Informed Headquarters and Socialistic Internal Capital Markets
Daniel Hoang and Martin Ruckesy
Karlsruhe Institute of Technology
October 10, 2011
Abstract
This paper develops a theory of resource allocation in internal capital markets that is consistent with the empirical
…nding that multi-division …rms bias their investment levels in favor of divisions with weaker investment prospects.
Headquarters has private information about the capital productivities of its divisions and capital allocations in the
present serve as a signal to divisional managers about those in the future. As competition between divisions is only
productivity-enhancing if the perceived di¤erences in productivity are not too pronounced, headquarters biases current
investment in favor of ex-ante weaker divisions either to not disclose productivity di¤erences or to credibly signal their
absence. The model o¤ers an explanation for why new CEOs allocate relatively less capital to the divisions through
which they advanced.
JEL Classi…cation: G31, D82
We thank Hendrik Hakenes, Ulrike Malmendier, Andrew Postlewaite, Jörg Rocholl and Adam Szeidl for many helpful
comments. We also bene…ted from comments by participants at the 2008 CEPR European Summer Symposium in Gerzensee, the
2008 Annual Meeting of the German Finance Association and the 2009 Annual Meeting of the Swiss Society for Financial Market
Research and seminar participants at the University of California, Berkeley, ESCP-EAP Paris, ESMT Berlin, and Humboldt
University Berlin. Hoang gratefully acknowledges …nancial support from The Monitor Group.
y
Hoang is with The Monitor Group and the Karlsruhe Institute of Technology, email: daniel.hoang@kit.edu. Ruckes is with
the Karlsruhe Institute of Technology, email: martin.ruckes@kit.edu. Contact information for both authors: Dept. of Finance
and Banking, Karlsruhe Institute of Technology, Building 20.13, Kaiserstraß
e 12, 76131 Karlsruhe, Germany, Phone: +49 721
608 43427, Fax: +49 721 359 200.
Informed Headquarters and Socialistic Internal
Capital Markets
Abstract
This paper develops a theory of resource allocation in internal capital markets that is consistent with the empirical
…nding that multi-division …rms bias their investment levels in favor of divisions with weaker investment prospects.
Headquarters has private information about the capital productivities of its divisions and capital allocations in the
present serve as a signal to divisional managers about those in the future. As competition between divisions is only
productivity-enhancing if the perceived di¤erences in productivity are not too pronounced, headquarters biases current
investment in favor of ex-ante weaker divisions either to not disclose productivity di¤erences or to credibly signal their
absence. The model o¤ers an explanation for why new CEOs allocate relatively less capital to the divisions through
which they advanced.
1
Introduction
Well-functioning internal capital markets channel scarce …nancial resources into their most productive uses. In
multi-division …rms, headquarters has ownership rights and is therefore able to allocate capital across divisions
(Gertner, Scharfstein and Stein, 1994). This allows headquarters to steer funds towards divisions with
relatively favorable investment opportunities (Stein, 1997). However, the value of such internal capital markets
has been questioned. Empirical research points to the distortion of capital allocation, such that headquarters
favors divisions with poor growth prospects at the expense of those with good growth opportunities (Shin
and Stulz, 1998, Rajan, Servaes and Zingales, 2000, and Ozbas and Scharfstein, 2010).1
These …ndings have led to a number of theoretical characterizations of the workings of internal capital markets,
consistent with such behavior based on agency problems between top management and divisional managers
(Rajan, Servaes and Zingales, 2000, Scharfstein and Stein, 2000, and Bernardo, Luo and Wang, 2006). This
paper provides an alternative explanation of “socialistic” allocation of …nancial resources arguing that top
management uses capital allocation as an instrument to communicate with divisional managers.
Divisional managers exert costly e¤ort to increase their own divisions’investment productivities. Whereas in
case of identical ex-ante investment productivities of two divisions, the entire capital budget is allocated based
on divisional managers’productivity improvements, this is not the case if ex-ante investment productivities
di¤er markedly. Then, a large fraction of the capital budget remains earmarked for the more productive
division, which limits the e¤ect of productivity improvements by divisional managers on capital allocation.
This weakens either divisional manager’s incentives to engage in productivity improvements.
Top management bene…ts from the productivity improvements divisional managers make. As competition for
capital budgets is only e¤ective in stimulating productivity improvements if ex-ante productivities of divisions
are perceived to be similar, top management has an incentive to communicate such a productivity structure
to divisional managers. Possibly the most important means of communicating relative productivities are
the divisions’ capital budgets themselves. For example, a signi…cant di¤erence in capital budgets between
two divisions reveals a substantial productivity di¤erence. As this leads to little managerial e¤ort, top
management would like to avoid sending this message to divisional managers. It can do so by allocating
capital evenly across divisions even though capital productivities di¤er or by allocating more capital to an
on average weaker division when capital productivities turn out to be identical. This behavior implies that
1
These empirical studies are not free of measurement and endogeneity problems. Maksimovic and Phillips (2007) provide a
comprehensive discussion of these issues in the literature on internal capital markets.
1
divisions with better investment opportunities do not receive as much capital as their capital productivities
would imply.
The cost of a bias in favor of divisions with lower capital productivities is ine¢ cient capital allocation in the
present, but higher capital returns in the future due to stronger managerial e¤orts to improve productivities.
Thus, capital allocation favoring divisions with weaker investment prospects can be interpreted as an investment by top management. Consequently, biased capital allocation is predicted to be most prevalent when
the opportunities for top management to make such investments are best and its incentives for making them
are strongest. Among the predictions of the model are that the described bias in capital allocation is more
frequent during periods of temporarily relatively severe …nancial constraints and those when capital productivities of the divisions have changed. For example, the model implies that diversi…ed …rms respond later to
new product market developments than their standalone counterparts. It also o¤ers an explanation for why,
after CEO turnover, divisions not previously a¢ liated with the CEO display on average a signi…cantly larger
change in capital expenditures than the divisions through which the new CEO has advanced (Xuan, 2009).
The argument rests on the notion that top management possesses private information relative to divisional
managers. There are several reasons why this appears a sensible premise. First, headquarters is well-informed
about all the divisions of the …rm, whereas divisional managers have detailed knowledge only about their
own divisions. Thus, it appears natural that headquarters holds better information about the relative productivity of capital across divisions than do divisional managers. Second, top management is likely to be
better informed on issues in‡uencing the pro…tability of several divisions, such as general economic conditions,
political developments, potential merger opportunities, or possible spillovers across divisions.2 Such informational advantages often result from top managers’activities beyond the realm of the …rm including board
memberships, activities in professional associations, or the use of personal contact networks.3 Third, top
management in all likelihood holds a private assessment of divisional managers’level of ability to successfully
implement new projects.
In addition to holding “objective”private information, headquarters’private information may also be “subjective” in nature. Often top managers have strong convictions about the future of their respective …rms. Such
2
The literature on strategic management recognizes the informational advantages of CEOs and other higher-ranking individuals. Mintzberg (1975), for example, sums it up as follows: “The manager may not know everything but typically knows more
than subordinates do. Studies have shown this relationship to hold for all managers, from street gang leaders to U.S. presidents.”
3
In the sample of Mintzberg (1975), chief executives spent an average of 44 percent of their contact time with individuals
outside the organization. He writes that “... liaison contacts expose the manager to external information to which subordinates
often lack access. Many of these contacts are with managers of equal status, who are themselves nerve centers in their own
organization. In this way, the manager develops a powerful database of information.” For a recent, more extensive study of
CEOs’allocation of time see Bandiera, Guiso, Prat and Sadun (2011).
2
“visions”may re‡ect a bias toward certain activities (Rotemberg and Saloner, 2000, and Hart and Holmström,
2002) and represent a determinant of …rms’investment policies. Also, top management may have privately
known preferences for certain types of investments resulting from career concerns. These preferences may
lead top management to tend to act too conservatively (Zwiebel, 1995), too aggressively (Prendergast and
Stole, 1996) or similarly to other relevant players (Scharfstein and Stein, 1990). Irrespective of whether top
management’s private information is objective or subjective in nature, its disclosure does a¤ect divisional
managers’incentives. Thus, top management has an incentive to manage the level of information disclosed
to divisional managers.
Related Literature
The empirical documentation of biased allocations of …nancial resources in favor of divisions with weaker
investment prospects have led to several theoretical characterizations of the workings of internal capital markets consistent with such behavior. Scharfstein and Stein (2000) argue that managers of divisions with poor
investment opportunities have stronger incentives to spend time lobbying to increase their capital allocations.
When there is a preference of top management to compensate these managers with capital allocations rather
than higher salaries, this behavior leads to larger–than–e¢ cient allocations to weaker divisions. Rajan, Servaes and Zingales (2000) show that a very uneven resource allocation can lead divisional managers to steer
their investment policies away from e¢ cient cooperative investments and towards those that bene…t only the
managers’own divisions. To avoid such ine¢ ciencies, headquarters tilts capital allocations towards divisions
with fewer investment opportunities. In a setting in which divisional managers have private information
about project quality and in addition need to be incentivized to provide e¤ort, Bernardo, Luo and Wang
(2006) show that headquarters optimally biases project choice in favor of weaker divisions. Managers of
high productivity divisions have an incentive to underreport their productivity to increase their …nancial
compensation for strong results. Making it harder for those divisions to obtain capital improves managerial
information disclosure, thus permitting less expensive incentive provision for division managers. None of
these papers view biased capital allocation as an investment by top management and ignore an explicitly
dynamic dimension of capital allocation decisions.
While this paper focuses on information asymmetries within the …rm, these are not the only information
asymmetries that a¤ect capital allocation. De Motta (2003) and Goel, Nanda and Narayanan (2004) include
the impact of informational asymmetries between corporate insiders and …nancial markets on the distribution
of capital across divisions.
The argument advanced in this paper is based on the notion that headquarters’ability to reallocate capital
3
across divisions may sti‡e managerial initiative. This has also been noted by Brusco and Panunzi (2005) and
Gautier and Heider (2009), who assume that e¤ort leads to increased income in the period of its provision.
In contrast, Inderst and Laux (2005) model, like we do, managerial e¤ort directed at generating future
investment opportunities. Inderst and Laux (2005) show that managerial incentives of …nancially constrained
…rms tend to increase when divisions display similar capital productivities. Neither of these papers studies
the implications of a privately informed headquarters on capital allocation.
The …nance literature has long recognized the importance of communication issues (for example, Ross, 1977).
In this literature agents typically communicate private information to principals. A central feature of our
model is that it is the principal who possesses private information. Contracting problems in such settings
are studied in the economic literature on informed principals; see for example Maskin and Tirole (1990 and
1992) and Inderst (2001). Crémer (1995) and Aghion and Tirole (1997) demonstrate that better information
of principals may worsen agents’incentives. Ferreira and Rezende (2007) recognize that managers frequently
have an incentive to refrain from disclosing their strategic intentions. None of these papers are concerned
with the information conveyed by capital budgets.
In Section 2 we characterize the model. Section 3 contains the analysis of the model. We describe the model’s
implications and relate it to existing empirical evidence in Section 4. Section 5 provides an extension and
Section 6 concludes. Proofs of formal results are relegated to an appendix.
2
The Model
Players and investment opportunities
We model an internal capital market with three agents: headquarters and two divisional managers i, i = A; B.
There are two periods, t = 1; 2. Agents are risk-neutral. Headquarters distributes a …xed amount of funds It
based on expected performance, i.e. capital productivity qi;t , of divisions A and B. Available funds It > 0 are
deterministic and are derived from investments in previous periods. There is no access to external …nancing.
Expected investment returns are strictly positive with decreasing returns to scale and assume that divisional
periodical payo¤s
i;t
are given by
i;t
= qi;t Ii;t
1 2
kI
2 i;t
(1)
where Ii;t denotes the period t capital investment in division i and k > 0 parametrizes returns to scale.
Divisional capital productivity qi;t > 1 depends linearly on a …xed baseline productivity q i > 1 and a
4
productivity parameter xi 2 f0; xg. We refer to the sum of these productivity parameters as a division’s
intrinsic productivity. In addition, divisional managers can exert e¤ort during period 1, ei 2 f0; eg, e > 0,
in order to increase capital productivity of their divisions during the next period. In this formulation,
e¤ort can be interpreted as engaging in restructuring production or distribution, repositioning part of the
product portfolio, mentoring employees, furthering long-term relationships with customers or suppliers, or
simply searching for investment opportunities to be implemented during the upcoming period. Concretely,
divisional capital productivities are given by
qi;1 = q 1 + xi and qi;2 = q 2 + xi + ei :
(2)
Asymmetry between divisions and information
To model asymmetry between divisions in the simplest way, we assume that while for division B the productivity parameter xB takes on the value zero with certainty, for division A the probability of xA = 0 is
2 (0; 1) and that of xA = x is 1
. This implies that intrinsic productivities di¤er in favor of division
A or are equal across divisions. Headquarters is assumed to possess private information about division A’s
intrinsic productivity and observes xA perfectly. Division managers do not observe the realization of xA .4
We assume that divisions have su¢ ciently pro…table investment opportunities such that available funds are
fully invested during any period.5 For simplicity, we assume that payo¤s from investments in t = 1; 2 are
additively separable and do not accrue before the end of period 2. The interest rate is normalized to zero.
Let
t
2 [0; 1] denote the period t portion of available funds It invested in division A and
headquarters’ periodical payo¤ when allocating
t.
t( t)
denote
Thus, considering equation (1), for all t = 1; 2;
t( t)
equals
t( t)
= qA;t
t It
1
k( t It )2 + qB;t (1
2
t )It
1
k((1
2
2
t )It )
(3)
Managerial preferences
Divisional managers strictly prefer more capital to less. Concretely, we assume private bene…ts
to be
4
A more general and probably more natural approach would be an information structure in which headquarters is well-informed
about the true prospects of all divisions, whereas divisional managers have detailed knowledge only about their own divisions.
This would, however, require allowing for at least three states of the world which complicates the exposition. Our formulation
is su¢ cient to capture the idea that both managers do not know their position relative to each other. As will become clear, the
main implications of our analysis would be una¤ected with a more general structure.
5
A richer setting in which the intertemporal transfer of funds from the …rst to the second period is optimal would not
qualitatively change the conclusions.
5
proportional to assets under control (for example, Harris and Raviv, 1996, De Motta, 2003, and Brusco and
Panunzi, 2005). We consider the admittedly extreme case in which empire-building motives are su¢ ciently
strong that no feasible incentive payment can alter managers’behavior (see also Hart and Moore, 1995, and
Aghion and Tirole, 1997).6 E¤ort creates a private cost to the manager c(ei ) which is c > 0, if ei = e and 0,
if ei = 0. Consequently, in this two-period setting, managers seek to maximize utility, which is described by
the sum of private bene…ts derived from assets under control in both periods less the cost of exerting e¤ort
in period 1:
Ui (ei ) =
(Ii;1 + Ii;2 )
c(ei ).
(4)
Sequence of actions and events
The sequence of actions and events is shown in Figure 1.
Figure 1: Sequence of Actions and Events
1. Before any capital allocation occurs, headquarters observes xA which reveals the intrinsic capital productivity in division A in both periods. We refer to headquarters as being of type H if intrinsic
6
Even if divisional cash ‡ows are veri…able, providing e¤ective contractual incentives for the search for new investment
opportunities is di¢ cult to achieve. Due to the typically considerable time lag between search e¤ort and investment cash ‡ows,
divisional cash ‡ows in each period are in‡uenced by a multitude of factors that are frequently weakly related to the e¤ort in
question.
6
productivities di¤er in favor of division A and as being of type L if intrinsic productivities are equal
across divisions.
2. Headquarters distributes available funds I1 based on observation of qi;1 = q 1 + xi .
3. After observing capital allocation
1
(and possibly revising their assessments about xA ), divisional
managers simultaneously choose e¤ort ei .
4. After learning qi;2 = q 2 + xi + ei , headquarters allocates available funds I2 .
5. At the end of period 2, payo¤s
3
i;t
from investments made in the previous periods are realized.
Analysis
In the following, we examine optimal capital allocation of headquarters and equilibrium behavior of divisional
management. We decompose the analysis of two-period capital allocation into three stages: a …rst stage,
in which headquarters chooses …rst-period capital allocation; a second stage, in which divisional managers
choose their levels of e¤ort; and a third stage, in which headquarters makes its second-period capital allocation
choice after productivity-enhancing activities of divisional management have been realized. Since equilibrium
behavior is sequentially rational, we solve the game backwards, beginning with headquarters’second-period
capital allocation. We restrict our attention to pure strategy equilibria.
3.1
Capital Allocation in Period 2
By the beginning of period 2, headquarters learns about second-period productivity of its divisions qi;2 with
certainty. Hence, headquarters solves:
max qA;2
2
2 I2
1
k(
2
2
2 I2 )
+ qB;2 (1
subject to
2
2 [0; 1].
7
2 )I2
1
k[(1
2
2
2 )I2 ]
(5)
Considering the strict concavity of (5), the optimal rule for capital allocation in period 2 is:
2
8
>
>
0
>
<
=
1
>
>
>
: xA +eA
eB +kI2
2kI2
if xA + eA
eB
if xA + eA
eB
kI2
(6)
kI2
otherwise.
Headquarters shifts all funds to division i if its productivity advantage is su¢ ciently large, and headquarters splits funds evenly if productivities are identical. (6) reveals that it is division As relative productivity
advantage, xA +eA eB , that matters for its second-period capital allocation. Exerting e¤ort weakly increases
a manager’s own capital allocation and thereby weakly decreases the other manager’s allocation. In addition,
second-period capital allocation to division A,
3.2
2,
weakly increases in xA .
Managerial E¤ort in Period 1
When choosing …rst-period e¤ort levels ei , division managers anticipate that headquarters reacts optimally
given pro…tabilities qi;2 , and that it allocates capital according to (6). Since funds I2 are scarce, managers
compete for their share of the limited total capital budget. This competition for funds represents a game
of incomplete information: each manager chooses whether to exert e¤ort or not, while the (type-dependent)
value of xA and the (unobservable) e¤ort choice of her counterpart are uncertain.
In contrast to headquarters, managers do not know the true productivities ex ante (either their “opponent’s”
or their own), which implies that managers are unable to distinguish one type of headquarters from the other.
Before choosing ei , managers observe headquarters’current capital allocation
in divisions is persistent, this is relevant information and
1
1.
When capital productivity
is indicative of future allocations. Hence,
divisional managers may learn from current allocations about headquarters’ private information and may
update prior probabilities about headquarters’type. For example, a particular capital allocation may reveal
to managers that headquarters is type L, leading managers to exert e¤ort. However, other allocations may
not disclose such additional information. We denote the resulting posterior beliefs as p(Lj
p(Hj
1)
=1
(
1)
= (
1)
and
1 ).
To focus the analysis, we make the following parametric assumptions:
Assumption 1 Signi…cance and bene…t of e¤ ort: e
kI2 and c
1
2
I2 .
The …rst part of the assumption implies that in case of equal intrinsic productivity a division manager is
8
able to attract the entire period-2 capital budget by exerting e¤ort given that her counterpart does not. The
second part ensures that in such a situation the division manager …nds it attractive to indeed exert e¤ort as
her cost of providing e¤ort is not too large.
Assumption 2 Diversity in intrinsic productivities dominates e¤ ort: x
e
kI2 .
This assumption has as a consequence that if division A is intrinsically more pro…table, it receives all funds
regardless of whether its manager works hard or not. In other words, if intrinsic productivities turn out to
di¤er, the competition for internal resources has been decided irrespective of the managers’e¤ort choices. The
central implication of this assumption is that a division’s second-period capital allocation is less sensitive to its
manager’s …rst-period e¤ort when intrinsic productivities di¤er than when they are identical (see Inderst and
Laux, 2005). As a consequence, ex-ante inequity diminishes divisional managers’incentives to exert e¤ort.7
The disincentive e¤ect of inequity characterized here is similar to that in tournaments when contestants have
unequal chances of winning (see, for example, Lazear and Rosen, 1981, O’Kee¤e, Viscusi, and Zeckhauser,
1984, and Schotter and Weigelt, 1992).
The strategic considerations of each manager can be represented in the matrix depicted in Figure 2. Managers
A and B represent the column and row players, respectively.
0
0
e
I2
1
2
I2
(
1)
e
1
2
I2
c
(
0
1)
I2
(1
I2
1
2
(
1 ))
(
1)
I2
I2
(
1
2
c
1)
(
1)
I2
I2
c
c
Figure 2: Competition for funds under incomplete information.
Utility of manager A on the left, utility of manager B on the right of each box.
Given the action of the other manager, a divisional manager’s increase in utility by switching from no e¤ort
to positive e¤ort is given by
1
2
(
1)
I2
c, which is linearly increasing in
(
1 ).
Thus, a perceived high
probability of identical intrinsic productivities across divisions increases managers’incentive to exert e¤ort.
It is now straightforward to derive equilibrium levels of e¤ort:
7
While Assumption 2 is su¢ cient to generate the disincentive e¤ect of inequity, it is not necessary to assume that the entire
budget is allocated to the more intrinsically productive division. It su¢ ces to assume that a large enough fraction of the budget
is earmarked for investment in the stronger division. Relaxing the assumption in that way would render the exposition somewhat
more complicated.
9
Lemma 1 Posterior beliefs re‡ect any information conveyed by headquarters’ capital allocation in period
1. Equilibrium levels of e¤ ort (eA ; eB ) are sensitive to these beliefs and weakly increase with the belief that
headquarters is type L:
(eA ; eB ) =
8
>
>
< (e; e)
>
>
: (0; 0)
if 2
if 0
c
I2
(
(
1)
1
1)
<2
c
I2 .
When managers with empire-building tendencies choose to engage in productivity improvements, they do so
based on the expected increase in capital allocation that results from such e¤orts. The incentive to choose a
high level of e¤ort is su¢ ciently strong if (
1)
is equal or above a certain threshold. The result establishes the
positive role that the existence of the potentially weaker division has in leading to productivity improvements
in the stronger division. This role is limited to situations, however, in which the posterior belief suggests that
heterogeneous productivities across divisions are not too likely.
3.3
Capital Allocation in Period 1
We now move to the …rst stage of the game in which headquarters decides on the optimal capital allocation in
period 1. We begin by studying optimal capital allocation in the case of complete information about division
A’s intrinsic productivity. This characterization is then used to examine capital allocation in situations
in which information on productivities is private to headquarters and managers are unable to distinguish
headquarters’type.
3.3.1
Benchmark Case: Complete Information about Intrinsic Productivity
Since periods are additively separable we can derive the optimal capital allocation
formation simply by maximizing
1( 1)
+
2( 2)
with respect to
1.
1
1
under complete in-
depends on marginal returns in
divisions A and B. The di¤erence is that returns are exogenously given and therefore independent from other
decisions. Considering that
1
1
2 [0; 1], we obtain
=
8
>
>
<
1
2
n
o
>
>
: min x+kI1 ; 1
2kI1
if headquarters is type L
(7)
if headquarters is type H:
10
Hence, if headquarters is type L, headquarters’ e¢ cient allocation is to split funds evenly, since marginal
divisional returns are identical and strictly decreasing. If headquarters is type H,
the exposition of the analysis, we set x
1
2 (0:5; 1]. To simplify
kI1 . Thus, headquarters invests all available funds in division A.
Using the …ndings of the previous section, we can establish the following result.
Proposition 1 Under the assumptions previously imposed, there is a unique subgame perfect equilibrium under complete information about intrinsic productivity. Subgame perfect equilibrium behavior (
is given by
(
1 ; (eA ; eB );
2) =
8
>
>
< ( 21 ; (e; e); 12 )
1 ; (eA ; eB );
2)
if headquarters is type L
>
>
: (1; (0; 0); 1) if headquarters is type H
Using these results, second-period …rm pro…ts yield
L;2
=
L;2 ( 2 ; eA ; eB )
H;2
=
H;2 ( 2 ; eA ; eB )
= (q 2 + e)I2
= (q 2 + x)I2
1
2
4 kI2
1
2
2 kI2
if headquarters is type L
if headquarters is type H
and total expected payo¤ s result in
L
H
=
=
H;1
L;1
+
+
L;2
H;2
= q 1 I1 + (q 2 + e) I2
1
k(I 2 + I22 )
4 1
= (q 1 + x)I1 + (q 2 + x)I2
1
k(I 2 + I22 ):
2 1
Consequently, when productivities of divisions are common knowledge among headquarters and managers, our
model implies: if divisions di¤er in their investment opportunities (type H), headquarters uses its allocative
authority and consistently steers all funds to its strongest division A. Managers foresee headquarters’optimal
strategy, anticipating that e¤ort has no impact on ex ante predetermined capital allocation. Hence, there is no
incentive for either manager to be productive. In contrast, if divisions have similar investment opportunities
(type L), headquarters’right to allocate funds to the most productive use creates the incentive for managers
to work hard. Headquarters allocates capital evenly in both periods.
11
3.3.2
Capital Allocation with Incomplete Information: E¤ort in the Absence of Learning
Under incomplete information, our model conceptually de…nes a signaling game. An informed headquarters
moves …rst with its …rst-period allocation, which may reveal additional information. Then, uninformed
managers update their beliefs about headquarters’ type and react to these allocations, according to the
policy described by Lemma 1.
To characterize the outcome of this game we apply the concept of Perfect Bayesian Equilibrium (PBE) and
determine the set of separating and pooling equilibria in pure strategies. In a separating equilibrium, both
types of headquarters choose di¤erent allocations, and managers can learn headquarters’type. In contrast,
in a pooling equilibrium, both types of headquarters set the same allocation and managers can infer nothing
from the allocation. As usual, a multiplicity of equilibria arises since PBE does not impose any restrictions on
managers’beliefs following out-of-equilibrium allocations. To provide sharp predictions on likely equilibrium
outcomes, we restrict the set of out-of-equilibrium beliefs by jointly applying two well-known re…nements:
the concept of Undefeated Equilibrium introduced by Mailath, Okuno-Fujiwara and Postlewaite (1993) and
the concept of D1 proposed by Cho and Kreps (1987).8
We assume that the ex-ante probability of identical intrinsic productivities across divisions is not too small:
2
such that managers’best response after observing
p
c
,
I2
(8)
is to exert e¤ort ei = e. We examine the case in which
condition (8) is violated in Section 5.
The following proposition characterizes the equilibrium.
Proposition 2 Suppose
2
c
I2 .
a) If x < 12 kI1 + 2e II21 , there is a unique (Undefeated Equilibrium and D1) pooling equilibrium outcome, in
which both types of headquarters split funds evenly according to
(
1 ; (eA ; eB );
2)
=
8
>
>
< ( 12 ; (e; e); 12 )
p
= 0:5. Equilibrium strategies are given by
if headquarters is type L
>
>
: ( 1 ; (e; e); 1) if headquarters is type H.
2
8
See also Nöldeke and Samuelson (1997) for the joint relevance of both re…nement concepts in an evolutionary model of
job-market signaling.
12
The …rst-period allocation
equal their prior, p(Lj
p)
deviation on the interval
p
is uninformative with respect to divisional productivity, hence managers’beliefs
=
. Managers assign zero probability to type L following an o¤ -equilibrium
2 (0:5; 1] and form arbitrary beliefs otherwise.
Equilibrium payo¤ s to headquarters equal
L;1
+
H;1 (
if headquarters is type L
L;2
p
= 0:5) +
H;2
+ eI2
if headquarters is type H.
b) If x > 12 kI1 + 2e II12 , there is a unique (Undefeated Equilibrium and D1) separating equilibrium outcome,
which is the complete information outcome described in Proposition 1.
Proposition 2 establishes that the incentive of headquarters not to disclose information on divisional productivity through capital allocation can be important enough to determine the equilibrium outcome. Avoiding
disclosure is accomplished by allocating the …rst-period capital budget evenly between divisions. Any larger
capital allocation to division A would be interpreted by managers as evidence that division A is intrinsically
more productive and would in turn lead to the absence of e¤ort.
The incentive of headquarters to avoid disclosing information is su¢ ciently strong when heterogeneous productivity across divisions is not too likely ex ante. Then, uninformed managers expect their e¤ort to have an
impact on second-period capital allocation and they therefore engage in value-enhancing e¤ort, regardless of
their relative rank with respect to productivities. In addition, the bene…t of increased second-period capital
productivity must be su¢ ciently large to a type H headquarters relative to …rst-period cost due to ine¢ cient
investment, in order for pooling to be pro…table.
Corollary 1 Under the assumptions of Proposition 2a, type H headquarters with private information about
the productivity of the …rm’s divisions allocates more (fewer) …rst-period funds to the division with relatively
poor (good) investment prospects than under complete information about divisional productivity. Type L
headquarters’ capital allocation does not depend on whether or not it has private information.
Corollary 1 follows from Propositions 1 and 2 and implies a “socialistic”capital allocation in internal capital
markets for the speci…ed parameter set. If investment opportunities across divisions di¤er (headquarters is
type H), the …rm invests …rst-period capital less pro…tably in its “lower q” division B than it would if allocating that capital to its “higher q”division A. Thus, the model provides an alternative explanation for such
13
a distortion of capital allocations based on headquarters’management of divisional managers’expectations
about the sensitivity of capital allocation to changes in e¤ort. The bias in favor of divisions with weaker
investment prospects has been documented in empirical studies by Shin and Stulz (1998), Rajan, Servaes
and Zingales (2000), and Ozbas and Scharfstein (2010). For example, Rajan, Servaes and Zingales (2000)
…nd that divisions that are in high Tobin’s-q industries relative to the …rm’s other divisions tend to invest
less than comparable stand-alone …rms, and that low Tobin’s-q divisions relative to the …rm’s other divisions
tend to invest more.
Corollary 2 The pooling equilibrium under incomplete information about intrinsic productivity renders a type
H headquarters better o¤ than its complete information equilibrium. For a type L headquarters, equilibria
under complete and incomplete information are payo¤ -equivalent.
Thus, private information improves the equilibrium outcome for headquarters. From the perspective of
the two-period investment cycle, either type of headquarters is (weakly) better o¤ following a policy of
nondisclosure (via capital allocation), which implies that the pooling equilibrium outcome dominates the
complete information outcome for both homogeneous and heterogeneous intrinsic productivities. The reason
for this is that when the assumptions of Proposition 2a are met, the bene…ts of productivity-enhancing
competition for funds between divisions outweigh the cost of a temporarily distorted capital allocation.
Thus, if headquarters’ preferences are aligned with investors, withholding information about true capital
productivities increases …rm value.9 Consequently, our model provides an argument for limiting access to
information about other divisions’business opportunities and, in this respect, for strategic lack of transparency
within multi-business …rms. It also may serve as a rationale for why …rms may oppose regulation that increases
transparency about individual units such as detailed segment reporting.
4
Empirical Implications
Overall, biased capital allocation can be interpreted as an investment decision by headquarters: it accepts an
on average lower quality of …rst-period investment to achieve higher quality second-period prospects. In this
section we identify a number of additional implications of the model and discuss related empirical evidence.
9
Note that this result does not necessarily imply that the value of the diversi…ed …rm is higher than the aggregate value if
the divisions were run and …nanced separately. For example, if aggregate budgets are held constant and outside investors are
assumed to be uninformed about intrinsic productivities, it is possible that separately run divisons have a higher aggregate value
than the joint incorporation of both divisions.
14
a) Top Management’s Private Information and Capital Allocation
Our line of argument is based on the existence of private information of top management about relative capital
productivities of divisional managers. It appears reasonable to assume that a given divisional manager’s lack
of information about the other divisions’productivities is more pronounced when those divisions operate in
di¤erent industries from the manager’s own division. Then, managers may rely less on their own knowledge
about industry, technology, products, and regulations when assessing the quality of other divisions’investment
opportunities precise assessments are more challenging to achieve. Thus the model predicts that socialistic
capital allocations are more prevalent in multi-business …rms operating strictly unrelated businesses. This
implication is consistent with Ozbas and Scharfstein (2010) who …nd that it is predominantly unrelated
segments of conglomerate …rms that tend to invest less than stand-alone …rms in high Tobin’s-q industries.10
If top management pursues objectives di¤erent from maximizing shareholder value it is unlikely to communicate about its true objectives. Thus, a weak governance of a …rm in which top management is not prohibited
from pursuing its private objectives typically adds an additional level of private information of top management. As top management’s private information is a prerequisite in our model for an investment bias in
favor of weaker divisions, this bias is predicted to be more prevalent in …rms with weaker governance. This
prediction is also made by Scharfstein and Stein (2000) and Bernardo, Luo and Wang (2006) and is consistent
with empirical evidence in Ozbas and Scharfstein (2010) who document that “socialistic” behavior is more
pronounced in diversi…ed …rm in which top management has small ownership stakes.11
b) Heterogeneity of Productivities and Capital Allocation
Assumption 2, (x
kI2 + e), and Proposition 2, x < 21 kI1 + 2e II21 , provide a lower and an upper bound for
the productivity parameter x that lead to biased capital budgets, respectively. This implies that biased capital
allocations only occur when the di¤erence in intrinsic productivities is neither to small nor too pronounced.12
The latter requirement suggests that an investment bias in favor of weaker divisions is more prevalent when
a …rm’s divisions have reached at least a certain level of maturity.
For each …rm, the parameter range of x in which an investment bias takes place depends on I1 , I2 , e and
10
See also Khanna and Tice (2001) who …nd that …rms with operations in related industries do not appear to subsidize weaker
divisions.
11
Other empirical studies that …nd that better governance leads to higher investment e¢ ciency are Durnev, Morck and Yeung
(2004), Datta, D’Mello and Iskandar-Datta (2009) and Sautner and Villalonga (2010).
12
The empirical …ndings in Rajan, Servaes and Zingales (2000) suggest that “socialistic” behavior is more pronounced the
larger the spread in the industry Tobin’s-q across divisions. However, due to the assumed linearity, their approach would not be
able to detect the inverse U -shape that our model generates.
15
k. While changes of I2 , e and k have ambiguous e¤ects on the parameter set of x that lead to “socialistic”
cross-subsidization, a decrease in I1 strictly expands this set. Thus, an investment bias in favor of weaker
divisions is predicted to be stronger in time periods of temporarily constrained funds, i.e. when free cash
‡ows are low.
c) Intertemporal Pattern of Investment and Inertia
While during periods of stable productivity levels, divisional managers learn about their relative productivity
levels over time, times of possible structural changes in at least one of the divisions frequently renders this
information obsolete and provides room for top management’s own assessment about possible changes in
corporate direction. In the model, headquarters only responds to new developments when the intrinsic
productivity of division A exceeds a certain threshold. Thus, information about changes in productivity
needs to be more precise in order to make diversi…ed …rms alter their capital allocation. As information about
structural productivity changes are revealed over time, the model implies that diversi…ed …rms respond later
to developing structural changes in one of their businesses than single-segment …rms do. When they do, the
shift in investment distribution is substantial as top management’s assessment of productivity changes are
signi…cant to trigger such a shift. The economic literature has long documented that major innovations in
industries are made by new entrants rather than industry incumbents (see, for example Scherer, 1980). The
model predicts that this is even more pronounced if the incumbents are predominantly diversi…ed …rms.
5
Capital Allocation with Incomplete Information:
No E¤ort in the Absence of Learning
The analysis of the preceding sections focuses on the situation in which divisional managers choose to exert
e¤ort in case they don’t learn anything from …rst-period capital allocation
discuss the situation in which this condition is violated
<2
c
I2
2
c
I2
. In this section, we
and, hence, managers do not exert e¤ort,
if …rst-period allocation is uninformative.
In this case, as long as x < 12 kI1 +2e II12 , a pooling equilibrium does not exist, since pooling is not an attractive
proposition for either type of headquarters. In addition, the complete information outcome characterized in
Proposition 1 is not an equilibrium outcome, since a type H headquarters has still an incentive to mimic a
type L headquarters’complete information allocation of
L
= 0:5. For brevity, we omit a detailed analysis
here, but it can be shown that under some additional parametric restrictions, there exists a unique separating
16
equilibrium outcome in which
L
2 (0; 0:5) and
H
= 1. The reason is that a type L headquarters has
a strong incentive to signal its type to restore managerial e¤ort incentives. It does so by allocating more
…rst-period capital to division B than to division A, despite equal capital productivities. This renders it too
costly for a type H headquarters to mimic L’s strategy.
This result implies that, on average, division B obtains a larger …rst-period capital allocation than it would
under complete information. As in turn the frequently stronger division A receives less capital than it
would under complete information, the internal capital market displays “socialistic” behavior also under
circumstances in which pooling does not lead to e¤ort provision.
Xuan (2009) documents that, after CEO turnover, divisions not previously a¢ liated with the CEO display
on average a signi…cantly larger change in capital expenditures (relative to assets) than the divisions through
which the new CEO has advanced. If newly appointed CEOs are perceived to prefer investment in those
divisions they were previously a¢ liated with a high probability (p(L) low), it is optimal for those CEOs to
signal the absence of such a preference by allocating an amount of capital to the other divisions that is beyond
what is justi…ed under symmetric information.
One di¤erence to the pooling equilibrium outcome characterized in Proposition 2 is that in the separating
outcome described here, ex ante expected pro…ts are lower than under complete information. This implies
that ex ante headquarters has an incentive to commit to creating transparency about investment opportunities
across divisions.
6
Conclusion
This paper provides a novel explanation for the bias of investment levels in internal capital markets in favor
of weaker divisions. We present a model based on the communication aspects of capital budgeting. Top
managements of diversi…ed …rms can only generate productivity-enhancing competition between divisional
managers when the perceived di¤erences in productivity between divisions are not too pronounced. E¤orts
either to not disclose it if such productivity di¤erences exist or to credibly signal their absence can induce
top management to choose “socialistic” capital allocations for certain periods of time. Thus, biased capital
allocations can be viewed as an investment decision by headquarters, which yields a number of novel empirical
implications.
The key argument of our analysis is that communication from top management to divisional management
17
(or in more technical terms: private information of top management) is useful in understanding how …rms
allocate capital to its business units. We believe that this notion may also contribute to the understanding
of related areas of capital management, such as the design of budgeting procedures, delegation of authority,
reporting practices, and corporate restructuring. The exploration of these topics may provide interesting
avenues for future research.
18
Appendix 1
Proof of Proposition 2
Perfect Bayesian Equilibria and Re…nements
We …rst de…ne the notion of Perfect Bayesian Equilibrium.
De…nition 1 In our model a Perfect Bayesian Equilibrium is a set of strategies and a belief function (
1)
2
[0; 1] satisfying each of the following conditions:
1. For each type
= H; L, headquarters’ strategy is optimal given managers’ strategies and managers’
posterior beliefs.
2. Both managers share a common posterior belief derived from the prior belief p(L) =
allocation
1;
and headquarters’
following Bayes’ rule where applicable.
3. For each choice of
1,
managers’e¤ ort levels following
1
constitute a Nash equilibrium of a simultaneous-
move game in which the probability that managers face a headquarters of type L is given by their posterior
belief (
1 ).
Condition (2) implies that when
1
is not part of headquarters’optimal strategy for any type, any belief (
is admissible, since in equilibrium observing
1
1)
is a zero probability event and beliefs cannot be derived from
Bayes’rule. Thus, any e¤ort pair (e1 ; e2 ) may be chosen as long as it is a best response for some beliefs. In our
model, beliefs are common knowledge between all players. In addition, managers’beliefs are identical after
any message, not just an equilibrium allocation. Condition (3) says that, given headquarters’allocation
and given their updated posterior beliefs (
1)
1
about , managers react optimally to headquarters’allocation
1.
We …rst determine the set of separating and pooling PBE in pure strategies. Subsequently, we by jointly apply
the equilibrium re…nement concepts of Undefeated Equilibrium and D1. Re…nement D1 puts restrictions on
out-of equilibrium beliefs focusing on one single equilibrium, while Undefeated Equilibrium compares among
equilibrium outcomes and therefore requires a characterization of the full set of PBE, considering all degrees
of freedom with respect to out-of-equilibrium beliefs. Consequently, we start with the analysis of Perfect
Bayesian Equilibria.
Pooling Perfect Bayesian Equilibria
We begin with a characterization of the set of pooling equilibria. It is helpful to recall that
19
;2
refers to type
’s second-period equilibrium pro…t under complete information. Let
;1 (
) denote type ’s …rst-period
pro…t when it allocates .
p
In a pooling equilibrium, both types of headquarters choose
and managers learn nothing from capital
allocation.13 Bayesian updating implies that managers’ beliefs after observing
p(Lj
p)
p
equal the prior belief,
= . O¤-equilibrium beliefs p(Lj^ ) are arbitrary as long as beliefs and corresponding o¤-equilibrium
allocations ^ 6=
p
p.
deter both types from deviating from
response after observing
p
The easiest way to support
Because of assumption (8), managers’ best
is to exert e¤ort ei = e. We thus obtain type ’s pooling pro…ts
p
P
H
=
H;1 (
p
P
L
=
L;1 (
p
)+
H;2
)+
L;2
P:
+ eI2
as an equilibrium allocation is to restrict o¤-equilibrium beliefs such that
p.
managers do nothing unless they observe
Then, o¤-equilibrium payo¤s are lowest and deviating is least
bene…cial for all types of headquarters. We set p(Lj^ ) = (^ ) = 0 for any ^ 6=
p,
supports the largest set of pooling equilibria. To determine the set of admissible
since this belief function
p,
we maximize over all
potential o¤-equilibrium allocations to solve for the highest out-of-equilibrium allocation given these beliefs.
p,
Thus, for any pooling equilibrium choice
H;1 (
p
)+
H;2
L;1
p
+ eI2
H;1 (
H;1
L;1 (
the following conditions must apply:
)+
L;2
L;1 (
p
)
p
max
)
H;1 (
)+
H;2
eI2
max
L;1 (
(9)
)+
L;2
eI2
eI2
Both conditions characterize an interval of permissible
(10)
p
2[
p;
p ],
where
p/ p
denotes the lower/upper
bound of the interval solving (11)/(12).14 We illustrate this formulation in Figure A1, for the interesting case
in which
H;1
13
H;1 (
p
= 0:5) < eI2 :
(11)
We disregard index t since we made second-period allocations implicit in managers’contest for funds.
14
The proof is quite straightforward, given the strict convexity of the left-hand side of inequalities (11) and (12), type H’s and
p
type L’s full information choices at 1 = 1 and 1 = 0:5 as well as the resulting single-crossing point of L;1
) and
L;1 (
p
(
)
on
the
interval
(0:5;
1):
H;1
H;1
20
When (11) does not hold, type H has no incentive to imitate L’s full information allocation
the cost of moving away from its full information optimum,
H;1
H;1 ( 1
1
= 0:5, since
= 0:5), outweighs the gain from
imitating type L, eI2 . In this case, both types of headquarters are better o¤ following their full information
strategy.15 If condition (11) is met, a pooling equilibrium always exists, since eI2 is su¢ ciently high relative
to headquarter’s cost of ine¢ cient investment at the crossing point of both curves, which also implies that
condition (12) is non-binding.16 Hence, we obtain a continuum of pooling equilibrium allocations
interval [
p ; 1],
where
p
p
on the
< 0:5.
Figure A1: Interval of Pooling Allocations
P.
p
p
The conditions H;1
eI2 (11) and L;1
eI2 (12) characterize the set of feasible
H;1 ( )
L;1 ( )
p
p
pooling equilibrium allocations
: The left-hand side of these conditions,
2 fL; Hg,
;1 ( ),
;1
depicts a type’s cost from ine¢ cient investment if it chooses the pooling equilibrium allocation p compared
to its …rst-period pro…t
;1 under full information. eI2 denotes the second-period productivity gain
induced by managerial e¤ort provision.
The …ndings to this point can be summarized in the following lemma.
15
Although a pooling PBE may exist, it can easily be shown that it will not survive the application of any of the standard
re…nements.
16
It can be easily shown that
assumption that x kI1 :
H;1
H;1 (
p
= 0:5)
L;1
21
L;1 (
p
= 0) , 21 xI1
1
kI12
4
1
kI12
4
always holds given the
Lemma 2 Let p(L) =
2
c
I2 .
p
Any …rst-period pooling equilibrium allocation
must belong to an interval
de…ned by (11) and (12). The associated pooling PBE can be supported by p(Lj^ ) = (^ ) = 0 for any o¤ equilibrium allocation ^ 6=
p
are also permissible. If
p.
Other beliefs that do not motivate some type of headquarters to deviate from
H;1 (
H;1
p
headquarters split funds according to
p
= 0:5) < eI2 , a pooling PBE always exists, and both types of
p ; 1],
2[
p
where
< 0:5.
Separating Perfect Bayesian Equilibria
We have so far considered equilibria in which managers remain uninformed after observing headquarters’…rstperiod choice. Let us now characterize the set of separating equilibria.
allocation, if headquarters is type L, and
H,
equilibrium a type H headquarters chooses
L
denotes a separating equilibrium
if headquarters is type H. We show that in any separating
H
= 1, i.e. distributes all funds to its most pro…table division
A, while a type L headquarters selects an allocation
L
that belongs to an interval.
In a separating equilibrium, headquarters’private information is revealed by its …rst-period allocation. Posterior beliefs yield (
L)
= 1 and (
H)
= 0 and managers react optimally as under complete information. For
the equilibrium to be separating, we must guarantee that
L
6=
H
and assure that allocations are incentive
compatible. This implies that a type H headquarters does not want to pick type L’s allocation and vice
versa. In addition, o¤-equilibrium allocations (i.e. allocations that di¤er from
L
and
H)
and corresponding
beliefs must deter both types from deviating from their equilibrium action.
In a separating equilibrium, each type prefers its own allocation as long as the following incentive-compatibility
constraints apply:
H;1 ( H )
+
H;2
H;1 ( L )
+
H;2
L;1 ( L )
+
L;2
L;1 ( H )
+
L;2
+ eI2
(12)
eI2
(13)
Under incomplete information, a type H headquarters, for instance, could deploy type L’s allocation
L
to
induce e¤ort and thereby raise divisional payo¤ in period 2 by eI2 . However, if (12) holds, H has no incentive
to do so. Condition (13) follows from similar reasoning.
In any separating equilibrium type H selects its full information allocation
to division A. The intuition is that any other putative equilibrium allocation
H
= 1 and distributes all funds
H
6= 1 would motivate type H
to deviate from the equilibrium strategy and increase allocations to the more pro…table division A with no
further negative e¤ect on managers’e¤ort levels.17 Using this …nding, H’s incentive compatibility constraint,
17
Any putative equilibrium allocation
H
6= 1 would yield a strictly smaller payo¤ than a putative out-of-equilibrium strategy
22
condition (12), simpli…es to:
H;1
H;1 ( L )
Condition (14) has a straightforward interpretation: for
its own allocation
H
L
eI2 :
(14)
to be incentive compatible, such that H prefers
= 1, H’s …rst-period cost due to ine¢ cient investment,
H;1
H;1 ( L ),
must be larger
than its second-period gain, eI2 , earned by mimicking a type L headquarters.
We now analyze type L’s incentive-compatibility constraint. A type L headquarters would never want to
imitate H since
H
= 1 makes managers believe that headquarters is type H, inducing them to do nothing.
This immediately lowers productivity in period 2 by eI2 . At the same time,
H
clearly makes L’s …rst-period
investment weakly less e¢ cient than any other allocation. Hence, (13) holds for any
L
2 [0; 1]. Consequently,
the sole rationale for headquarters to move away from its full information optimum and to select separating
allocation
L
is to prevent type H from deviating and to make pooling su¢ ciently costly.18
However, in order to credibly signal its type, type L generally cannot select arbitrary
L ’s
satisfying (14) as
for any out-of-equilibrium allocation, there must exist (at least) some belief that would prevent type L from
deviating from
L.
Hence, analogous to the previous analysis of pooling equilibria, in order to determine
the maximum set of admissible
L,
we need to maximize over all o¤-equilibrium allocations to solve for the
highest out-of-equilibrium allocation under beliefs that do not induce e¤ort and impose
L;1 ( L )
+
L;2
max
L;1 (
L;1
L;1 ( L )
)+
L;2
eI2
which yields
This result has an interesting yet simple interpretation: for
eI2 :
(15)
L
to be an equilibrium candidate, L’s cost
due to ine¢ cient investment in period 1 must be weakly smaller than the productivity gain from defending
second period gain from managerial e¤ort. Also, if condition (15) is violated, the cost of ine¢ cient investment
relative to eI2 is “too high”, such that type L may be better o¤ not to signal its type. Consequently, in a
separating equilibrium, type L chooses an allocation
L
which belongs to the interval [
= 1; even if most “favorable” o¤-equilibrium beliefs, namely ( H = 1) < 2
equilibrium, since: H;1 ( H ) + H;2 < H;1 ( H = 1) + H;2 = H;1 + H;2 :
H
18
c
I2
L;
L ],
where
L
and
(which would induce ei = 0), sustained this
Thereby, type L’s ability to separate stems from the fact that type L …nds ine¢ cient investment marginally less costly than
[ H;1 ( )
L;1 ( )]
does type H, while both types of headquarters prefer more managerial e¤ort to less:
> 0: Thus, for type L;
the incentive to separate (i.e. to defend higher period 2 productivity) and the ability to separate (due to low signaling cost) are
aligned.
23
L
denote the lower bounds of the interval solving (14) and (15), respectively. For exposition, we resume the
case of the previous section in which condition (11) holds, and we depict the set of separating equilibrium
allocations in Figure A2.
L
is on the interval [0;
L ],
where
L
< 0:5. Our …ndings can be summarized as
follows.
Lemma 3 In any separating equilibrium, a type H headquarters’optimal …rst-period choice equals its choice
under full information,
H
= 1. A type L headquarters chooses to allocate
L,
which must belong to the
interval de…ned by (14) and (15). The associated separating PBE can be supported by p(Lj^ ) = (^ ) = 0
for any o¤ -equilibrium allocation ^ . Other beliefs that do not motivate some type of headquarters to deviate
from
H
and
L
are also permissible. If a separating PBE exists and
headquarters splits funds according to
L
2 [0;
L ],
where
L
H;1 (
H;1
p
= 0:5) < eI2 , a type L
< 0:5.
Figure A2: Interval of Separating Allocations
L
The conditions H;1
eI2 (14) and L;1
eI2 (15) characterize a type L’s set
H;1 ( L )
L;1 ( L )
of feasible separating equilibrium allocations L : The left-hand side of these conditions,
;1 ( L ),
;1
2 fL; Hg, depicts a type’s cost from ine¢ cient investment if it chooses allocation L compared to its
…rst-period pro…t
;1 under full information. eI2 denotes the second-period productivity gain induced by
managerial e¤ort provision.
24
Equilibrium Re…nement
In the previous sections we have shown that there are two kinds of PBE in pure strategies for the case in
which condition (11) holds. Pooling equilibria are given by
and
L
2 [0;
L ],
where
p
=
L
p
2[
p ; 1]
and separating equilibria by
H
=1
= z < 0:5. We show that jointly applying the notions of Undefeated
Equilibrium and D1 eliminates all equilibria except the pooling equilibrium in which
p
= 0:5.
The rationale behind the equilibrium re…nement is straightforward. We require headquarters and managers
to reason “forward” in such a way, that any deviation from a conjectured equilibrium would lead managers
to form beliefs according to some hierarchy. By applying Undefeated Equilibrium, we require that managers
initially interpret an o¤-equilibrium allocation as an attempt by some type of headquarters to consciously shift
to another, preferred equilibrium, thereby leading managers to adjust their o¤-equilibrium beliefs accordingly.
Concretely, Undefeated Equilibrium applies in our model intuitively as follows. Consider a proposed PBE19 ,
some out-of-equilibrium allocation
not chosen in this equilibrium as well as an alternative PBE in which
some set T of headquarters’types plays
in equilibrium. If each member of T strictly prefers the alternative
equilibrium to the proposed one, the latter is said to be defeated.20 If such an interpretation is not possible,
managers ask which type of headquarters is more likely to gain from this deviation relative to the conjectured
equilibrium, applying the notion of D1. Speci…cally, D1 tests whether an out-of-equilibrium deviation
is more likely to come from some headquarters’ type i than from type j and, if so, managers should put
zero probability on j, p(jj ) = 0. Applying D1, an out-of-equilibrium deviation is said to be more likely to
occur from type i if the set of managers’best responses that motivate i to deviate is strictly larger than the
corresponding set of type j.
Once all o¤-equilibrium beliefs have been restricted according to this hierarchy, a conjectured equilibrium is
reasonable only if neither of the informed headquarters’types has an incentive to deviate.
Applying the re…nement requires several steps. Without loss of generality, we focus on the case in which
both pooling and separating PBE exist. It is helpful to recall that both pooling allocations
separating allocations
L
p
and type L’s
induce managerial e¤ort e. First, diminishing returns to scale and L’s optimum at
= 0:5 make any separating equilibrium allocation
than the least-cost separating equilibrium in which
L
L
< z strictly less pro…table from type L’s perspective
= z. Hence, L has an incentive to shift to its least-cost
19
In general, Undefeated Equilibrium is applied to the notion of Sequential Equilibrium (Kreps and Wilson, 1982). In our
game, PBE and Sequential Equilibria coincide (see Fudenberg and Tirole, 1991).
20
This de…nition of Undefeated Equilibrium is valid, since our model allows us to avoid issues connected with payo¤ ties of
headquarters’types. For a general de…nition, the reader is referred to the original work.
25
separating equilibrium, which defeats any other separating equilibrium. Second, notice that if headquarters
is of type L, marginal productivities of divisions A and B are equal, which implies that any capital allocation
=
is payo¤-equivalent to an allocation
=1
;
2 [0; 1]. Hence, pooling equilibria at
are not reasonable: if headquarters turns out to be L; the separating equilibrium at
L
= 0:5. Managers infer that the pooling equilibrium at
p
>1
z
= z yields a strictly
higher payo¤ to this type. Third, consider any conjectured pooling equilibrium in which
deviation to
p
p
< 0:5 and a
= 0:5 is being played, since both
p
types’payo¤ function strictly increases on the interval [z; 0:5]. Since pooling at
= 0:5 also renders either
type strictly better o¤ than the least-cost separating equilibrium, the latter is also defeated. Undefeated
Equilibrium therefore leaves an interval of pooling equilibria
Let us now show that pooling equilibria at
p
2 (0:5; 1
p
2 [0:5; 1
z].
z] do not survive D1. Consider any conjectured
Undefeated Equilibrium on this interval and also a deviation to
= 0:5. Following D1, managers immediately
eliminate H as the potential defector. By defecting, type H strictly loses, regardless of managers’ beliefs
(and corresponding e¤ort levels) as the cost of ine¢ cient investment increases whereas managerial e¤ort in
equilibrium is already at a maximum. In other words, the set of managers’ best responses inducing H to
deviate is empty. On the other hand, type L clearly deviates to
= 0:5 (its full information optimum)
if managers form a belief that causes them to exert e¤ort. Therefore, D1 requires that managers’ beliefs
following such defection should put all the weight on type L, which in turn forces type L to deviate from the
conjectured pooling equilibrium.
Finally, we show that there exists a unique Undefeated Equilibrium satisfying D1: the pooling equilibrium
at
p
= 0:5. By following its equilibrium strategy, L is strictly better o¤ than with any other allocation,
regardless of managers’ beliefs. Type H may obtain a higher payo¤ by defecting to
causes managerial e¤ort. Consequently, since H has a greater incentive to allocate
requires that the posterior belief conditioned on
2 (0:5; 1] only if
(whereas L has none), D1
should be concentrated on type H. This argument in fact
restricts o¤-equilibrium beliefs, but does not rule out the equilibrium. H prefers to stick to the equilibrium,
since any allocation
induces managers to reduce e¤ort and condition (11) holds.
Proof of Corollary 2
Equilibrium outcome under complete and incomplete information for type H yields
eI2 and
H;1
+
H;1 +
H;2
H;2 ,
respectively; whereas payo¤ equals
follows from condition (11).
26
L;1 +
H;2
for type L.
H;1 (
H;1 (
p
p
= 0:5) +
= 0:5) +
H;2
+
H;2 +eI2
>
Appendix 2
The following table provides an example for parameters that satisfy the assumptions for the result in Proposition 2a.
Parameter
Numerical Value
q1
15
q2
5
I1
80
I2
100
k
0:05
e
10
c
3
x
16
0:1
0:7
27
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