (10) and (11), 2

  θ˜H α + 1 2α+1 ; = α θ˜L from equation (4), θ˜G = θ˜D

(α + 1)θ˜G − α −αθ˜G + α + 1 19

1 ! 2α+1

.

Suppose that θ˜H /θ˜L ≥ θ˜G /θ˜D . Manipulation of the above expressions shows that this would mean that θ˜G ≤

α3 + (α + 1)3 . α(α + 1)(2α + 1)

Similarly, from equation (6), θ˜G = θ˜D

−αθ˜D + α + 1 (α + 1)θ˜D − α

1 ! 2α+1

.

θ˜H /θ˜L ≥ θ˜G /θ˜D then implies that α(α + 1)(2α + 1) . θ˜D ≥ α3 + (α + 1)3 Hence θ˜H /θ˜L ≥ θ˜G /θ˜D implies that θ˜G /θ˜D ≤ 1—a contradiction (since θG ≥ θD ). Therefore θ˜H /θ˜L ≤ θ˜G /θ˜D . The proposition follows.



Proposition 4 tells us that there is too much delay in equilibrium. The efficient payoff is πE ≡ max{λθ, η}; the equilibrium payoff of firm G is πG ≡ max{λθ − η, 0}, and of firm D is πD ≡ max{η − λθ, 0}. Equilibrium payoffs are lower, therefore, because competition creates negative externalities: πi = πE − s−i < πE for i ∈ {G, D}. Notice that the equilibrium payoff of a firm is lower than the efficient payoff, both when it wins the target and when it loses. The reduction in payoff on winning the target leads the firm to bid later (relative to the efficient trigger); the reduction on losing leads to an earlier bid. When an firm chooses its trigger point optimally, the former effect dominates, so that negative externalities lead to more delay in equilibrium. To see why, consider first the choice of trigger point by firm G. The equilibrium payoff to firm G is max{0, λθ − η}; the efficient payoff is max{η, λθ} i.e., an upward shift of the equilibrium payoff. The equilibrium payoff of firm G is therefore lower than the efficient payoff by an amount η (i.e., the valuation of firm D) both when it wins the target and when it loses to firm D. Firm G will be the first to bid only when θ ≥ θ ∗ i.e., when it has the higher valuation. Hence, the payoff reduction on winning is more important than the reduction on losing because the former occurs immediately, while the latter occurs in 20

the future (when θ falls below θ ∗ ) and so is discounted. Since the undiscounted payoff reductions are equal, the reduction on winning has greater weight. So, for a given lower trigger point (θL , say), the equilibrium upper trigger is greater than the efficient upper trigger. The shape of the reaction functions (established in proposition 1) then ensures that θG ≥ θH ≥ θL ≥ θD . This argument for firm G relied on firm D’s valuation being a constant, η. A more general intuitive argument can be made. Firm G’s value function in equation (3) has two components. The first OLG (θ) ≡ AG θ −α < 0 is an option-like term anticipating firm D’s successful bid; the second OG (θ) ≡ BG θ α+1 > 0 is an option term relating to its own successful bid. At firm G’s optimally-chosen trigger point θG , value-matching and smooth0 pasting imply that OG (θG ) > −OLG (θG ) > 0 and OG (θG ) > −OL0G (θG ) > 0 (where

the prime denotes the derivative with respect to θ). Hence the option term is greater in terms of both level and slope. In short, the value from winning is more important than the value from losing.8 Hence firm G’s trigger point lies above the efficient level. This argument suggests that the result in proposition 4 should generalize beyond linear valuation functions, although we have been unable to show this analytically.9 In general, a second effect may arise: the negative externalities may not only shift equilibrium payoffs, but also change the degree of convexity, relative to the efficient payoff. If the externalities create more (less) convexity, then they will tend to lead to more (less) delay in equilibrium, other things equal. Firm G’s equilibrium payoff function is (weakly) more convex than the efficient payoff function iff s00D (θ) s00G (θ) ≥ ∀ θ, s0G (θ) s0D (θ) assuming non-zero first derivatives. This condition is satisfied with equality with linear valuation functions (since s00G = s00D = 0); hence this effect does not arise in our analysis. 8

Equivalently, for firm D: its value function in equation (5) has two components: the option term OD (θ) ≡ AD θ−α > 0 relating to its own successful bid; and the option-like term OLD (θ) ≡ BD θα+1 anticipating firm G’s successful bid. Value-matching and smooth-pasting at firm D’s trigger point θ D 0 imply that OD (θD ) > −OLD (θD ) > 0 and −OD (θD ) > −OL0D (θD ) > 0. Again, the option term is greater in level and slope. 9 Numerical investigation supports this optimism.

21

θG

θH θ∗ θL

θD σ Figure 2: Triggers against the degree of uncertainty σ

5

Summary of Results

In figure 2, the various triggers are shown against different values of σ. We use assumption 1 for the firms’ valuation functions, and set the interest rate r to 5%, firm D’s value η to 10 and the slope of firm G’s value λ to 1. With these values, θ ∗ = 10. Summarising, our analytical results are that 1. θG is greater than θ ∗ and increasing; θD is less than θ ∗ and decreasing (propositions 1 and 2); 2. θH is increasing and θL decreasing (proposition 3); 3. θG ≥ θH and θD ≤ θL (proposition 4). These results are all confirmed by the numerical analysis shown in figure 2. The figure points to a further result, which we are unable to verify analytically, but is borne out by numerical investigation. The extent of equilibrium inefficiency, measured by the gap between the equilibrium and efficient triggers, increases in the degree of uncertainty σ.

22

6

Project Choice

In this section, we consider the incentives that the target firm faces when choosing its projects. We analyse this by supposing that the target can choose the volatility parameter σ, along the lines of e.g., Holmstrom and Milgrom (1987) and Sung (1995).10 The owner of the target receives the payoff of the losing bidder when acquisition occurs i.e., its payoff is

T (θ) =

   λθ   η

θ ≤ θD , θ ≥ θG .

Using familiar calculations, the target’s value function can then be shown to be    λθ     U = AT θ −α + BT θ α+1      η

θ ≤ θD , θD < θ < θ G , θ ≥ θG .

The coefficients AS and BS are determined by value-matching conditions at the triggers θG and θD ; there are no corresponding smooth-pasting conditions as the trigger points involve no optimality on the part of the target. The first observation is immediate: the target’s payoff function S(θ) is concave. Thus greater delay (i.e., a widening of the trigger points θG and θD ) is harmful to the target. Note that this finding does not require a functional form assumption, such linearity of the valuation functions. It depends only on the assumption that the difference in the valuations is (weakly) increasing; bidding is competitive, so that the target receives the lower of the valuations. 10

Note that when the acquiring firms’ payoffs satisfy assumption 1, there is an alternative interpretation of volatility. The stochastic process θ drives the gap between the valuations of the target firm and acquirer G, with this difference being scaled by λ. Thus, the volatility σ may be interpreted as the degree of correlation between the value of these assets when controlled by the target and under the alternative ownership of firm G—a higher value of σ is a reduction in the correlation between these valuations. Under this interpretation, the target’s choice of a lower value of σ corresponds to a higher degree of correlation between itself and the acquirer with lower marginal cost. (The degree of correlation with firm D, with a fixed cost reduction, is not affected.)

23

From proposition 2, we know that the acquiring firms’ triggers widen when the degree of uncertainty σ increases. An increase in σ reduces the target’s pre-bid value: for any two parameters such that σ1 < σ2 , the associated value functions U1 and U2 are such that U1 > U2 . Hence the target always prefers a lower value of σ. This is illustrated in figure 3, in which an increase in σ from 0.2 to 0.4 shifts θD to the left and θG to the right, and results in a downward shift in the target’s value function.

Figure 3: The target’s value function against σ These arguments are summarised in the next proposition. Proposition 5 (Target’s project choice) If the owner of the target can choose the volatility parameter σ of the state variable process (1), then it will choose the lowest possible value of σ. To complete this section, we consider the efficient choice of the parameter σ, given the behaviour of the acquiring firms G and D. The efficient value function is of the same form as in equation (9), but the value function coefficients AE and BE are now determined by 24

value-matching conditions at θG and θD . We are unable to obtain analytical results in this case. Numerical analysis shows that the efficient choice of σ depends on the initial level θ0 of the process in equation (1). If θ0 is close to θ ∗ , the efficient choice of σ is high, as illustrated in figure 4. But when the initial value is further away from θ ∗ , the efficient choice of σ is low. This result is due to the tension between two factors. Since the efficient payoff function is convex, greater uncertainty increases the (option) value of takeover. This effect is particularly strong close to the kink at θ ∗ . However, in non-cooperative equilibrium, the firms tend to delay too much; greater uncertainty worsens this inefficiency. At values of θ further away from θ ∗ , this inefficiency dominates and the efficient choice of σ is low, despite the convexity of the efficient payoff, in order to avoid excessive delay.

Figure 4: The efficient value function against σ

25

7

Extensions

In this section, we consider how our main results would be changed if certain features of the model are extended. We consider two extensions; we allow for more than two bidders; and for alternative forms of takeover. We have fixed exogenously the number of bidders to equal two.11 Whatever the number of bidders, the critical assumption for our analysis is that the bidders’ valuations can be ordered with respect to a single state variable. With just two bidders, this means that the relative valuations of the bidders must be single-crossing. There is then a single value of the state variable so that the identity of the highest valuation is bidder is determined by whether the state variable is above or below that level. With more than two bidders, the same basic assumption is required: that the set of values of a single state variable can be sectioned into disjoint and contiguous intervals. A particular bidder then has the highest valuation only if the level of the state variable is in a particular interval. Subject to this assumption, a number of possibilities arise. Suppose first that there are two broad types of bidders: growth-acquirers (G-bidders) and decline-acquirers (Dbidders). For simplicity, suppose that there is a single D-bidder, whose valuation of the ¯ target is η > 0. But there are two G-bidders: one with a valuation of the target of λθ ¯ (we label this bidder as the G-bidder), the other (the G-bidder) with a valuation λθ, ¯ ¯ ¯ The G-bidder never acquires the target in equilibrium; but it acts as a where 0 < λ < λ. ¯ ¯ ¯ the G-bidder ¯ ¯ constraint on the G-bidder when it sets its bid. When θ ≥ θ¯∗ ≡ η/λ, has 11

The empirical evidence (which is not extensive) provides some support for this feature. In De, Fedenia, and Triantis (1996), out of 660 contests over the period 1962–1988, 460 involve just one bidder, and 200 involve more than one bidder. De, Fedenia, and Triantis (1996) do not report how many bidders are involved in contests in which there are multiple bidders; but in those 200 contests, only 279 bidders were involved. This indicates that, where there are multiple bidders, there are relatively few bidders; and that some bidders are involved in more than one contest. A similar story emerges from Boone and Mulherin (2002), who study a sample of 50 US firms acquired by private auction over the period 1989– 1998. The mean (median) number of firms initially contacted by the target was 63.2 (50). Of the firms initially contacted, an average of 28.7 firms (median of 18) indicated interest by signing confidentiality agreements. An average of 6.3 firms submitted preliminary proposals. An average of 2.6 potential bidders (median of 2) submitted binding written offers. The authors contrast these statistics with the number of competing public bids for the firms in the auction sample. On average, there were 1.3 competing public bids; the median was 1. Of course, little can be said about the presence of potential bidders, who do not submit bids but affect the bidding behaviour of firms that do actually bid. Nevertheless, these findings provide informal support for our modelling decision to limit the number of bids to two.

26

the highest valuation. When θ ≤ θ ∗ ≡ η/λ, the second highest valuation bidder is the ¯ ¯ ¯ D-bidder; and so the G-bidder must pay η in order to acquire the target. When θ > θ ∗ , ¯ ¯ the G-bidder has the second highest valuation, and the G-bidder must pay λθ > η to ¯ ¯ acquire the target. Clearly, if λ is very low, then the presence of the G-bidder has no effect on the ¯ ¯ equilibrium and efficient solutions. But for higher values of λ, the equilibrium behaviour ¯ ¯ of the G-bidder is constrained by the G-bidder. In particular, the equilibrium bid of ¯ ¯ the G-bidder is higher, and hence its payoff is lower, because of the G-bidder. A direct ¯ ¯ consequence of this is that, for a given trigger of the D-bidder, the trigger of the G-bidder is increased, since its payoff on winning is lower. Since triggers are strategic complements, this means that the trigger of the D-bidder will be lower. In other words, the increase in competition among G-bidders increases the extent of equilibrium delay. Hence our main result is robust to this type of extension. The picture is more complicated picture when there are more than two possible winning bidders. To illustrate the issues, suppose that there are three possible winning bidders, labelled D, M, and G. The D-bidder has a valuation of the target of η¯; the ¯ (The M-bidder has a target valuation of η + λθ; the G-bidder values the target at λθ. ¯ ¯ M-bidder can be viewed as having a combined strategy: some asset stripping, and some ¯ The bidders’ valexploitation of synergies.) Suppose that 0 < η < η¯ and 0 < λ < λ. ¯ ¯ uations are shown in figure 5. There are three critical values of the state with these valuations functions; see the figure. Standard arguments suggest that the D-bidder bids in the interval [0, θD ], where θD < θ ∗ . The B-bidder bids in the interval [θG , +∞), where ¯ θG > θ¯∗ . The M-bidder bids in the interval [θM , θ¯M ], where θ ∗ < θM < θ¯M < θ¯∗ . The ¯ ¯ ¯ bidding intervals are illustrated in figure 5, in bold along the horizontal axis. In this case, therefore, takeovers occur at moderate values of the state variable, as well as at extreme (large or small) values. Nevertheless, a general feature remains: given an initial value of the state at which no takeover occurs, a sufficiently large positive or negative shock to the state can lead to a takeover occurring. Does a third bidder reduce the size of the shock that is required to induce takeover? There are two effects pulling

27

¯ λθ η + λθ ¯ ¯ η¯ η ¯

D

M θ ¯

∗

θ

∗

G θ¯∗

θ

Figure 5: Valuations with 3 bidders in opposite direction. Takeover, by the M-bidder, can occur for values of θ around θ ∗ ; with just the D-bidder and G-bidder, no takeover occurs in this interval. But because the presence of the M-bidder decreases the equilibrium payoffs of the D- and G-bidders, the takeover triggers of these bidders are widened (i.e., θD falls and θG rises), relative to the 2-bidder case. Either effect can dominate, depending on parameter values. In summary: allowing for additional bidders does not change the general feature that a sufficiently large positive or negative shock to the state can lead to takeover occurring. Furthermore, it can be that more competition for takeover can lead to greater delay. Our model, in which the target is entirely passive and is acquired by the highest bidder, is effectively that of a hostile takeover. A hostile bid is defined as one in which the target board’s initial reaction is to recommend target shareholders to reject the offer. This accounts for 20–25% of takeovers in the UK and the US; see e.g., Schwert (2000). In contrast, in friendly takeovers, the bidder may engage in bargaining with the target to set the terms of takeover. Our model could be adapted relatively easily to cover this case. To see how, suppose that the successful bidder has not only to out-bid the rival bidder, but also share the surplus from takeover, with the target receiving a fraction t ∈ [0, 1] of the surplus. Hence the G-bidder’s payoff from successful takeover is (1 − t)(λθ − η), and the D-bidder’s payoff is (1 − t)(η − λθ). These changes to the bidders’ payoffs would

28

in fact make no difference to the equilibrium triggers. More generally, as long as some proportion of the takeover surplus accrues to the winning bidder, the effects that we find go through.

8

Conclusions

We have developed a model of the timing of takeovers, in order to explain the empirical fact that takeovers occur after both positive and negative shocks to industry-specific conditions. Our model relies on two key assumptions: that there are two types of acquirers distinguished by their different comparative advantages in running the acquired firm; and that the takeover market is competitive. We show that in equilibrium, the acquiring firms make takeover offers only when there is a large shock to industry conditions. We show that this delay in making offers is inefficient. And we show that the extent of the delay increases in the degree of uncertainty about industry conditions. This last result (which is not found in e.g., Bernile, Lyandres, and Zhdanov (2006)) suggests an empirical test of our model. In future work, we plan to examine in more detail the owner’s decision of when to offer the target for sale. When the target has incomplete information about acquirers’ valuations, the choice of the selling mechanism is likely to influence the timing of takeover. The separation of ownership and control implies that managerial, rather than shareholder, incentives determine the behaviour of the target; in this context an analysis of managers’ payoffs becomes relevant.

Appendix A In this appendix, we derive the value function of firm G when firm D uses a CCT strategy. The derivation of firm D’s value function is very similar and so is omitted. Firm G’s value function VG (θt ) given a level θt of the state variable is given by equation (2). In the ‘continuation’ region before either firm has bid, in any short time interval dt starting at time t firm G receives a flow payoff of 0 and experiences a capital gain or loss 29

dVG . The Bellman equation for the value of the entry opportunity is therefore

VG = exp (−rdt)Et [VG + dVG ] .

Itˆo’s lemma and the GBM equation (1) gives the ordinary differential equation (ODE) 1 2 2 00 σ θ VG (θ) − rVG (θ) = 0. 2 The general solution of this homogeneous ordinary differential equation is VG = AG θ α + BG θ α+1 , where AG and BG are constants, and α > 0 is the positive root of the quadratic equation Q(z) = 21 σ 2 z(z + 1) − r. We now show that a CCT equilibrium exists. Consider first the bidding stage when at least one of the firms has made a bid. By the standard ‘Bertrand’ argument in common knowledge bidding games, when one firm bids competitively and cautiously, then the best response is a competitive, cautious bid. Given this behaviour in the bidding stage, now consider the prior stage when the firms must decide when to bid; and suppose that firm D bids at the first time that the state variable hits the interval [0, θD ]. Firm G’s value function V0 when it is not bidding is

V0 =

   0

θ ≤ θD ,

  AG θ −α + BG θ α+1

θ > θD .

Its value function when it makes a bid is given by

λθ − η −  sG (θ) > sD (θ) 0

otherwise.

By a standard argument (see e.g., Dixit and Pindyck (1994)), the continuation region (i.e., ˆ while the stopping in which no bid is made) for firm G is the half-open interval [0, θ), ˆ ∞), for some θ. ˆ region (in which a bid is made) is the interval [θ, An analogous argument holds for firm D’s best response when firm G uses a CCT 30

strategy. In summary: the best response to a CCT strategy is itself a CCT strategy. Hence Proposition 6 A CCT equilibrium exists, and is given by the solutions to equations (4) and (6). Of course, non-CCT equilibria exist—there are Markovian equilibria that are nonCCT (for example, the continuum of equilibria with non-cautious bidding in the bidding stage of the game); and there are non-Markovian equilibria. As we explain in the text, our focus on Markovian CCT equilibrium is motivated by our interest in the timing of non-collusive bids.

Appendix B In this appendix, we indicate how the analysis of the CCT equilibrium can proceed without the assumption of linear valuation functions. Instead of linearity, assume the following: Assumption 2 The functions si (θ) : R+ → R+ , i ∈ {G, D} are continuously differentiable and positive. Let δ(θ) ≡ sG (θ) − sD (θ). (i) δ(θ) is a (continuously differentiable) increasing function of θ; (ii) there is a unique θ ∗ > 0 such that δ(θ ∗ ) = 0, and δ(θ) < (>)0 when θ < (>)θ ∗ .

Continuous differentiability of valuations simplifies the analysis. The fact that the firms’ valuations are non-negative means that we focus on the case in which it is efficient to sell the target; moreover, with two credible bidders at every value of θ, bidding occurs competitively in equilibrium. Finally, the single-crossing assumption on relative valuations implies that firm G’s valuation of the target is greater than firm D’s when θ is high (above θ ∗ ); and that firm D has the higher valuation when θ is low (below θ ∗ ).

31

The derivation of the value functions is unchanged. Following the same steps, we arrive at the best-response correspondence of firm G: 2α+1 θG 2α+1 = θD , Θ(θG ) αδ(θ) + θδ 0 (θ) . where Θ(θ) ≡ −(α + 1)δ(θ) + θδ 0 (θ)

(12)

Likewise for firm D: 2α+1 θD 2α+1 , = θG Θ(θD )

(13)

In order for the firms’ best responses to be well-defined and non-empty, we assume Assumption 3 Let γG (θ) ≡ −(α + 1)δ(θ) + θδ 0 (θ), γD (θ) ≡ αδ(θ) + θδ 0 (θ). ¯ = 0 and γG (θ) > 0 for all θ ∈ [θ ∗ , θ). ¯ And There exists a θ¯ ∈ (θ ∗ , +∞), such that γG (θ) there exists θ ∈ (0, θ ∗ ), such that γD (θ) = 0 and γD (θ) > 0 for all θ ∈ (θ, θ ∗ ]. ¯ ¯ ¯ A CCT equilibrium with θG > θ ∗ > θD exists if there is a solution to the simultaneous equations (12) and (13) that satisfies these inequalities. The next proposition establishes the existence of such a solution. Proposition 7 (CCT Equilibrium) Given assumptions 2–3, there exists a solution to equations (12) and (13) with θ¯ > θG > θ ∗ > θD > θ. ¯ Proof. The proof uses five facts about the reaction functions defined by equation (4) and (6). Note that the functions θG (θD ) and θD (θG ) defined by the two equations are ¯ continuously differentiable for θ ∈ (θ, θ). ¯ 1. The reaction functions intersect at θG = θD = θ ∗ . At these values, Θ(θG ) = Θ(θD ) = 1, since δ(θ ∗ ) = 0; and θG /θD = 1.

32

2. As θD → 0, the solution to equation (12) must be such that Θ(θG ) → ∞ i.e., θG → θ¯ < ∞ (the latter because of assumption 3). 3. As θG → ∞, the solution to equation (13) must be such that Θ(θD ) → 0 i.e., θD → θ > 0 (the latter because of assumption 3). ¯ 4. The derivative of firm G’s reaction function, when it exists, is obtained by total differentiation of equation (12): 

 2α+1 2α θG θG 0 2α (2α + 1) − Θ (θG ) dθG = (2α + 1)θD dθD . Θ(θG ) (Θ(θG ))2

Straightforward calculations show that

lim ∗

θG ↓θ , θD

↑θ ∗

∂θG = −∞. ∂θD

5. Similarly, the derivative of firm D’s reaction function, when it exists, is obtained by total differentiation of equation (13). Evaluated at θG = θD = θ ∗ , this gives the derivative of firm D’s reaction function as

lim ∗

θG ↓θ , θD

↑θ ∗

∂θG = 0. ∂θD

Hence the solutions of equations (12) and (13) are equal at θG = θD = θ ∗ (fact 1); for any given value of θD less than but close to θ ∗ , the solution of equation (12) is greater than the solution of equation (13) (facts 4 and 5); and the solution of equation (12) is less than the solution of equation (13) for θD sufficiently close to θ (facts 2 and 3). ¯ Therefore, by the intermediate value theorem, a solution to the equations exists that satisfies ∞ > θ¯ > θG > θ ∗ > θD > θ > 0. ¯



The details of the proof of proposition 2 make clear that the comparative static with respect to uncertainty can be established in the more general case dealt with in this appendix. We cannot, however, establish (i) uniqueness of CCT equilibrium; (ii) inefficiency of the equilibrium triggers (proposition 4. 33

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