THE ACCOUNTING ART OF WAR: A MULTI-PERIOD MODEL OF EARNING
MANIPULATION AND INSIDER TRADING
The Rotman School of Management
University of Toronto
105 St. George Street
Phone: (416) 978-3827
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THE ACCOUNTING ART OF WAR: A MULTI-PERIOD MODEL OF EARNINGS
MANIPULATION AND INSIDER TRADING
If managers use earnings manipulation to maximize their payoffs, why would they not also
trade securities based on their inside information? To address this question, this study examines
how trading by managers on inside information is related to earnings management and managers'
effort in a multi-period setting in the presence of capital markets.
Unlike previous studies, we show that managers' marginal costs of effort are correlated
positively with earnings management. The results of the model also indicate that if managers'
marginal costs of effort are observable, market traders can then infer the optimal level of managerial
effort and, in turn, the true earnings of the firm. If, however, managers' effort costs are
unobservable, depending on the role the regulatory agency assumes, a partial or a full separating
equilibrium ensues with respect to reported income. Furthermore, along the optimal trajectory of
control variables, both earnings management and insider trading are negatively correlated with the
level of audit intensity chosen by the regulatory agency.
“Appearance and intention are fundamental to the Art of War. Appearance and intention
mean the strategic use of ploys, the use of falsehoods to gain what is real.” (Yagyu
Munenori, The Book of Family Traditions on The Art of War)
Earnings manipulation, which, like the quote above, is strategic and uses false information
to gain something real, has interested accounting researchers for a while (see for example, Dye
, Merchant , and Bruns and Merchant , Elitzur and Yaari , Evans and
Sridhar , Demski , Arya et al. , Fischer and Verrecchia , Liang ,
Ewert and Wagenhofer , Goldman and Slezak , Guttman et al. , Ronen and
Yaari , and others). Trading by insiders on private information is an interesting dimension of
this phenomenon that, despite various related empirical findings (for example, Beneish and Vargus
, Bartov and Mohanram , and Cheng and Warefield ), has not been modeled 1.
This paper, in contrast with other models of earnings management, bridges the gap between the two
phenomena by incorporating endogenously strategic trading of securities by managers (based on
inside information) and analyzing it in tandem with earnings management and managers' effort.
The study also analyzes the signaling aspects of the financial statements under insider trading.
The contribution of this study lies in the new insights it provides on earnings management
and its relation to insider trading and managers' effort. One of the criticisms with respect to
earnings management models is that any meaningful analysis of the managers' decisions on earnings
management, insider trading and their efforts cannot be captured in a single period model. Almost
all of modeling studies on the topic of earnings management use single period models (the only
notable exceptions are Dye  and Elitzur and Yaari ). To address this concern, a multi-
period dynamic model is used to capture the long-term aspects of the relationship between the
The exception is Ronen et al. , which focuses on insider trading by directors, not managers, and uses a single
manager and the firm. Another criticism of earnings management models, particularly those based
on the principal-agent model, which is addressed in this study, is the absence of a capital market
from the analysis. The use of a capital market in this study also provides a means to examine the
informational aspects of insider trading. In addition, the model here also provides for analysis of
regulatory issues involving insider trading, such as how the choice of inspection intensity and
penalty function by the regulatory agency affects the market.
The results indicate that, along the optimal path of control variables, the managers' marginal
cost of effort is correlated positively with earnings management and negatively with the level of
managerial effort. We distinguish between two cases. First, if managers' marginal costs of effort
are observable, market traders can infer managers’ optimal level of effort and, in turn, the firm’s
true earnings. Second, when managers' effort costs are unobservable, a partial separating
equilibrium ensues with respect to reported income. Moreover, it is shown that a regulatory agency
such as the SEC can design a penalty function that yields a fully separating equilibrium with respect
to managers' types and, in turn, their efforts and the firm’s true income. This result implies that if
the market holds rational expectations, models of earnings response coefficients that ignore
differences in earnings quality among firms should then be reexamined. Another interesting finding
is that, along the optimal trajectory of control variables, both earnings management and insider
trading are negatively correlated with the level of audit intensity chosen by the regulatory agency.
The paper proceeds as follows: Section 2 describes the model; Section 3 examines the
relationship between earnings management, insider trading and managers' effort; Section 4 deals
with the informational aspects of reporting by managers; Section 5 contains an analysis of
regulatory issues related to insider trading. Concluding remarks and summary are offered in Section
2. THE MODEL
2.1 The Time Line
The firm is a principal-agent contract between owners and managers. The time line of their
relationship is depicted in Exhibit 1. The setting in this study is a multi-period dynamic setting
where the manager is hired by the owner to operate the firm (event 0 in Exhibit 1 below). The
expected tenure of the manager with the firm is T periods. The manager's compensation consists of
a cash bonus which is based on a fraction, k, of reported income, R(t), and equity holdings in the
firm whose value is linked to the capital market (see Appendix 1 for a complete description of the
notations we use in this study). In each period, the manager operates the firm (event 1 in Exhibit 1)
and then assesses future streams for the firm (event 2 in Exhibit 1). Based on his or her
observations, the manager decides on the optimal paths over time of earnings management, θ(t),
insider trading, I(t) and managerial effort, a(t) (event 3 in Exhibit 1). These three decision variables
are the control variables in the manager's problem. The manager then implements the trading and
effort strategies (event 4 in Exhibit 1). The manager’s effort affects the firm's true income,
π(t),which can be observed only by the manager (event 5 in Exhibit 1). Based on his pre-
determined reporting strategy, the manager then (event 6 of the Exhibit) releases his report of
income, R(t), which consists of true income, π(t), plus an earnings management accrual, θ(t), that
can assume any sign. The report is then analyzed by all players other than the manager (event 7),
and is followed by the market’s reaction (event 8). The last event in the time line is the awarding of
cash bonus and equity holdings to the manager (event 9). This sequence of events is repeated for
every period until T (node 10 of the Exhibit).
Exhibit l here
2.2 The Setting:
The Manager's Preferences:
It is assumed that the manager is risk-neutral and that (s)he discounts the periodic cash flow at a
discount rate, denoted as r. The manager selects his level of effort, a(t), which carries a marginal
cost for the manager, μ. That is, the manager in this setting is risk-neutral but effort-averse. We
assume that he has limited wealth and cannot purchase the firm from shareholders. Similar to
Elitzur and Yaari (1995), I assume that the manager does not go bankrupt, i.e., W(T) > 0. Short
sales, however, are allowed in periods prior to T. The possibility of the manager deliberately
causing the dissolution of the firm is also ruled out. This last assumption can be justified based on
the detrimental effects of bankruptcy on the manager's reputation and, consequently, on his or her
market value (i.e., the assumption is that the managers' cost of a dissolving the firm, deliberate or
not, exceeds his or her gain from it).
The Capital Market:
The capital market reaction to reported earnings is based on the following equation
[πˆ(t) + θ (t) - Fˆ(t)] (1)
P t -1 P t -1
where P(t ) denotes the change in price in period t; π(t) and θ(t), as defined above, are the
components of reported income; F(t ) is the earnings forecast for the period, i.e., the term in
brackets is the earnings surprise; β is the earnings response coefficient; and Pt -1 denotes the
previous price (note that P(t ) divided by Pt -1 is the rate of return in period t). Multiplying both
sides by Pt -1 provides for the following equation of motion governing the periodic rate of change in
& [ ˆ ]
P(t) = β π (t) + θ (t) - F(t)
Consistent with the literature on accruals mispricing [Sloan, 1996], we first study a naive market, as
it does not distinguish between the two components of reported income and thus has the same price
reaction with respect to firms with the same reported earnings, irrespective of the underlying
earnings quality. This assumption will subsequently change to investigate the effect of various
degrees of market efficiency. This formulation is useful to answer the question of whether inside
trading, under such a setting, improves market efficiency. Furthermore, similar to Ronen et al.
(2003) and Ewert and Wagenhofer (2005), this equation of motion, in essence, constitutes the
market’s response function, and this, in turn, allows us to find the equilibrium of the game.
Trading on Inside Information
The manager's decision to trade securities based on superior information takes place in
addition to his or her decisions on the choice of earnings management and level of effort. One can
distinguish between the terms insider trading, which refers to trading on inside information, and
trading by insiders, which is not necessarily based on inside information. For our purposes the term
"Insider Trading" denotes the former. The net number of securities sold (bought) in time t is I(t)
leading to the net cash in-flow (out-flow), or P(t)I(t) . The payoffs for such activities can be quite
"Although the reasons why corporate insiders choose to trade or not to trade in their
firm's securities remain unclear, the conclusion that insiders on average earn superior
returns on their trading activity is well documented (Fowler and Rorke, 1988;
Seyhun, 1986; Givoly and Palmon, 1985; Finnerty, 1976; and Jaffe, 1974). This
result suggests that trading by insiders, on average, is based on superior information
about their firms." (Allen and Ramanan, 1990, pp. 519)
The findings above indicate that managers often choose to engage in insider trading
activities (as is also shown by recent US cases). This activity, however, carries the risk of being
found out and suffering a significant (though not unbounded) penalty, on top of the potential legal
Let δ be the probability that the manager's trading on inside information will be detected by the
regulatory agency, say, the SEC, and let the associated penalty imposed by the agency be ω(.). The
penalty paid, ω, is a function of the trading on inside information, i.e., it is an increasing function of
the absolute value of both insider trading, I(t) , and inside information θ(t). In order to insure an
interior solution, we assume that the second derivative of this penalty with respect to either I(t) or
θ(t) is positive, respectively, ωII(t) > 0 and ωθθ(t) > 0. Note that since the penalty is defined on the
absolute value of the units traded, I(t) can have any sign.
The Owner-Manager Relationship and the Manager’s Program
The manager is hired by the owner to operate the firm. At the time of contracting, the owner
knows that the manager's payoff is defined over his or her expected periodic cash flow, C(t) , and
wealth, W(T ) , but the former does not know the values of these variables. The contract between
the parties in its most general form is made of: (l) a cash bonus as a fraction, k, of reported income,
R(t)= [π (t) + θ (t)] ; and (2) equity holdings in the firm. Such compensation schemes are quite
common in practice. The true income of the firm, π(t), is based on a Cobb-Douglas production
function where A is a constant; γ is the elasticity of a, which is assumed to be positive but less than
one; and b(t ) represents exogenous factors whose elasticity is η. The Cobb-Douglas production
function is often used in economics and is used here to enhance mathematical tractability without
loss of generality.
Consistent with the above discussion, the manager's problem is to maximize his or her objective
functional (3) subject to constraint (4) and (5):
Max ∫ e- rt C(t)dt + e- rT W(T) (3)
Subject to the following equation of motion:
& [ ˆ ]
P = β π (t) + θ (t) - F(t)
The components of (3) are defined as follows:
C(t) = k(t)[π (t) + θ (t)]+ P(t)I(t) - ω(.)δ - μa(t)
ˆ ˆ ˆ
π (t) = Aa(t )γ b(t )η
W(t) = P(T)α (T)
The objective functional (3) describes a decision by the manager with an uncertain payoff. The
expected payoff of the manager comes from two sources: his or her expected periodic cash
flow, C(t) , and the present value of his or her terminal holdings, W(T) . The manager's expected
periodic cash flow, C(t) , as described in constraint (5), is equal to an annual bonus, as a percentage,
k(t), of expected reported income, plus (less) net sales (purchases) of securities at the expected going
market price, less the expected penalty to insider trading, ω(.)δ, and less the cost of managerial
effort, μa(t). The results here can be obtained with any function mapping a(t) to π(t). Constraint (4)
describes the equation of motion that governs the periodic rate of change in price according to the
above formulation of the capital market. This equation of motion, in essence, provides the response
function of the market to earnings management, and thus, the manager, a Stackelberg leader,
formulates his strategy given the response function of the market, a Stackelberg reactor. The
manager's problem is analyzed in Appendix 2.
3. THE RELATIONSHIP BETWEEN INSIDER TRADING, EARNINGS
MANAGEMENT, AND MANAGERIAL EFFORT
Appendix 2 provides the analysis and solution of the manager's program described in
objective functional (3) subject to constraints (4) and (5). Because the problem here takes place in a
multi-period setting, the natural approach is optimal control theory and Pontryagin's Maximum
T, the optimal net number of units traded in a period, is at the point where the net gain from
the marginal unit trade equals the expected penalty on it.
[P(t) - δ ω (.)]= 0
ˆ I (6)
[See (A5) in the Appendix for derivation of (6)] Equation (A11) in Appendix 2 demonstrates that
the optimal level of θ(t) is based on the tradeoff between the manager's marginal cost of effort, μ,
and the marginal penalty to earnings management, πa(t)δωθ(.).
- rt ⎡ μ ⎤
Hθ = e ⎢ - δ ωθ (.)⎥ = 0 → μ = π a (t)δ ωθ (.) (7)
⎣ π a (t) ⎦
It also follows from (7) that the optimal level of a(t) must be positive when μ is positive since δ and
ωθ(.) are positive, thus leading to a positive πa(t), and in turn to a positive a(t). This, however, does
not imply that firm will always have profits, since exogenous factors (denoted as b in this study)
may still cause the firm to lose money.
Next, it is shown that the manager's marginal cost of effort, μ, and the level of earnings management
are positively correlated.
Proposition 1: Along the optimal trajectories of the control variables there is a positive correlation
between the manager’s effort cost, μ, and the level of earnings manipulation, θ(t).
Proof: Follows from equation (A14) in Appendix 2, Panel B.
The intuition behind Proposition 1 stems from (7) which provides the optimal level of θ(t).
According to (7), the optimal level of earnings management is at the point where the marginal cost
of effort for the manager, μ, equals the marginal income of a(t) times the marginal penalty of
earnings management, πa(t)δθ(.). πa(t)δθ(.) slopes upward, thus, as μ increases, it intersects with
πa(t)δωθ(.) at a higher level of θ.
The following Proposition establishes that the manager’s marginal cost of effort and his choice of
managerial effort level are negatively correlated.
Proposition 2: Along the optimal trajectories of control variables there is a negative correlation
between the manager’s cost of effort, μ, and effort, a(t).
Proof: Follows directly from (A17), Appendix 2, Panel B.
The intuition behind Proposition 2 is rather straightforward. As expression (A16) demonstrates, the
optimal level of effort is inversely related to μ because a higher level of managerial effort carries a
higher cost when compared to a situation with a lower μ, thus leading to a lower level of a(t).
Proposition 3: Along the optimal trajectories of control variables there is a negative correlation
between earnings manipulation, θ(t), and the manager’s effort, a(t).
γπ (t)δ ωθθ (.)
Proof: Differentiating a(t) (provided by A.16) with respect to θ(t) yields <0
The intuition behind Proposition 3 is straightforward: managers will compensate for their shirking
on the job by manipulating the accounting numbers.
4. THE INFORMATIONAL ASPECTS OF INSIDER TRADING AND FINANCIAL
This section examines the informational aspects of insider trading and the financial reports.
First, the first-best solution is investigated where market participants can costlessly observe the
costs of effort, μ, of managers then. This will be changed later where we move to a scenario with
unobservable μ's. When market participants can costlessly observe the managers’ cost of effort, μ,
the following Proposition obtains:
Proposition 4: If managers' costs of effort, μ's, are observable then their effort, a(t), and in turn
true income, π(t), can be inferred.
Proof: Equation (A12) in Appendix 2, Panel A, provides the solution for a(t), and thus it can be
reverse-engineered if μ is observable, since all other factors are known.
Such a world, where one can observe how effort-averse managers are, is obviously unrealistic, and
hence we will now relax this assumption. Under this scenario, if the regulatory agency does design
a penalty function in the manner described in section 5 below, the following Proposition holds:
Proposition 5: When the manager’s cost of effort, μ, is observable solely by the manager, and the
slope of the marginal penalty to a unit change in earnings manipulation, ωθθ(.), is sufficiently small,
along the optimal trajectory of control variables there is a threshold point of managerial effort, a(t),
above which the reported income, R(t), increases in the manager’s cost of effort, μ, and below it
the reported income, R(t), decreases in the manager’s cost of effort, μ.
Proof: Relegated to Appendix 2, Panel C.
Proposition 5 is important because it provides a partial separation with respect to reported income,
which is consistent with Guttman et al. (2006), as a result of the kinks in financial reporting despite
the very different setting. This implies that if ωθθ(.) is sufficiently small, then, up to a certain level of
effort, the reported income is positively correlated with μ, i.e., the effect of μ on θ(t) is far greater
than on π(t). After this threshold point the relation of reported income to μ is reversed. This
proposition is useful for the analysis of regulatory aspects of insider trading.
5. THE REGULATORY ASPECTS OF INSIDER TRADING
The establishment of the new regulatory body of the Public Company Accounting Oversight
Board (PCAOB) by Section 105 of the Sarbanes-Oxley Act of 2002 has changed the audit intensity
of firms because auditors now also attest to the internal control over financial reporting. In this
section some of the regulatory aspects of insider trading are analyzed in lieu of the above findings.
In particular, the study addresses the question of how the regulatory agency's choice of its audit
intensity affects insider trading and earnings management. I also attempt to examine here whether a
regulator such as the SEC can come up with a penalty function that will induce a full separation, as
opposed to partial separation.
Let us relate the regulatory body's audit intensity to δ, the probability that the manager's
insider trading will be caught; specifically, any increase in δ is due to the board's decision to
increase its audit intensity.
Proposition 6: Both insider trading, I(t), and earnings manipulation, θ(t), decrease in relation to
the regulatory board's level of inspection.
Proof: Relegated to Appendix 2, Panel C.
The intuition behind Proposition 6 is straightforward. If, for example, the board decides on δ=0,
then both first order conditions for I(t) and for θ(t) (expressions (6) and (7), respectively) will
collapse, thus resulting in both I(t) and θ(t) tending to infinity. If, on the other hand, δ tends to
infinity, these first order conditions will result in both θ(t) and I(t) converging to zero. This result is
consistent with Ewert and Wagenhofer (2005), who found that tightening accounting standards
would result in enhanced earnings quality (but, in turn, could potentially have some significant
Next I examine how the regulatory agency can affect the signaling properties of the financial
statements through its policy on penalties for insider trading.
Proposition 7: If managers’ cost of effort, μ's, are the private information of the managers, the
regulatory body can then set up a such penalty function, ω(.), that will induce a fully revealing
Proof: Relegated to Appendix 2, Panel C.
Proposition 7 stems from the regulatory agency's ability to design such a penalty function with a
sufficiently high ωθθ(.) to cause the threshold point from Proposition 5 to essentially vanish,
therefore yielding an entire range of earnings reports that decrease in μ. This design of the penalty
for insider trading makes μ's effect on θ(t) relatively weaker than its effect on true income, π(t),
through managers' choice of optimal a(t).
Furthermore, if the regulator indeed exercises its capacity to choose ω and the market holds
rational expectations, the market is then bound to find the separation of managers by their μ and
their choice of a(t). Consequently, a naive market with the same earnings response coefficient with
respect to either π(t) or θ(t) is not possible, a finding that may motivate researchers in the earnings
response coefficients area to rethink the way they specify their tests.
6. CONCLUDING REMARKS
This study examines the relationship between insider trading, earnings management, and managerial
effort in a multi-period setting. The findings indicate that, along the optimal path of control
variables, managers' marginal effort cost is correlated positively with earnings management and
negatively with the level of managerial effort. Further, it is shown that the optimal level of
managerial effort is always positive. When examining the informational aspects of reporting by
managers, the results indicate that if managers' marginal effort costs are observable, market traders
can infer the optimal level of effort chosen by the managers and, in turn, the true earnings of the
firm. If, however, managers' effort costs are unobservable, it is then shown that a partial separation
with respect to reported income ensues. Moreover, it is shown that the regulatory agency can, if it
chooses to, design a penalty function yielding a fully separating equilibrium. This last result implies
that if the market holds rational expectations, models using the same earnings response coefficients
for true and managed earnings should be rethought. Another interesting finding is that, along the
optimal trajectory of control variables, both earnings management and insider trading are negatively
correlated with the regulatory agency level of inspection.
Appendix 1 Some Useful Notations
r – Discount rate
C - Manager's cash flow in period t
- Expectation operator
W - Manager's terminal wealth
P - Market price of an equity security
P - Rate of change in P
β - Earnings response coefficient
π - True income of the firm
θ - Earnings management accrual
R - Reported income
F - Forecast of income
k - Fraction of reported income awarded as cash bonus
I - Net number of securities sold (purchased) by the manager
ω - Penalty on insider trading
δ - Probability of insider trading being found out
μ - Marginal cost of managerial effort
a - Managerial effort
x - Joint trajectory of control variables
A - A constant in the production function
γ - Elasticity of effort
b - Exogenous factors in the production function
η - Elasticity of b
α - Total number of equity securities held by the manager
λ - Shadow cost of the state variable
H - Hamiltonian
Appendix 2: Mathematical Addendum
A. Some Properties of the Optimal Solution
Max ∫ e-rt C(t)dt + e-rT W(T) (A1)
subject to the following equation of motion:
P = β [ π (t) + θ (t) - F(t)]
& $ (A2)
The components of (A1) are defined as follows:
C(t) = k(t)[ π (t)+ θ (t)] + P(t)I(t) - ω (. )δ - μa(t)
π (t) = Aa(t )γ b(t )η
$ $ (A3)
W(t) = P(T)α (T)
The above problem is solved using optimal control theory and, in particular, Pontryagin's Maximum
Principle. The Hamiltonian, H, is defined as follows:
[ $ $
H = e-rt k(t)[ π (t)+ θ (t)] + P(t)I(t) - ω (. )δ - μa(t) ]
+ λ (t)β [ π (t) + θ (t) - F(t)]
The Hamiltonian is first differentiated with respect to each of the control variables to find the
optimal trajectories. The following first order condition yields the optimal trajectory of insider
H I = e-rt P(t) - δ ω i (. ) = 0 ] (A5)
Expression (A5) states that the optimal insider trading takes place at the point where the marginal
gain from such transaction, the expected price, is equal to its marginal cost, the incremental
The second order condition, (A6) below, provides us with a maximum
H II = - e-rt δ ω ii (. ) < 0 (A6)
Next, (A4) is differentiated with respect to θ in order to obtain the optimal trajectory of earnings
Hθ = e-rt k(t) + Pθ (t)I(t) - δ ω θ (. ) + λ (t)β = 0
In order to find λ(t) first differentiate (A4) with respect to effort, a:
[k(t)π a (t) + Pa (t)I(t) - μ ]+ λ(t)β π a (t) = 0 (A8)
Isolating λ(t)β from (A8) to substitute in (A7) the following expression obtains:
⎡ Pa (t) ⎤
Hθ = e-rt ⎢ k(t) + Pθ (t)I(t) - δ ω θ (. ) - k(t) -
$ + μ⎥ = 0 (A9)
⎣ π a (t) ⎦
- rt ⎡ μ ⎤
H θ = e ⎢ I(t)[Pθ (t) - Pπ (t)] - δ ωθ (.) +
ˆ ˆ ⎥=0 (A10)
⎣ π a (t) ⎦
This leads to the following first-order condition jointly for the optimal trajectory θ and a:
- rt ⎡ μ ⎤
Hθ = e ⎢ - δ ωθ (.)⎥ = 0 → μ = π a (t)δ ωθ (.) (A11)
⎣ π a (t) ⎦
The equation above states that the optimal level of earnings management is based on the tradeoff
between manager's marginal cost of effort and marginal penalty to earnings management.
The relationship above states that the optimal level of earnings management is based on the tradeoff
between manager's marginal cost of effort and marginal penalty to earnings management.
If there is only one type of manager or if the μ is observable, the optimal effort can be reverse-
engineered using (A11) and the production function in (A3):
γ -1 μ
a(t ) = (A12)
γAb δ ω θ (. )
B. The Relationship Between Manager’s Cost of Effort, μ, And Reported Income
The relationship between reported income and the marginal cost of effort, μ, along optimal
trajectories of the control variables depends on the effect of μ on the 'true income', π, and the
earnings management component, θ. To avoid cumbersome notations let's denote the joint
trajectories of the control variables as x, thus, x=x* relates to the optimal paths of θ, a and I over
dR(t) dπ (t) dθ (t
= + (A13)
dμ x = x * dμ x= x * dμ x= x *
Utilizing the implicit function theorem we can find how μ is related to θ
e − rt
dθ (t ) H θμ π a (t ) 1
=− =− = >0 (A14)
dμ x = x*
H θθ e δωθθ
π a (t )δωθθ
The second derivatives in (A14) above are derived using (A11). Relation (A14) indicates that
there is a positive correlation between μ and earnings management, i.e., the higher the marginal
cost of effort the more she or he are going to manage reported earnings.
The second component in reported income is π. This portion represents the 'true' income of the
firm which is observable only by the manager. In order to find how π relates to μ invoke the chain
rule as follows:
dπ (t) dπ (t) da(t)
dμ x=ñ* da(t) dμ x=ñ*
From the production function in (A3) it follows that πa(t) = gπ(t)/a(t),
consequently, from (A11) the following obtains:
γπ (t)δ ω θ (. )
a(t) x = x * = (A16)
This expression represent a(t) along the optimal joint trajectory of all control variables. In order to
find the relationship of this a(t) and μ we differentiate (A16) with respect to μ,
da(t) γπ (t)δ ω θ (. )
= - < 0 (A17)
dμ x = x * μ2
Rearranging (A16) provides us with π(t) along the optimal path of control variables as follows:
γa(t)δ ω θ (. )
π (t ) x = x * = (A18)
dπ (t) γδ ω θ (. )
a(t) x = x * μ
The relationship between μ and π(t) can be now found using (A17) and (A19) as follows:
dπ (t) dπ (t) da(t) [ γδ ω θ (. )] 2 π (t)
= = - < 0 (A20)
dμ x = x * da(t) dμ x= x * μ3
Expression (A20) indicates that the manager's marginal cost of effort, μ, and the resulting π(t) are
The question is then since μ is correlated positively with earnings management and negatively with
the true income what is the net effect on reported income, R(t). To answer this question we
substitute (A20) and (A14) for their counterparts in (A13). The composite effect is as follows:
dR(t) [ γδ ω θ (. )] π (t)
= - + (A21)
dμ x = x * μ 3
δ ω θθ (. )π a (t)
Finding μ explicitly from (A16), substituting for it in (A21), creating a common denominator and
rearranging yields the following expression:
2⎡ μ ⎤
δ [γ ω θ (. )π (t)] ⎢ - γδ ω θθ (. )⎥
dR(t) ⎣ a(t) ⎦
dμ x = x * μ ω θθ (. )π a (t)a(t)
The above relationship is interesting because it indicates that regardless of the sign of the true
income, π(t), (be it a profit or a loss) its sign is the same as the sign of the following expression:
- γδ ω θθ (A23)
C. Proofs of Some Propositions
Proof of Proposition 5:
The relation of R(t) to μ, along the optimal trajectory of control variables, depends on the sign of
(A23). From (A16) the following obtains:
γπ (t)δ ω θ (. )
μ x = x* = (A24)
Substituting (A24) for μ in (A23) yields,
γ 2 δ ω θ (. ) - γδ ω θθ (A25)
Using the production function from (A3) and rearranging,
[ $ η
γδ Aa(t )γ -2 b ω θ (. ) - ω θθ ] (A26)
Consequently, the correlation between optimal reported income and μ has the same sign as (A26).
Since γ is positive but below one it is obvious that the first expression in brackets declines in a(t),
thus, it will exceed ωθθ(.) up to a certain point of a(t) after which it will fall below it, when ωθθ(.) is
sufficiently small. It thus follows that up to this threshold point of a(t) optimal reported income
increases in m and after it declines in μ.
Proof of Proposition 6:
To prove the Proposition, the implicit function theorem is invoked to find how θ(t) and I(t) are
related to δ. First, the following second cross derivatives are found:
Hθδ = − e − rtωθ (.) < 0 and H Iδ = − e − rtωI (.) < 0 (A27)
Next, the relationship between δ and θ is found:
dθ (t) H
= − θδ < 0 (A28)
dδ x = x * Hθθ
The relationship between δ and I(t) is as follows:
= − Iδ < 0 (A29)
dδ x = x * HII
Proof of Proposition 7:
The proof follows directly from Proposition 5. Since the first expression in brackets in (A26) is
declining as a(t) increases, the regulator only needs to produce such ω(.) whose second derivative
with respect to θ, ωθθ(.), is above the first expression in brackets for a(t)=0. Consequently, such a
penalty will be characterized as follows:
( γ -2) η dR(t)
ω θ (. ) > 0 and ω θθ (. ) > Aa(t ) b ω θ (. ) a(t)=0 ⇒
$ < 0 (A30)
dμ x = x *
Consequently, such a penalty function results in a ranking of reporting income that is decreasing in
μ. It thus follows from Propositions 1 and 2 that along the optimal trajectory of the control
variables the greater is reported income, the larger is π(t) and the smaller is θ(t).
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EXHIBIT 1- TIME LINE
0 1 2 3 4
| | | | |
Setting contract; Operation True income Manager assesses Manager decides
Bonus and shares observed by future streams on the optimal
manager report and on
5 6 7 8
| | | |
Manager sells or buys Manager releases Owner and Manager is awarded
securities his/her report other traders bonus and equity
analyze the report.
Owner either rehires the manager, back
to 0, or terminates the game, end.