Optimality of the 51:49 Equity Structure1
Susheng Wang
and
Tian Zhu2
June 2016
Abstract: As an extension of Wang & Zhu (2005), this short paper shows that the popular
51:49 equity structure can be optimal. This equity structure in joint ventures (JVs) has puzzled
economists the world over. We find that, when the two parties are highly asymmetric in their
abilities to acquire private benefits from their JV, the 51:49 equity structure is optimal and as
efficient as joint control.
Keywords: Income Rights, Control Rights, 51:49 Equity Structure, Joint Control
JEL Classification: L22, D23, D82.
1
We gratefully acknowledge the helpful comments and suggestions from a referee.
2
Wang: Hong Kong University of Science and Technology. E-mail: [email protected]. Zhu: China Europe In-
ternational Business School, [email protected].
1. Introduction
We interpret the 51:49 equity structure as a contractual arrangement in which the two
partners share revenue equally but only one partner is given control rights. Our goal is to show
why this equity structure is optimal.
In practice, most JVs allocate equal or almost equal equity stakes among partners. According to Hauswald & Hege (2009), about two-thirds of two-partner JVs adopt the 50:50 equity split and about 12% adopt the 51:49 (or 50.1% and 49.9%) split. This is intriguing. As
Holmström (1999) points out, the observed popularity of partial ownership is at odds with the
property rights view of sole ownership in the case of complementary assets. Researchers have
explained asymmetric partial ownership by differences in partner characteristics such as resource costs (Belleflamme & Bloch, 2000), private information (Darrough & Stoughton, 1989),
and incentives (Bhattacharyya & Lafontaine, 1995; Chemla et al., 2004). Wang & Zhu (2005)
show that with incomplete contracting both asymmetric and symmetric ownership can be optimal. Marinucci (2009) finds that the firm whose effort has a higher impact on the JV’s profits should be entitled to a larger profit share, while we show that only majority shareholders,
including the 50% shareholders, should be given control rights.
We use Wang & Zhu’s (2005) two-period model, with incomplete contracting and separate income and control rights. A unique feature of this model is that allocations of both income and control rights are allowed in the initial contract. However, Wang & Zhu (2005) fail
to show the optimality of the 51:49 equity structure. We use the Shapley value to decide on the
revenue reallocation in ex post renegotiation and show that the 51:49 equity structure is optimal when the two parties’ abilities to acquire private benefit are highly asymmetric.
2. The Model
Project
Following Wang & Zhu (2005), consider two partners,
and
who are engaged in a
JV. They are risk neutral in income but have convex costs. There are two periods. In period 1
(ex ante), the two partners invest unverifiable investments (efforts)
ly with private costs
and
and joint effort
and
defined by a function
In period 2 (ex post), the controlling party takes an action
.
. Here, the value of
uncontractible ex ante, but its control right (the right to decide on the value of
ble ex ante. Further, the value of
simultaneousis
) is contracti-
is contractible ex post.
2
Uncertainty realized
Renegotiation
e1 , e2
Contracting
q
Allocating
revenue
1
0
Ex Ante
2
Ex Post
Figure 1. Timing of Events
There is uncertainty in the first period and this uncertainty is realized at
The two par-
Revenue is produced at
ties are allowed to renegotiate their contract at
and it is
allocated based on the existing contract.
Private Benefits
Let
be the ex post revenue. Denote by
ante conditional on
the ex ante revenue, which is random ex
. The controlling party can appropriate part of the revenue. Specifically,
the controlling party has the right to choose
where
is
such that
’s ability to appropriate revenue—a measure of corruptibility. That is,
the controlling party reports only a fraction
of
The announced revenue
is
contractible ex ante.
Contract
Suppose that the two parties negotiate an ex ante contract at
renegotiate an ex post contract at
control rights over
and, if necessary, they
Given that the announced revenue
and the
are contractible ex ante, at the beginning of period 1, the two parties
sign an ex ante contract for
•
allocation of public revenue
•
allocation of control rights over
(income rights)
(control rights)
That is, an ex ante contract has the following form:
This contract can be renegotiated ex post after the investments are sunk but before the action
is taken. Following the literature, renegotiation ensures ex post efficiency. Since ex post social welfare is
if
has the control rights, ex post efficiency means
.
This means that an ex post contract has the following form:
3
We want to identify the optimal ex ante contract. Let
be the set of admissible reve-
nue-sharing schemes defined by
A revenue-sharing scheme
allocates
to
and
There are three possible allocations of control:
control, and
and
where
has sole control,
has sole
have joint control. For joint control, each party is entitled to half of
the announced revenue
agreement on
to
and has the veto rights over
. If the two parties cannot reach an
, no revenue is generated For single-party control, the controlling party can
unilaterally choose
to maximize its ex post payoff.
We impose the following three assumptions on the functions as in Wang & Zhu (2005).
is strictly increasing in
Assumption 1.
Assumption 2.
and
is convex and strictly increasing in
, for
is concave and strictly increasing in
Assumption 3.
and
3. The Solution
If
is ex ante contractible, the solution is called the second-best (SB) solution. If
is
not ex ante contractible, the solution is called the third-best (TB) solution. We solve for the TB
solution only. The TB solution is generally less efficient than the SB solution. When the TB
solution is as efficient as the SB solution, we say that the TB solution is SB. When the TB solution is strictly less efficient than the SB solution, we say that the solution is TB.
The General Solution
If
has sole control over
, which implies
∗
ley value, implying
but to choose
in the second period,
. The ex post social welfare
for each party. Alternatively,
can choose to renegotiate with
is then divided based on the Shapcan choose not to renegotiate,
unilaterally based on the ex ante revenue-sharing scheme. According to Wang
& Zhu (2005), since both parties are risk neutral in income, an optimal revenue-sharing
scheme involves a pair
the latter case,
of revenue shares, with
and
. Hence, in
receives payoff
4
while
receives
. This implies
if
and
decides whether or not to renegotiate by comparing
if
with
This im-
plies Lemma 1.
Lemma 1. Suppose that
,
is given sole control and revenue share
’s ex post payoff is
if
Then, if
’s ex post payoff is
.
Since the contract is renegotiable, two aspects must be considered when trying to determine the optimal revenue-sharing scheme. The controlling party may try to maximize her private benefits by asking for at least revenue share
and ask for revenue share
ex post, or it may demand renegotiation
ex post according to the Shapley value. This means that a re-
negotiation-proof revenue share
for the controlling party must satisfy
.
This explains Lemma 1.
∗
Denote by
∗
the SB revenue shares. Wang & Zhu (2005) offer
∗
∗
and Lem-
ma 2.
Lemma 2. Suppose
must satisfy
∗
is
has sole control and a renegotiation-proof revenue share for
for some
∗
but if
. Then, if
∗
the optimal revenue share for
the optimal revenue share for
is
Lemma 2 is intuitive. It simply states that the further a revenue share deviates from the
SB
∗
, the less efficient it is. Hence, the optimal share under condition
∗
if
∗
satisfies condition
the latter case,
is closest to
∗
or
if
∗
under condition
is either
does not satisfy the condition. In
Lemma 1 indicates
.
To determine the optimal contract, we consider three possible allocations of control:
has sole control,
allocation
has sole control, and
and
have joint control. Conditional on each
of control, using Lemmas 1 and 2, we can find the conditional optimal reve-
nue-sharing scheme
which implies a value of social welfare
. By com-
paring the three values of social welfare from the three possible allocations of control, we can
find the optimal allocation
of control. Then, the optimal revenue-sharing scheme is
This solution is stated in Proposition 1 and illustrated in Figure 2.
The proof is available upon request. Without loss of generality, assume throughout the paper
that
∗
∗
suggesting that
is more important than
for the JV.
5
Proposition 1. Suppose
∗
∗
. If the ex post action
is not contractible ex ante but its
control rights are, and
(1) If
∗
, the optimal ex ante contract gives
∗
(2) If
sole control and revenue share
. This solution is SB.
∗
, single party control and joint control can both be optimal. For single party
control, if
thermore, if
then giving
has control, the optimal revenue share for
, then it is not optimal for
is
. Fur-
to have sole control; if
;
sole control yields the same outcome as joint control. These solutions are
TB.
∗
Figure 2. Regions of the General Solution
Figure 2 divides the possible combinations of
into three regions. In each region,
the possible optimal allocations of control and income rights are indicated.
Parametric Solutions
In regions I and II of Figure 2, we do not identify the optimal solution; instead, we narrow
down to two possible solutions. To pin down the optimal solution in regions I and II, we consider a simple parametric case with
(1)
where
is a random variable with
Let
∗
. We find the
optimal solution in each of four regions as illustrated in Figure 3. The optimal solution is
6
unique in each region, except in region I. In region I, there are two optimal solutions, which
are equally efficient. The derivation of these solutions is available upon request.
∗
Figure 3. Regions of the Parametric Solution
To show that the results in Figure 3 do not depend on the special functions in (1), consider
more general functions:
(2)
, and
where
is a random variable. There is a well-defined
,
∗
,
such that the solution is again defined in Figure 3.
4. Main Results
The solution presented in Figure 3 implies the following conclusions. First, intuition suggests that since
is by assumption more important for the JV i.e.,
∗
∗
, if
is not
very corruptible, it should be given sole control. Indeed, as shown in regions III and IV, when
is not very corruptible
, giving it sole control is optimal. Further, when
∗
ficiently incorruptible
control will motivate it to invest the SB effort
moderately corruptible
∗
, offering it the revenue share
∗
compatible for the SB effort, since
∗
∗
the optimal revenue share under condition
is
is no longer incentive
can obtain a larger revenue share
Hence, the revenue share must be
together with sole
, implying a SB outcome. However, if
, the revenue share
is suf-
from corruption.
to ensure incentive compatibility. By Lemma 2,
is
, which is a TB outcome.
7
Second, intuition suggests that if
is very corruptible, it should not be given sole con-
trol. Indeed, as shown in regions I and II where
Third, intuition suggests when both
,
and
is not given sole control.
are both highly corruptible, neither
should be given sole control. Join control can be a good solution in this case since it gives each
party veto power and prevents the other party from appropriating revenue. Indeed, Figure 3
indicates that joint control is optimal when both
and
and
are highly corruptible
).
Fourth, intuition suggests that if
corruptible,
is sufficiently incorruptible and
is sufficiently
should be given sole control. Indeed, as shown in region I, if
ciently incorruptible
and
is sufficiently corruptible
,
is suffican be giv-
en sole control. A simple and practical way to implement this solution is to give
and
51% and 49% of shares (or 50.1% and 49.9% of shares), respectively. Hence, the 51:49 equity
structure is optimal when the two parties are highly asymmetric in their abilities to gain private benefits.
Fifth, when
is given sole control, a revenue share
is not renegotia-
tion-proof, since this partner can demand renegotiation to obtain the revenue share
ex post. Subject to the renegotiation-proof condition
enue share for
is
negotiation will ensure
, by Lemma 2, the optimal rev-
if this partner has sole control. Given the fact that ex post re, giving
sole control with
implies the same out-
come as joint control. Indeed, Figure 3 indicates that these two arrangements are equally efficient in region I.
Finally, all the solution regions in Figure 3 show that a minority shareholder is never given control rights; only majority shareholders, including the 50% shareholders, can be given
control rights. Hence, in our solutions, income and control rights are bundled together. Hart &
Moore (1990) observe the same in practice. In their model, control and income rights are bundled together by assumption; in our model, income and control rights are not bundled together in the setting. However, we show that it is optimal to bundle them together in equilibrium.
Appendix
Proof of Lemma 1
If
has the control rights over
in the second period,
can choose to renegotiate
∗
with
and ex post social welfare maximization implies the choice of
pie
is then divided based on the Shapley value given sunk investments
which means that each party obtains
Alternatively,
. The fixed-sized
and
,
can choose not to renegotiate,
8
but to choose
unilaterally and then divide the public revenue with
revenue-sharing rule. In the latter case,
while
receives
if
based on the ex ante
receives payoff
. This implies that
if
decides whether to renegotiate by comparing
decision follows from the following poblem, implying that
and
with
That is,
’s
receives payoff
This immediately implies Lemma 1.
Proof of Proposition 1 (Figure 2)
For Part (1)
∗
Consider a contract that offers revenue share
∗
sumption of
∗
∗
implies
and
∗
∗
∗
and
∗
Since
ex post payoff
∗
∗
and sole control to
Together with
, by Lemma 1,
by the definition of
∗
∗
. The as-
∗
we have
∗
’s payoff is
. With
, the ex ante efforts will be the SB efforts
. That is, the solution is SB. Since no solution can do better than SB, this solution is
optimal.
If
is given sole control instead, by Lemma 1, any renegotiation-proof revenue share
∗
which is greater than
must satisfy
will not invest the SB
∗
. Hence, giving
. Hence, the solution is inferior to giving
Hence, the best solution is to give
∗
∗
. Since
sole control is inferior to giving
∗
If the two partners are given joint control, since
∗
as
,
sole control.
will not invest the SB
sole control.
sole control with revenue share
∗
.
For Part (2)
Since
Also, since
Suppose
∗
, condition
∗
we have
∗
implies
∗
; hence
for both
has sole control. If
is
’s payoff is
manage to obtain revenue share
ante contract with such a revenue share
and 2.
, by Lemma 2, the optimal revenue
share under the constraint that
by Lemma 1,
∗
On the other hand, if
This means that
can
by corruption or renegotiation. Hence, an ex
is not renegotiation-proof. But the outcome with
9
’s payoff being
is achievable using a renegotiation-proof revenue share
We have thus proven that when
is given sole control, the optimal rene-
gotiation-proof revenue share is
∗
i.e.,
∗
Further, because
, the optimal solution in this case is not SB.
when
We now need to determine optimal control. If
optimal sharing scheme is
has sole control, the
. By Lemma 1, since
contract is inferior to a contract offering joint control with
∗
, this
Hence, it is not optimal
sole control in this case. Thus, the optimal solution is either joint control or
to give
control with
On the other hand, if
when
then the optimal revenue-sharing scheme is
has sole control,
This contract obviously
yields the same outcome as a contract offering joint control. Hence, the optimal solution is either to give
control with
or to give
control with
where the former is equally efficient as joint control.
Derivation of Figure 3
The following proposition is from Wang & Zhu (2005, Proposition 1).
Proposition 2. If the ex post action
∗
the solution is
is contractable ex ante, then under Assumptions 1-3,
and a linear revenue-sharing scheme
∗
where
∗
(
∗
(
∗
,
)
∗
)
which induces efforts
,
We have
∗
∗
∗
determined by
∈
and this solution is SB.
Consider the simple parametric case with
(5)
where
is a random variable with
and
We have
which satisfies Assumptions 1 and 3. The cost function
satisfies
Assumption 2.
The SB investments defined by (4) are
10
∗
∗
(6)
∗
(7)
and the SB revenue-sharing rule defined in (3) is
∗
∗
To have
∗
we need
In region I of Figure 2, if
then
has sole control with optimal sharing rule
will choose investment
(and
to solve the following problem:
∈
and
will choose investment
to solve the following problem:
∈
The Nash equilibrium is
which yields social welfare
Symmetrically, in region II of Figure 2, if
(and
has sole control with optimal sharing rule
investments are
and
and social wel-
fare is
∗
Under joint control, the sharing rule is
which is the same as that when giving
sole control is optimal. Hence, the solution is the same and social welfare is
In region I of Figure 2, we compare the case where
where
has sole control. The social welfares
spectively. The inequality
∗
∗
we have
∗
and
is equivalent to (11). Hence, if
∗
otherwise
In region II of Figure 2, we compare the case where
The social welfares
are defined in (8) and (9), re-
is equivalent to
∗
That is, if
and
has sole control with the case
has sole control with joint control.
are defined in (9) and (10), respectively. The inequality
∗
we have
otherwise
We thus
have the solution in Figure 3.
11
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