In this section, we formulate our model. Subsection 2.1 describes the model framework. Subsection 2.2 formulates the asymmetric information problem. Subsection 2.3, as a benchmark, provides the solution to the symmetric information problem.
The owner of a firm has the option to invest in a single project. We assume that the owner (principal) delegates the investment decision to a manager (agent). Throughout our analysis, we assume that the owner and the manager are risk neutral and aim to maximize their expected pay-offs.
If the investment option is exercised at time t, the firm pays the one-time fixed cost 
and receives cash flow
, which follows a geometric Brownian motion:
,
(1)
where 
denotes a standard Brownian motion, and where 
and 
are positive constants. For convergence, we assume that 
where 
is a constant interest rate.4 We assume that the one-time fixed cost, 
, takes one of two possible values: 
or 
with
, where 
for all
. We denote
. We assume that 
represents “lower cost” expenditure and 
“higher cost” expenditure. The probability of drawing 
equals
, an exogenous variable.
Let 
denote the project value for 
(
). The value is defined as
, 
, (2)
where 
denotes the expectation operator given that
, and 
is the stopping time at which the investment is exercised once 
arrives at the threshold 
for
, i.e., 
Using standard arguments, the project value is rewritten as
,
(3)
where
. In this paper, the current state value 
is sufficiently low that the investment is not undertaken immediately.
The cash flow, 
, is assumed to be observed by both the owner and the manager. However, the one-time cost, 
, is observed privately only by the manager.5 Immediately after making a contract with the owner at time zero, the manager observes whether the cost expenditure is of “lower cost” or “higher cost.” On the other hand, the owner cannot observe the true value of
. Therefore, the owner must induce the manager to reveal private information truthfully at the time when the manager undertakes the investment. Otherwise, the owner suffers from further losses. Suppose, for example, the manager observes 
as the realized value of I. Then the manager diverts the difference 
to himself/herself by reporting 
to the owner. To prevent the diversion, the owner must encourage the manager to report the true value by providing incentives.
In this subsection, we formulate the owner’s maximization problem under asymmetric information. As explained earlier, under asymmetric information, the owner must induce the manager to reveal the manager’s private information truthfully.
In this paper, the owner designs a contract at time zero that commits the owner to give the incentive-disincentive to the manager at the time of investment. Renegotiation is not allowed. While commitment may cause ex post inefficiency in investment timing, it increases the ex ante owner’s option value. To reveal private information, we assume that the owner provides a bonus-incentive 
and/or audits the manager with a probability 
at the time of investment.
The audit technology allows the owner at the cost to verify the state announced by the manager, and to impose a penalty on the manager for cheating when a false report is detected. We assume that a penalty 
is an exogenous constant, and that the cost of auditing 
is a function of probability 
with
, 
,
, and
. These assumptions are intuitively reasonable. The first assumption is that there is no cost incurred if the owner does not use the audit technology. The second and third assumptions imply that 
is strictly increasing and convex in
. The final assumption is that complete auditing incurs a huge cost that the owner cannot pay.
Thus, the contract in the asymmetric information problem is modelled as a mechanism:6
,
.
Let superscript “A” refer to the asymmetric information (agency) problem.7
Then, the asymmetric information problem is to maximize the owner’s option value through choice of the mechanism
, i.e.,
subject to
, 
, (8)
,
. (9)
Here, the objective function (4) is the ex ante owner’s option value.
Constraints (5) and (6) are the ex post incentive-compatibility constraints for the manager under states 
and
, respectively. Consider, for example, constraint (5). The manager’s payoff in state 
is 
if he/she tells the truth, but it is 
if he/she instead claims that it is state
. Thus, he/she tells the truth if (5) is satisfied. Constraint (6) follows similarly. Constraint (6) will be shown not to be binding. Thus, only constraint (5) is relevant to our discussion.
Constraints (7) and (8) are the ex ante participation constraint and the ex post limited-liability constraints, respectively. Constraints (8) ensure that the manager makes an agreement about employment. For example, if 
, then the manager would refuse the contract on learning that
. In addition, it is straightforward to show that constraint (8) implies constraint (7). Thus, only constraint 
will be relevant to our discussion.
Constraint (9) is obvious, where 
is the probability of an audit.
Before analysing the asymmetric information problem, we first review briefly the symmetric information problem.
In this subsection, we consider the optimization problem when the owner observes the true value of
. This problem is equivalent to the problem in which there is no delegation of the investment decision because the manager has no informational advantage. Then we have 
and 
for all 
(
). Thus, the contract 
in the symmetric information problem is modelled as
,
.
Let superscript “asterisk” refer to the symmetric information (no-agency) problem.
In the symmetric information case, the owner’s maximization problem is defined as
where 
(
). The solutions are
By substituting these solutions, we have
We employ these results as a benchmark which is the same as those in the standard (MS) model.
does not have the incentive to tell a lie as a manager in state
, because the manager in state 
suffers a loss from such a false announcement. Thus, (6) is satisfied automatically, and 
and 
are obtained at the optimum. Finally, 
in (9) is satisfied automatically. This statement is shown by 
and 
for any
.
, 
, 
, (14)
for all 
(
).
. The contract can be derived in three possible regions: the incentive-only (bonus-incentive only, Ab) region, the incentive-disincentive (combination, Ac) region, and the disincentive-only (audit-penalty only, Aa) region. Let superscripts “Ab”, “Ac”, and “Aa” refer to the optimums for the three feasible regions, respectively.
is as follows:
, 
where 
and 
are the investment trigger for 
and the bonus for 
in GW model as shown in Appendix B.
, 
where
.
,
.
is decided by
, and 
is decided by
. Similarly, in Region Aa, 
is decided by
.
(
). In particular, we have
,
.
and 
for any
. It is less costly for the owner to distort 
away from 
than to distort 
away from 
in equilibrium.
and 
if
, and that 
and 
otherwise. In other words, a decrease in 
is equivalent to the associated decrease in
.
where 
can be regarded as the information rent for the manager in state
. The owner gives the manager in state 
a portion of the information rent. Importantly, note that the information rent is decreasing with
.
is unimodal with
. The reason is that 
is in-creasing and concave with
, while 
is de-creasing and convex with
. The first statement is straightforward because 
is increasing and convex with
. The second statement is shown by 
at the optimum.
moves the contract 
from the incentive-only (Ab) region to the disincentive-only (Aa) region via the combination (Ac) region. This property is intuitive as follows. For example, the larger the penalty 
is, the more available the disincentive mechanism. Then, an increase in the penalty encourages the owner to use the disincentive mechanism rather than the incentive mechanism.
, 
,
) of the six components of the contract, and that we have
, 
, 
for
. It is straightforward to show that 
as
. Thus, these results are summarized as follows.
, we have
, 
,
.
in the combination (Ac) region turns out to be that in the S model. As the penalty 
becomes sufficiently large, the solution in the disincentive-only region in the asymmetric information model converges to in the symmetric information model developed by MS.
is endogenously decided in the S model as shown in Appendix B. Thus, we have the second statement from Proposition 1. The third statement is given by Corollary 2. As a result, Remarks 1 implies that the solutions in our model include those in the three related papers.
and
, respectively, yields
, the total value 
is
. Obviously, inefficiency is caused by the second term in state
.
, 
, and 
as the owner’s, manager’s, and total values in the GW model, as shown in Appendix B. Then, we obtain the following results.
,
.
,
.
,
.
is monotonically increasing in
, while 
is monotonically decreasing in
. The second is that asymmetric information always leads to a decrease in total value for any finite penalty
. The third is that we do not always have 
although we always have 
. In summary, because the set of the incentive-disincentive contracts is enlarged by that of the incentive-only contracts in the GW model, the owner’s (manager’s) value is always larger (smaller) than in the GW model. However, the sum of these values is not always larger than in the GW model. Thus, depending on the parameters, we have
. These imply that the owner’s rationality does not correspond to the total rationality. We will discuss this result in the next section by using numerical calculations.
as
.
is strictly positive for any finite penalty P. This definition is exactly the same as in Subsection 4.3 of the GW model. We define the total social loss in the GW model as
. Obviously from the definition, the properties of total value 
are equivalent to that of total social loss
.8
is sufficiently large, the following results are obtained.
, we have
, 
, 
and
.
. In particular, the owner’s value is increasing monotonically in
. However, the total value (i.e., total social loss) is non-monotonic with
.
as
,
(18)
is interpreted as a measure of “efficiency” for the auditing cost function. Assume that the basic parameters are
, 
, 
, 
, 
, 
, 
and
.
, the incentive-only (Ab) region is 
, the combination (Ac) region is 
, and the disincentive-only (Aa) region is
. First of all, recall that the contracts in the GW model are the incentive-only contracts. The contracts in the incentive-only (Ab) region are the same as those in the GW model because 
for 
(
). Next, note that the penalty is equal endogenously to the managers’ diverted cash flow, i.e., 
. The contract for 
in the combination region turns out to be the same as that in the S model. Finally, as
, the contracts in the disincentiveonly (Aa) region converge to the symmetric information contracts in the MS model.
with respect to the penalty 
(
). We see that 
is decreasing mo-
with
.
and
, with respect to
. On the one hand, 
is decreasing with 
although 
is constant with
. On the other hand, 
is unimodal with
. In particular, as explained earlier, 
is increasing and concave with
, while 
is decreasing and convex with
.
and 
with respect to
. The important result is that 
is in between two values, 
and
. In addition, 
is increasing monotonically with
. This property corresponds to the “maximal penalty principle” in the microeconomics literature.9
and 
with respect to
. Recall that the manager’s value is zero under symmetric information while it is strictly positive constant under the GW model. The value 
is decreasing monotonically with
. The reason is that an increase in 
decreases the information rent for the manager in state
, which leads to the decrease in the bo-nus-incentive.
and 
with respect to
. The most interesting result is that 
is not decreasing monotonically with 
although 
is constant with P. Here, 
is constant with 
because it is exactly the same as
, while 
is always decreasing with
. On the other hand, 
is increasing or decreasing with
. Thus, an increase in 
does not necessarily lead to a decrease in 
although it always increases the owner’s option value. Consequently, an owner’s (individual) rationality does not necessarily lead to total social rationality.
(
). Under
, Region Ab is
, Region Ac is
, and Region Aa is
. Under
, Region Ab is
, Region Ac is
, and Region Aa is
. An decrease in the parameter 
does not necessarily decrease
. Consider, for example,
. Then, 
under 
is smaller than under
. Thus, a reduction in the inefficiency of the auditing cost always benefits the owner, while it does not necessarily increase total social welfare.
reaches the trigger
, the market does not know the true value of
. The market believes that 
with probability 
and 
with probability
.
, the private information is fully revealed. The manager’s investment decision signals 
to the market. If the manager undertakes the investment at 
, then the stock price instantly jumps upwards to
,
.
where
. Then the stock price instantly jumps downwards to
.
. Thus, the sizes of upwards and downwards jumps are the same under
.
. Then the stock price is 63.42 just prior to 
. If the manager undertakes the investment at
, the stock price jumps upwards to 73.13. Otherwise, the stock price jumps downwards to 53.70. Thus, the jump size in the stock price reaction to investment is 9.71 under
.10
on the size of the stock price reaction to investment. Consider a change in the penalty 
from 0 to 100. Our intuitive conjecture is as follows. The jump size in the stock price reaction is decreasing monotonically in P
, shown in the left-hand side panel of 
, shown in the upper right-hand side panel of 
is non-monotonic with P. This result is also new and not shown by the GW model where
. Note that the jump size in the GW model corresponds to that in the incentive-only region (
) in our model.
.
, shown in the lower right-hand side panel. The volatility parameter 
is changed from 0 to 0.4. The other parameters are unchanged. We see that the jump size in our model and the GW model are 9.71 and 7.24 under
, respectively. Here interestingly, the jump size in our model is increasing in
, while in Grenadier and Wang [
. The reason is that the probability is a Λ-shaped curve with
.
on the auditing cost function, shown in the lower right-hand side panel. The efficiency measure 
for the auditing cost function is changed from 10 to 40. Again, contrary to our intuition, the jump size in the stock price reaction to investment is decreasing in
.
, the Lagrangian can be formulated as:
(
) denotes the multiplier on the constraints. The Kuhn-Turker conditions are
, 
, 
, 
and
, 
,
.
and 
are equal to zero. First, suppose that 
and
. Then we have 
and 
implying
, which contradicts
. Thus, at least one of 
and 
must be binding. Second, suppose that 
and
. Then since
, we have the solution in the bonus-only (Ab) region with
. Third, suppose that
. Then we obtain 
and
. It is straightforward to obtain the solution in the combination (Ac) region. Finally, suppose that 
and
. Then we have 
, and 
because of 
. Thus, we get the solution in the auditonly (Aa) region with 
.
. First, it is easy to obtain 
from
and
. Second, 
can be proved because of
and
. Finally, we prove
. The trigger 
is equal to
.
at the optimum. Because 
is strictly increasing and convex with
, the second term is negative. Thus, the proof is complete.
, 
, 
, 
,
.
, compared with the symmetric information problem defined by (10). The optimal contracts are obtained by
,
.
, 
where
.
, 
, 
, 
, 
, 
, 
, 
,
.
, 
, 
and 
where the superscript “S” refers to the optimum in the S model, and reduce the number of constraints to only one:
. Equation (B.2) is reduced by the term
, compared with (10) in the symmetric information problem. The optimal contracts are obtained as follows. If
, 
turns out to be:
, 
is substituted. The owner’s, manager’s, and total values become
, 
.