THE ACCOUNTING ART OF WAR A MULTI-PERIOD MODEL OF EARNING

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					THE ACCOUNTING ART OF WAR: A MULTI-PERIOD MODEL OF EARNING
            MANIPULATION AND INSIDER TRADING




                             Ramy Elitzur
                 The Rotman School of Management
                        University of Toronto
                         105 St. George Street
                           Canada, M5S-3E6
                        Phone: (416) 978-3827
                         Fax: (416) 971-3048
               E-Mail: elitzu@fmgmt.mgmt.utoronto.ca

                            May 2007
    THE ACCOUNTING ART OF WAR: A MULTI-PERIOD MODEL OF EARNINGS
                 MANIPULATION AND INSIDER TRADING


                                               Abstract

        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.




                                                   1
1.       INTRODUCTION

         “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

[1988], Merchant [1990], and Bruns and Merchant [1990], Elitzur and Yaari [1995], Evans and

Sridhar [1996], Demski [1998], Arya et al. [1998], Fischer and Verrecchia [2000], Liang [2004],

Ewert and Wagenhofer [2005], Goldman and Slezak [2006], Guttman et al. [2006], Ronen and

Yaari [2007], 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

[2002], Bartov and Mohanram [2004], and Cheng and Warefield [2005]), 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 [1988] and Elitzur and Yaari [1995]). To address this concern, a multi-

period dynamic model is used to capture the long-term aspects of the relationship between the


1
 The exception is Ronen et al. [2006], which focuses on insider trading by directors, not managers, and uses a single
period model.


                                                           2
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

6.




                                                  3
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).




                                                   4
                                              -------------------------

                                                      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


                                         &
                                         P(t)
                                                  =
                                                       β
                                                               [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
      &

                               ˆ
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



                                                                  5
                          &
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

P:


                                       &        [               ˆ  ]
                                       P(t) = β π (t) + θ (t) - F(t)
                                                 ˆ                                                 (2)

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)

                                               ˆ
leading to the net cash in-flow (out-flow), or P(t)I(t) . The payoffs for such activities can be quite

substantial:

        "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


                                                    6
          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

issues.

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
        ˆ

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


                                                    7
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):

              T
                    ˆ               ˆ
        Max ∫ e- rt C(t)dt + e- rT W(T)                                                        (3)
              t=1


Subject to the following equation of motion:

        &      [              ˆ  ]
        P = β π (t) + θ (t) - F(t)
               ˆ                                                                               (4)

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


                        ˆ
        π (t) = Aa(t )γ b(t )η
         ˆ                                                                                     (5)


         ˆ
        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,



                                                       8
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

Principle.

        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.

        HI=e
               -rt
                     [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.



                                                        9
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).




                                                     10
                                                                             γπ (t)δ ωθθ (.)
Proof: Differentiating a(t) (provided by A.16) with respect to θ(t) yields                   <0
                                                                                    μ

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

       REPORTING

       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



                                                  11
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.



                                                     12
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

disadvantages).

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

equilibrium.



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),


                                                    13
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.




                                                   14
                                 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



                                                     15
α - Total number of equity securities held by the manager

λ - Shadow cost of the state variable

H - Hamiltonian




                                                16
                                      Appendix 2: Mathematical Addendum

       A. Some Properties of the Optimal Solution

                            T
                            $              $
                 Max ∫ e-rt C(t)dt + e-rT W(T)                                             (A1)
                            t=1


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) ]
                                                                                             (A4)
                                                           $
                                + 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

trading:


                      $ [
           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

expected penalty.



                                                           17
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

management,


                           [                               ]
           Hθ = e-rt k(t) + Pθ (t)I(t) - δ ω θ (. ) + λ (t)β = 0
                            $                                                                    (A7)

In order to find λ(t) first differentiate (A4) with respect to effort, a:


 Ha=e
         -rt
               [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)    ⎦

Rearranging,


                                         - 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-




                                                               18
engineered using (A11) and the production function in (A3):


                                                           γ -1       μ
                                                      a(t ) =                                        (A12)
                                                                  γAb δ ω θ (. )
                                                                      η




        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

time,

        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
                                               − rt
                                                         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:




                                                                 19
         dπ (t)     dπ (t) da(t)
                  =                                                                             (A15)
          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)
                                                                μ


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)
                                                                  μ


Consequently,


                                         dπ (t)          γδ ω θ (. )
                                                       =                                       (A19)
                                          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



                                                     20
negatively correlated.



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)
                                                                          2
                                                                                         1
                                                      = -                      +                       (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)               ⎦
                                                        =                                              (A22)
                                           dμ x = x *               μ ω θθ (. )π a (t)a(t)
                                                                       3




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:


                                                    μ
                                                          - γδ ω θθ                                    (A23)
                                                   a(t)




                                                    21
        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)
                                                                           a(t)


Substituting (A24) for μ in (A23) yields,


                                                          π (t)
                                                      γ        2   δ ω θ (. ) - γδ ω θθ            (A25)
                                                          a(t )


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:



                                                          22
  dθ (t)         H
              = − θδ < 0                                                                      (A28)
   dδ x = x *    Hθθ

The relationship between δ and I(t) is as follows:

  dI(t)          H
              = − 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).




                                                   23
                                  References


        Allen, S., and Ramanan, R. 1990. “Earnings Surprises and Prior Insider Trading:
Test of Joint Informativeness," Contemporary Accounting Research 6 (2-I): 518-543.

       Arya, A., Glover, J., and Sunder, S. 1998. “Earnings Management and the
Revelation Principle.” Review of Accounting Studies 3, 1-2 (March): 7-34.

       Bartov, E., and Mohanram, P. 2004. “Private Information, Earnings Manipulations,
and Executive Stock-Option Exercises.” The Accounting Review 79 (4): 889-920.

       Beneish, M.D, and Vargus, M.E. 2002. Insider Trading, Earnings Quality, and
Accrual Mispricing,” The Accounting Review 77(4): 755-, 38 pgs.

       Bruns, W.J. and Merchant, K.A. 1990. The Dangerous Morality of Managing
Earnings." Management Accounting (August): 22-25.

       Cheng Q., and Warefield, T.D. 2005. Equity Incentives and Earnings Management.
The Accounting Review 80 (2): 441-476.

       Collins, D.W., Kothari, S.P. and Rayborn, J.D. 1987. Firm Size and the
Information Content of Prices With Respect to Earnings," Journal of Accounting and
Economics (July): 111-138.

                 , and       . 1989. “An Analysis of Intertemporal and Cross-Sectional
Determinants of Earnings Response Coefficients." Journal of Accounting and Economics
(July): 143-181.

      Demski, Joel S. 1998. “Performance Measure Manipulation.” Contemporary
Accounting Research 15, 3 (Fall): 261-285.

       Dye, R.A. 1988. “Earnings Management in an Overlapping Generations Model.”
Journal of Accounting Research 26 (2): 195-235

      Elitzur, R. and Yaary, V. 1995. “Executive Incentive Compensation and Earnings
Manipulation in a Multi-Period Setting.” The Journal Of Economic Behavior and
Organization 26: 201-219.

      Evans, J. H. and Sridhar, S. S. 1996. “Multiple Control Systems, Accrual
Accounting, and Earnings Management.” Journal of Accounting Research 34 (1): 45-65.

       Ewert, R., Wagenhofer, A. 2005. “Economic Effects of Tightening Accounting
Standards to Restrict Earnings Management.” The Accounting Review 80 (4): 1101-1124.




                                      24
      Fischer, Paul E. and Robert E. Verrecchia.       2000.    “Reporting Bias.” The
Accounting Review 75 (April): 229-245.

        Goldman, E., and Slezak,. S. L. 2006. “An Equilibrium Model of Incentive
Contracts in the Presence of Information Manipulation.” Journal of Financial Economics
80, 3 (June): 603-626.

       Guttman, I., Kadan, O., and Kandel. E. 2006. “A Rational Expectations Theory of
Kinks in Financial Reporting.” Accounting Review 81 (4): 811-38.

       Liang, P. J. 2004. “Equilibrium Earnings Management, Incentive Contracts, and
Accounting Standards.” Contemporary Accounting Research 21 (3): 685-717.

      Merchant, K. A. 1990. “The Effects of Financial Controls on Data Manipulation
and Management Myopia.” Accounting, Organizations and Society 15 (4): 297-314.

      Ronen, J., Ronen, T. and Yaari. V. 2003. “The Effect of Voluntary Disclosure and
Preemptive Preannouncements on Earnings Response Coefficients (ERC) When Firms
Manage Earnings.” Journal of Accounting, Auditing & Finance.18 (3): 379-409.

        Ronen, J., Tzur, J., and Yaari, V. 2006. “The Effect of Directors’ Equity
Incentives on Earnings Management.” Journal of Accounting and Public Policy 25 (4):
359-389.

       Ronen, J. and Yaari, V. 2007. “Demand for the Truth in Principal-Agent
Relationships.” Review of Accounting Studies 12 (1): 125-153.

       Sloan, R. 1996. “Do Stock Prices Fully Reflect Information in Accruals and Cash
Flows About Future Earnings?” The Accounting Review 71 (3): 289-316.

      Xin., B. 2007. “Earnings Forecast, Earnings Management, and Asymmetric Price
Response.” Working paper, University of Minnesota, January 2007.

       Yagyū, M. 1974. The Book of Family Traditions on the Art of War. In The Book of
Five Rings by Musashi, Miyamoto, translated by Victor Harris. Woodstock, NY: Overlook
Press.




                                      25
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
                                                                          trading
                                                                          securities




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.




9
_________________________________________
|

Owner either rehires the manager, back
to 0, or terminates the game, end.


                                             26