accounting measurement basis market mispricing and firm investment efficiency 2007

8/3/2019 Accounting Measurement Basis Market Mispricing And Firm Investment Efficiency 2007

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DOI: 10.1111/j.1475-679X.2007.00227.x Journal of Accounting Research

Vol. 45 No. 1 March 2007Printed in U.S.A.

Accounting Measurement Basis,Market Mispricing, and Firm

Investment Efciency

P I E R R E J I N G H O N G L I A N G A N D X I A O YA N W E N

Received 15 June 2005; accepted 29 April 2006

ABSTRACT

In this paper, we investigate how the accounting measurement basis affects

the capital market pricing of a rms shares, which, in turn, affects the ef-ciency of the rms investment decisions. We distinguish two broad bases foraccounting measurements: input-based and output-based accounting. We ar-gue that the structural difference in the two measurement bases leads to asystematic difference in the efciency of the investment decisions. In partic-ular, we show that an output-based measure has a natural advantage in align-ing investment incentives because of its comprehensiveness. The (rst-)best investment is achieved when the output-based measure is noiseless and ma-nipulation free. In addition, under an output-based measure, more account-ing noise/manipulation always leads to more inefcient investment choices.Therefore, if an output-based accounting measure is highly noisy and easy to

manipulate in practice, the induced investment efciency can be quite low.On the other hand, an input-based accounting measure, while not as compre-hensive, may induce more efcient investment decisions than an output-basedmeasure if some noise is unavoidable in either measure. The reason is twofold.First, input-based measures may be associated with less noise and limited ma-nipulation in practice. Second, and more importantly, we show that under

Carnegie Mellon University. The authors gratefully acknowledge the suggestions from ananonymous referee, Joel Demski, Jon Glover, Burton Hollield, Volker Laux, Wei Li, Tingjun(TJ) Liu, and members of an informal reading group at the Tepper School of Business,Carnegie Mellon University. Parts of the study were conducted while Pierre Jinghong Liang

was granted the BP America Research Chair (20032004) and Xiaoyan Wen was granted the William Larimer Mellon fellowship.

155

Copyright C , University of Chicago on behalf of the Institute of Professional Accounting, 2007


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156 P. J. LIANG AND X. WEN

an input-based measure, a slight increase in accounting noise/manipulationmay lead to more efcient investment choices. In fact, the (rst-)best result isachieved when the noise/manipulability is small but positive. In other words,for an input-based measure, being less comprehensive makes small but pos-itive accounting noise/manipulability desirable. Two extensions of the basicmodel are also explored.

1. Introduction

In this paper, we investigate how the accounting measurement basis af-fects the capital market pricing of a rms shares, which, in turn, affectsthe efciency of the rms investment decisions. We distinguish two broadbases for accounting measurements. The rst, an output-based accountingmeasurement, is designed to measure a rms activities by recording theestimated nancial benets of production. In contrast, the second basis, aninput-based accounting measurement, is designed to measure a rms ac-tivities by recording the estimated factor costs of production. We argue that the two bases give rise to different informational properties of accountingnumbers. In turn, the different properties induce different capital market pricing, which leads to a systematic difference in the efciency of the invest-ment decisions. In particular, we show that an output-based measure has anatural advantage in aligning rm investment incentives. That is, becausean output-based measure provides comprehensive information about boththe scale and protability of the investment, it reduces the negative impact of mispricing on investment efciency through a dampening effect. How-ever, output-based measures suffer a potential disadvantage in that they may inherit more measurement noise and may be subjected to more managerialmanipulation. We show that an output-based measure performs (rst-)best when it is noiseless and manipulation free. However, in cases where they arehighly noisy and easy to manipulate, the investment efciency can be quitelow as a result.

On the other hand, if some noise is unavoidable in any measure, aninput-based measure may induce more efcient investment decisions thanoutput-based measures, despite having a disadvantage of only providinginformation about the scale of the rm investment. The reason is twofold.

First, input-based measures are typically associated with less noise and lim-ited manipulation. Second, and more importantly, we show that with aninput-based measure, some low levels of noise and manipulability are tol-erated or even preferred in some cases. In fact, the (rst-) best result isachieved when the measurement noise is small but positive. In other words,for an input-based measure, being less comprehensive makes small but pos-itive measurement noise desirable.

The debate on measurement bases has a long and varied standing in ac-counting history. Paton and Littleton [1940] describe accounting numbersas price aggregates that measure the economic activities of a rm. The de-

bate on the basis of price aggregates centers on which price to use insuch measurements. Extensive subsequent writings describe the rationales


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ACCOUNTING MEASUREMENT BASIS 157

for using entry prices, exit prices, past and/or future market prices, andsimulated prices (see Edwards and Bell [1961], Sterling [1970], and Ijiri[1975]). We frame thedebate withina partial equilibrium investment setting with rational expectations and evaluate the different measurement baseson the scale of induced investment efciency. In other words, we adopt an information-economic paradigm and evaluate the alternative measure-mentsaccordingto their information content. (See Christensen andDemski[2002] for more on such a paradigm.)

The insights in this paper may be helpful in current discussions of fair value measurements. There has been considerable movement in public ac-counting policy toward measurements based on fair value. The move ispartially motivated by the desire to increase the relevance of accounting,arguably with some sacrice in reliability. The emerging concern is the in-troduction of considerably more estimates that are to be used in preparingfair value accounting measures. The Financial Accounting Standards Board(FASB) [2004] has recognized the reliability issue and has prescribed differ-ent levels of estimates (Level 1, 2, 3, etc.) accordingly. In this paper, we point to theeffects of noise and manipulation on investment efciency. In particu-lar, we show that using output-basedmeasuresmakes noise or manipulability a social bad to the economy because either of the two induces inefcient investments. In other words, one must always be careful about its negativereal effect on rm investments if the measure brings in an increased levelof noise and manipulability. This is because under output-based measures,moreaccounting noise/manipulationalways leads to moreinefcient invest-ment choices. On the other hand, with input-based accounting, the (rst-)best result is achieved when the measurement noise is small but positive. As a result, our analysis points to a certain attractiveness of an input-basedmeasurement basis when some measurement noise is unavoidable for ei-ther measure. Further, a slight increase in accounting noise/manipulationmay lead to more efcient investment choices. Generally, our results imply that the rational economic choice between an output-based measure andan input-based measure is not obvious in most cases where varying levels of noise and manipulability exist in both measures.

Specically, we consider a simple two-period model in which a rms in-

vestment decision is jointly affected by the total return of the investment and by the short-term capital market pricing of its ownership shares. Condi-tional on private information about the protability of the investment, therm makes an investment decision that generates cash ows in both the rst and second periods. The rms objective is to maximize a weighted averageof the life-time cash ows and the short-term share price. The share pricereects the rational expectation of rm value based on a public report of the rst-period aggregate cash ow (i.e., the sum of rst-period investment return and the cash ow from the rst-period ongoing activities). The mar-kets inability to identify the sources of the rst-period cash ow may induce

a systematic short-term mispricing of rm value from the rms perspective.In turn, this mispricing induces a suboptimal investment decision, ex ante.


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We show that the deviation from the rst-best investment level may be pos-itive (i.e., overinvestment) or negative (i.e., underinvestment) depending,in part, on the timing of the investment return.

Within this framework, accounting is introduced as an information systemthat provides (price-aggregate) measures about the investment. In pricingthe rms shares, the capital market uses the information in the accountingmeasure as a supplement to the information contained in the aggregatecash ow. We consider two accounting measurement bases. First, an output-based accounting measure is modeled as an unbiased estimate of the invest-ment return (which is a function of the actual investment made and the trueinvestment protability). Second, and alternatively, an input-based account-ing measure is modeled as an unbiased estimate of the actual investment made. Either accounting signal provides additional information, which may help the capital market improve the pricing of the rms shares. The im-proved pricing induces better rm investment decisions. As the quality of accounting measures (which is inversely related to the measurement noise)improves, one would expect less mispricing, which would lead to better in- vestmentdecisions. We show that this conjecture is, indeed, true for both theinput- and output-based measures when the noise in these measures is high.That is, when accounting measures are highly imprecise, any improvement in precision will lead to more efcient investment decisions.

When themeasurement noise is low, however, the two measurement basesare fundamentally different. For the output-based measure, investment ef-ciency continues to increase in measurement quality. At the limit, when theoutput-based accounting measure perfectly reveals the investment return,rst-best investment choices are made. However, for the input-based mea-sure with low measurement noise, the investment efciency is an invertedU-shaped function of the measurement noise. As measurement noise de-creases, the investment efciency rst increases up to a threshold point anddecreases afterwards. The rst-best investment choices are made when thenoise is small but not zero (at the threshold point).

Thereasonfor thedifference, andfor thesurprising result, is thedifferent mispricing structures that are generated by the two measurement bases. Forthe output-based measure, both the actual investment made and the actual

investment protability affect the accounting measure. As the rm deviatesfrom therst-best investment level, theeffecton its shareprice is dampened,at themargin, by themeasures built-in protabilityestimate. In other words,the rms ability to use real investment to change the market perceptionof its investment protability is mitigated by the independent protability estimate built into the output-based measure. In turn, the dampening ef-fect lessens the ex ante incentives to distort the investment decision. Theinvestment efciency increases monotonically as themeasurement noise de-creases. For the input-based measure, theaccounting measure reliesonly onthe actual investment made and does not directly rely on a separate estimate

of the investment protability. The dampening effect does not apply to the(mis)pricing of the accounting report; it only applies to the (mis)pricing of


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a portion of the cash ow measure. As the measurement noise decreases,the dampening effect is diminished as more valuation weight is shifted fromthe cash ow to the accounting report. This is the reason that a noiselessinput-based measure invites severe deviation from the rst-best investment.

Finally, we offer two extensions of the model. First, we add accountingmanipulation to the mix. Following Dye and Sridhar [2004a], we model ac-counting manipulation by giving the rm an option to privately modify ac-counting measurements before they are reported to the capital market. Thetotal cost of manipulation contains a random element. As a result, account-ing manipulation introduces additional noise into the reported accountingmeasures. Based on the existing economic forces, it is shown that account-ing manipulation always makes the rm worse off under an output-basedaccounting regime because adding more noise to an output-based account-ing measure always leads to a less efcient investment decision. However,accounting manipulation may make the rm better off under an input-based accounting regime. This is because there are regions in the invertedU-shapedrelation where a slightincrease in measurement noisewill improvethe investment efciency. This is most likely when theexisting measurement noise is small, and the noise introduced by accounting manipulation to theinput-based measure is also small.

In the second extension, we modify the model to allow an option forthe date-2 shareholders to utilize the technology in the rm to generatefuture cash ows. In this modied model, the market price is reective of two streams of cash ows: those generated by the initial investment andthose generated by future investments. As before, the mispricing affects theinitial investment choice and the relative performance of the input- versusoutput-based accounting measures. However, it is no longer true that theoutput-based measure performs best when noiseless. Comparing account-ing measurement bases is further complicated and requires knowing moredetails of the context including, in particular, the importance of future cashow relative to current cash ow.

The antecedent of studies on investment myopia in nance and eco-nomics are Holmstr om [1982] and Stein [1989]. Stein [1989] nds that the managers, facing stock market pressure, forsake good investments so

as to boost current earnings, even though the market is efcient and is not fooled in equilibrium. In spite of being unable to fool the market, managersare trapped into behaving myopically in the classic signal-jamming model.Dye and Sridhar [2004a] examine how investment decisions are affected by the reliable and relevant components of an aggregate accounting report.Their study focuses on the reliability-relevance trade-offs in accounting ag-gregation, the conditions under which aggregation improves efciency, andthe optimal weights in constructing an optimal accounting report. WhileDye and Sridhar [2004a] and our study share the feature of accounting ma-nipulation, our focus is on the comparison of two alternative accounting

measurements, each of which is disaggregated from a common cash owmeasure. In addition, Demski, Lin, and Sappington [2005] analyze a setting


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in which entrepreneurs invest before they learn whether they will be forcedto sell their assets. They study the optimal design of asset impairment regu-lations when the assets resale market suffers from the lemon problem.

Using a contracting setting, Prendergast [2002] examines input-basedand output-based measures and argues that input-based monitoring, cou-pled with a directed action, performs best in stable settings, while output-based monitoring is best in uncertain environments. The reason is that output-based contracts, coupled with a delegation of decisions, align so-cial and private incentives better in uncertain situations. In contrast, ourstudy focuses on the market incentives produced by input- and output-basedmeasures and nds that the output-based measure performs best when themeasurement is precise. Among other accounting studies on investment efciency, Kanodia, Singh, and Spero [2005] analyze the economic conse-quences of the interaction between noisy accounting measuresand informa-tion asymmetry regarding the investment protability. They nd that somedegree of accounting imprecision can be value-enhancing, which is consis-tent with our results on the input-based measure. In contrast, we focus onthe comparison of input- versus output-based measures and show that therole of accounting imprecision depends on the accounting basis: under anoutput-based measure, noiseless accounting is optimal.

The rest of the paper proceeds as follows. Section 2 describes the modeldetails. Section 3 presents the benchmark results of the basic setup whereonly an aggregate cash ow is reported. In section 4, we introduce account-ing measures and analyze the equilibria induced by accounting. Section 5introduces an extension that models accounting manipulation. Section 6offers another extension where the rm technology is reusable in the fu-ture. Section 7 concludes the paper.

2. The Model

Consider an economy with a risk-neutral rm in a competitive risk-neutralcapital market. There are two periods (and three dates), representing short-and long-term concerns. The rms normal ongoing activities generate apair of cash ows, denoted x t , realized on date- t ( t

=1, 2). We assume:

x 1x 2

N

, 2 2

2 2. (1)

On date-0, the rm is faced with an investment opportunity (called aproject) and observes a private signal, denoted

R , about the projectsprotability. Theprior distribution of is normal with mean 0 and variance 2 (i.e., N [ 0, 2 ]). The rm chooses an investment level, denotedI

R +, to invest into the project. The project generates cash ows on date-1 and date-2, denoted by f 1( , I ) and f 2( , I ), respectively. For tractability,

we assume the project returns f 1(, I ) =k I and f 2(, I ) = (2 k ) I , where k (0, 2). Thus, the total investment return (2 I ) depends only


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on the investment level I and the protability variable . The timing of theinvestment return depends on k : Higher k indicates that more investment return is realized in the short-term. 1

The realized periodic cash ows to the rm are denoted by z t ( t

=1, 2): 2

z 1 = x 1 + f 1(, I ) = x 1 +k I . (2)z 2 = x 2 + f 2(, I ) = x 2 +(2 k ) I . (3)

Following the literature (Dye [2002], Dye and Sridhar [2004a,b]), weassume that the investment is made privately (i.e., not directly observable)and that the rm is unable to directly communicate the private information( ) to outsiders, including the capital market. We also assume that isindependent of x 1 or x 2 and the ongoing cash ows ( x 1 or x 2) are not affected by the investment choice ( I ).

At the end of date-1, the rms shares are priced in a competitive risk-neutral capital market such that the market price, denoted P , is equal tothe expected value of the cumulative cash ow on date-2. Denote the pub-licly available information set on date-1 by (assuming no discounting ordividend payments):

P = E [z 1 +z 2 | ] . (4)The information set includes public information available for pricing.

In the basic setup, the rm publicly reports its aggregate cash ow only,so contains z 1 alone. Later, we consider additional signals created by an

accounting information system; so may include additional items such asa deferral or an accrual.The rm is motivated by both the long-term interest (i.e., cumulative

cash ows on date-2) and the short-term interest (i.e., date-1 stock price). We assume for life cycle or liquidity reasons, a portion of the rm, denoted(0, 1), must be sold on date-1. The remaining (1 ) portion willbe held by the date-0 shareholders. 3 As a result, the rms objective is to

maximize a weighted average of date-1 market price and date-2 cumulativecash ow (net of the investment costs I ). For a type- rm with investment function I ( ), the objective function is

I ( ) + P +(1 )( z 1 +z 2) . (5)1 All qualitative results remainif we assumethat f t (.) are the expected valuesof the short- and

long-term investment returns and that the noise in the returns is independent of the existingrandom variables.

2 Here we assume that z 1 is gross of the investment cost ( I ), which is appropriate given that we assume the investment is made privately. If z 1 is net of the investment cost, the quantitativeresults would change while the same qualitative forces would remain. See Dye [2002] and Dyeand Sridhar [2004a] for similar assumptions.

3 Another reason may be that the rm faces a probability of takeover on date-1, in whichcase the rm wishes to maximize its date-1 share price (see Stein [1989] for additional discus-

sions). In the nance literature, incentive to underinvest may be driven by a debt-overhangproblem (see Myers [1975]).


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t = 0 t = 1 t = 2

Firm privately Cash ows x 1 and f 1 ( , I ) Cash ows x 2 and f 2 ( , I )observes and are realized are realizedchooses I ( ) Firm releases reports Firm is liquidated

Market prices P based on

FIG. 1.Time line of events.

The sequence of the events is summarized in Figure 1. As a reference point for what follows, we provide a description of the

rst-best investment policy in Lemma 1 (proof omitted).

LEMMA 1 ( rst-best ). The socially optimal investment policy consists of a function I FB ( ), where for each , I FB ( )maximizes f 1(, I ) + f 2(, I ) I =2 I I .So we have I FB = when > 0; and I FB =0 when < 0.

We now dene the equilibrium where the investment policy is made inthe self-interest of the rm.

DEFINITION 1. An equilibrium relative to consists of an investment function I () and a market pricing function P (), such that:

(i) Given P (

), optimal investment function I (

) maximizes V (

|I (

))

= E x 1x 2 [I + P () +(1 )( z 1 +z 2)];(ii) Given I (), the pricing function P () satises P = E [z 1 +z 2 | , I ()]. We employ an approximation assumption to calculate the pricing func-

tion in closed-form.

Approximation Assumption.Let random variables x and y be jointly nor-mally distributed, and z is normally distributed and independent of x and y . Denote by f ( x , z | y + az ) the jointly conditional density function of x and z for some known constant a and denote G (z

| y

+az ) the conditional

cumulative density function of z . We assume for all realizations of y +az ,

z > 0 x ( x +z ) f ( x , z | y +az ) dx dz +G (0 | y +az ) x x f ( x | y ) dx = z x ( x +z ) f ( x , z | y +az ) dx dz . (6)

In words, the approximation assumes that the error due to censoring thelower tail of a normally distributed randomvariable is smallwhen calculatingthe conditional mean of the sum of the censored variable and another

normal random variable. In our context, the investment strategy functionI ( ) censors the underlying protability parameter by the fact I = 0 if


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< 0. The marketobserves some linearfunction of I ( ), not theuncensored . The assumption allowsus to calculate themarket inference in closed-formby assuming the censored is close enough to the uncensored . Becausethe censoring point is always held at zero, it is clear that this approximationbecomes increasingly accurate as E [z ] is large, that is, the probability massto the left of zero is increasingly small. 4 , 5

3. Basic Setup

Except in section 6, we assume throughout that on date-1 the realizedaggregate cash ow ( z 1) is always publicly observable. However, outsidersare not able to differentiate the cash ow components. In other words,cash ows generated from ongoing activities and from the investment are

aggregated. This aggregation feature plays an important role in this setting. We now analyze the equilibrium behavior of the rm in the basic setup.

THEOREM 1. Using the approximation assumption (6), there exists a unique linear equilibrium relative to ={z 1}. It is given by

(i) an equilibrium linear pricing function:

P (z 1) =a +b z 1, where (7)

b =(1 + ) 2 +2k 2

2

+k 2 2

, (8)

a = (2 b ) +(2 kb ) 0. (9)(ii) an equilibrium investment function:

I ( ) = , if 00 if < 0,

(10)

where = 1 +bk 2 1

2

. (11)

4 Note this approximation assumption pertains to calculating the conditional expectation(i.e., expected x + z conditional on the realization of y + az ). We acknowledge that this isa global requirement. That is, the approximation applies for each and every realization of y +az . For extremely negative realizations of the censored variable z , the approximation errormay be large. However, the approximation error for a particular realization of y +az , denotedAE ( y +az ), becomes smaller and approaches zero as the mean of z increases. In the appendix, we formally prove this claim.

5 Dyeand Sridhar [2004b] usean approximation assumption intheir analysis. While their ap-proximation applies to calculating the unconditional mean of an altered normally distributed

randomvariable, ours applies toa seriesofconditionalmeansofanaltered normally distributedrandom variable. Stocken and Verrecchia [2004] also use a similar approximation.


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Proof of Theorem 1. All proofs are placed in the appendix.

Compared with the rst-best investment, a key observation in Theorem 1is that the equilibrium investment choice ( I ) is generally a function of theshort-term market pressure (parameter ) and the timing of the investment cash ows (parameter k ), not just a function of the investment protability (variable ). This is because in pricing the rms shares, the market is unableto distinguish the individual components of the short-term cash ow ( z 1).To illustrate, suppose that the market is able to distinguish the cash owcomponents; then the appropriate response to every dollar of the ongoingcash ows ( x 1) would be (1 + ) and the appropriate response to the short-term investment return ( f 1(, I ) =k I ) would be 2k . However, becauseonly the aggregate ( z 1 = x 1 +k I ) is reported, the market response isthe weighted average of (1

+ ) and 2k . Therefore, the market may not

price the investment efciently. Because the rm cares about the short-termshare price ( > 0), the efciency of the pricing affects the investment incentives.

To understand thenature of themispricing, wereturnto thermobjectivefunction (5), which may be rewritten as

I + P +(1 )( z 1 +z 2)= I +z 1 +z 2 + ( P (z 1 +z 2))

= I +x 1 +x 2 +2 I + ( P ( x 1 +x 2 +2 I )) . (12)The rst term of (12) is the rst-best objective function. The second term

of (12), when > 0, is an increasing function of the mispricing (equal toprice minus the long-term rm value). When making the investment deci-sion ( I ), the rm doesnot know x t but knows . So, theexpected mispricing,given , is

E x 1x 2 [ P ( x 1 +x 2 +2 I ) | ] .Substituting the linear pricing function P = a +b ( x 1 +k I ) into theexpected mispricing, we have

E x 1 x 2 [ P ( x 1 +x 2 +2 I ) | ]= E x 1 x 2 [a +b ( x 1 +k I ) ( x 1 +x 2 +2 I ) | ]= (bk 2) I + E x 1x 2 [a +bx 1 ( x 1 +x 2)] . (13)

Given that the second term of (13) is not a function of I , the mispricing will only affect the equilibrium investment through the rst term of (13): 6(bk 2) I . So we have

6 Notice that the second term of (13), while not affected by investment ( I ), is still a mispric-ing caused by reporting the aggregate cash ow. This mispricing would induce (operating)inefciencies if the rm is able to control the timing of the ongoing cash ows ( x 1 and x 2).See Stein [1989].


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V ( | I ()) = E x 1x 2 [( I +x 1 +x 2 +2 I ) + ( P ( x 1 +x 2 +2 I )) | ]

= I + E x 1 x 2 [ x 1 +x 2] +2 I + (bk 2) I

+ E x 1x 2 [a +b x 1 ( x 1 +x 2)]= I +2 I + (bk 2) I +(2 + (b 2)) + a .

When the rm either does not care about the short-term share price( = 0) or the market correctly prices rm investments ( b = 2k ), the rmmakes the rst-best investment choices. If neither condition is met, the(mis)pricing is affected by the rms investment choice, so the rm has anincentive to deviate from therst-best level. In fact, thermfaces conictingincentives when making its investment decision:

r Incentive to underinvest.Given k =2, some investment returns are real-ized in the long-run. Given 0 < < 1, the rm receives only a fraction(i.e., (1 ) share) of the long-run marginal benet but must bear thefull marginal cost. This leads to underinvestmentbecause themarginalreturn is positive but decreasing in I . (See similar economic forces inDye [2002], Dye and Sridhar [2004a,b].)

r Incentive to overinvest.Given k =0, some investment returns are realizedin the short-run.Given 0 < < 1,thermhasanincentivetoinatetherst period cash ow ( z 1 = x 1 +k I ) because the pricing functionplaces a positive weight on z 1. In our model, the only way to inate z 1is to increase investment level ( I ). This leads to overinvestment as theshort-run marginal benet from the investment may be inated. (Seea similar tension in Stein [1989].)

We summarize the investment results with the following theorem.

THEOREM 2. In the basic setup = {z 1}, there exists a value for k, namely,k = 21 + , such that the linear equilibrium produces the rst-best investment level =1. If k > k , then > 1 or the rm overinvests; if k < k , then < 1, or the rm underinvests .

Intuitively, when k = k , the appropriate response to the short-term investment return ( 2k ) happens to be the same as the appropriate responseto the ongoing cash ow (1 + ). That is, the proportion of the short-terminvestment return perfectly matches the time-series correlation of cash owsfrom ongoing activities. Thus, even with an aggregated cash ow report, themarket price motivates the efcient investment decision.

If k = k , the market reaction to the aggregate cash ow is an aver-age of 2k and (1 + ). When k > k , the average is higher than 2k , whichplaces too high of a weight on the short-term investment return, leadingto overinvestment. When k < k , the average is lower, leading to underin-

vestment. In other words, the aggregation of information leads to market mispricing of the rm investment, which leads to suboptimal investment decisions.


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Using this basic model, we can show that (1) both the market responsecoefcient b and the investment coefcient increase as increases; (2)if k < k , both b and increase as the ongoing cash ows are less noisy (i.e., 2 decreases); and (3) as the short-term pressure increases, market response b decreases and rm investment coefcient decreases if k < k .(See Proposition 1 in the appendix for details.)

4. Accounting

Now we expand the information set that is available to the market. We areparticularly interested in howinformation containedin thenoncash compo-nent of the nancial statements affects the market pricing and the inducedinvestment incentives. We model this by introducing a public signal y , whichis produced by the accounting information system of the rm. We assumethat on date-1 both the realized cash ows ( z 1) and the accounting signal ( y )are publicly observable. The general idea is that y may provide additionalinformation beyond an aggregate cash ow report ( z 1) and in particular, y may help differentiate the individual components of cash ows. 7

4.1 INPUT -BASED AND OUTPUT-BASED ACCOUNTING MEASURESIn accounting practice, two broad accounting measurement bases domi-

nate how accounting deferrals/accruals are prepared. First, under an input- based measurement basis, accounting metrics are prepared to be estimatesof the effort (or costs) expended in various rm activities. The historical(exchange) cost principle reects this approach well. Ready examples arelong-lived assets and inventory, where book values are based on acquisitioncosts. Second, under an output-based measurement basis, accounting metricsare prepared to be estimates of the expected reward in return (for the costly activities). The fair value principle reects this approach well. Ready exam-ples are market value methods where assets and liabilities are measured at market value or based on an estimate of the expected negative predictive value (NPV) of future cash ows (which is designed to simulate a would-bemarket value). We study both accounting systems and examine the effect of alternative accounting reports on the investment efciency. We assumethat the rm can choose either an input-based approach, labeled IP, or anoutput-based approach, labeled OP. 8

7 This idea is consistent with some recognizable features of certain timing accruals. Forexample, the unearned revenue accruals help classify the timing properties of cash inow, andthe extraordinary-item category may help distinguish components of realized cash ows withdifferent serial correlations.

8 Here wehave limited ourattention toa single, one-timemeasurement.In practice,account-ingmeasurement systems canbe much more complex with an initial measurement,subsequent (date-2) re-valuation, and a nal measurement on the disposal of the item in question. A dy-

namic model with multiple information arrivals would make it possible to model these issues,and it is certainly an interesting extension of the current model.


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Denote the signal produced by an output-based measure y OP , and assumethat

y OP

=k I

+ OP , (14)

where OP N (0, 2OP ). The output-basedaccounting reportprovidesa noisy

measurement of the short-term investment return. 9For the input-based measure, denote the report y IP , and assume that

y IP = I + IP , (15) where IP N (0,

2IP ). The input-based accounting report provides a noisy

measurement of the investment cost. In the following, to simplify the nota-tion, we denote the accounting policy by m , m {OP , I P }.

4.2 EQUILIBRIA UNDER ALTERNATIVE ACCOUNTING REGIMES We now analyze the equilibrium behavior of the rm under both output-

based accounting and input-based accounting.

THEOREM 3. If y OP =k I + OP and y IP = I + IP , using (6), there exists a unique linear equilibrium relative to = {z 1, y m } (m {OP , I P }) and it is given by (i) an equilibrium linear pricing function:

P z 1, y m = a m +b m z z 1 +b m y y m , where (16)

b OP z =(1 + )k 2OP 2 2 +(1 + ) 2 2OP +2k OP 2OP 2

k 2OP 2 2 + 2 2OP +k 2OP 2OP 2, (17)

b OP y =[2 (1 + )k ]k OP 2 2

k 2OP 2 2 + 2 2OP +k 2OP 2OP 2,

a OP = 2 b OP z + 2 kb OP z kb OP y OP 0,

(18)

b IP z =

(1 + )2IP 2 2 +(1 + ) 2 2IP +2k IP 2IP 22IP 2 2 + 2 2IP +k 2IP 2IP 2

, (19)

b IP y =[2 (1 + )k ]

32IP

2 22IP

2 2 + 2 2IP +k 2IP 2IP 2,

a IP = 2 b IP z + 2 kb IP z IP b IP y IP 0.

(20)

9 For simplicity, we choose the short-term return as the expected value of an output-based

accounting report. Our results do not change if we assume that the report is scaled up toprovide a noisy measurement of the total investment return (i.e., y OP =2 I +OP ).


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(ii) an equilibrium investment function:

I m ( ) =m , if 0

0 if < 0, where (21)

OP = 1 + k b OP z +b OP y

2

2

, IP =1 +

kb IP z 2

1 b IP y

2

. (22)

To gain some insights into theresults, we make thefollowing observations. r If k =k = 21 + , the linear equilibrium under either accrual account-ing system produces the rst-best investment level ( m = 1) and the

market response coefcients b m z =

2k , b

m y =0(m {O P , I P }). In thiscase, the cash ow z 1 provides sufcient information for efcient pric-

ing, and the market ignores the accruals completely ( b m y =0). r If 2m +, the equilibrium is the same as the basic setup ( ={z 1}). The quality of accruals is so poor that the market ignores the

accounting signals ( b m y = 0), which is equivalent to a setting without accounting reports. r The direction of the response to accounting signals (i.e., the sign of

b OP y or b IP y ) can be positive or negative depending on the sign of [2 (1 + )k ].The following corollary summarizes the intuitive properties of the equi-

librium under either accounting regime.

COROLLARY 1. If y OP =k I + OP and y IP = I + IP , using (6), for any m {OP , I P },(i) if 2m +, the investment choice approaches that in the basic setup where

={z 1} ;(ii) m increases (decreases) in 2m when k > k (k < k );(iii) b m z decreases (increases) in 2m when k > k (k < k );(iv) b m y increases (decreases) in 2m when k > k (k < k ).

This corollary conrms an intuitive relation between measurement noiseand investment efciency. Recall that Theorem 2 shows that the aggrega-tion of ongoing and investment cash ows induces the suboptimal invest-ment. The combination of items (i) and (ii) of Corollary 1 indicates that thesuboptimal investment problem is alleviated by the accounting report. Forexample, when k < k and 2m +, we know m is the same as in the basicsetup and the rm underinvests. In this case, item (ii) implies that a lower 2m (than +) would induce a higher m , alleviating the underinvestment problem.

When the accounting quality is extremely poor ( 2m +), no valuation weight is placed on the accounting signals ( b m y =0). As the quality improves,


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more weight is shifted between the cash ow report and the accountingreport. For example, when k < k , the market underprices the investment. As 2m is lowered (from +), the market response to the accounting signalincreases (from zero) and the response to the cash report decreases (i.e.,the weight shifts from cash ow to accruals).

Overall, the results thus far showthat, when the noise level is high enough,an improvement in the quality of the accounting measures (i.e., a drop inthe noise) improves the communication between the rm and the market,and this benets the investment efciency. That is, the quality of accrualsis well dened and well behaved: the lower the variance, the higher theaccrual quality, the lower the market mispricing of rm investments, andmost importantly, the more efcient the investment decision.

4.3 COMPARING OUTPUT -BASED AND INPUT-BASED ACCOUNTING MEASURES When the noise level is low, the output-based and input-based measures

exhibit a fundamental difference. We nd a monotonic relation between theinvestment efciency and the quality of the output-based accounting signal.That is, the investment efciency under output-based accounting continuesto improve when measurement noise decreases.

In the extreme, output-based accounting achieves the rst-best result as 2OP is reduced to zero. If

2OP =0, the market is able to infer the short-terminvestment return perfectly from the accounting report (because y OP =k I for certain). Subtracting the accounting signal from the aggregate

cash ow reveals the cash ow from rst period ongoing activities. That is,the accounting report helps the investors clearly distinguish the cash owcomponents. In turn, the market response coefcients are b OP z =1 + andb OP y = 2k (1 + ) , leading to a combined reaction of 2k to the short-terminvestment return, which provides the rst-best investment incentive.

In contrast, input-based accounting is another story in which the resultsare not as straightforward. The relation between investment efciency andmeasurement noise ( 2IP ) is not monotonic. When k > k , the overinvest-ment problem exists only when 2IP is high enough. If

2IP is very low, the

rm underinvests. In the extreme, when 2IP = 0, outsiders are able to in-fer the actual investment made ( I ); the market response coefcients areb IP z = 1 + and b IP y = [2 (1 + )k ]

12IP , and the investment level IP =

(1 (k (1 + )2 1)) 2 < 1.The following theorem summarizes and compares the effects of the two

accounting systems on the investment efciency.

THEOREM 4. If y OP =k I + OP and y IP = I + IP ,(i) if 2OP =0, investment choice is the rst-best level, and (ii) if 2IP = 2 , investment choice is the rst-best level.(iii) There exists a (0 < < 2 ), such that a sufcient condition for input-

based accounting to be more (less) efcient than the cash ow reporting regime (of the basic setup) is 2IP > ( 2IP < ).


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(iv) There exists a ( < < 2 ), such that a sufcient condition for input- based accounting to be more efcient than output-based accounting is the combination of (a) b IP [ ,

2 ] and (b) 2OP > .

The two systems achieve the rst-best at different noise levels. Under theoutput-based system, the rst-best is achieved when the accounting measureis noiseless, which is very intuitive. Under the input-based system, the rst-best is achieved when the noise is small but not zero. More importantly,the output-based system does not dominate the input-based system in allsituations. Since some noise is unavoidable in practice, it is likely the input-based system may be preferable. Consider the following two comparisons.First, suppose an accounting item in question is well understood and easy tomeasure. So we assume both output-based and input-based measures sharethe same (small) noise level (e.g., less than 2

). In this case, Theorem 4

predicts the input-based system is preferred if the variance of the noise( 2m ) is between and

2 . Alternatively, suppose the accounting item is

not well understood and hard to measure. So we assume the noise level ishigh for both measures (i.e., greater than 2 ). In this case, it is more likely that output-based accounting may entail a more noisy measure than input-based accounting. 10 As a result, the investment efciency under a highly noisy output-based measure may be closer to the (benchmark) cash owsetting (by Corollary 1), while the efciency under a not-so-noisy input-based measure may be closer to rst-best (by Theorem 4). In this case,input-based dominates output-based as long as their variance difference islarge enough.

The key difference between input-based and output-based systems hasto do with the fundamental difference between the two accounting ap-proaches. The input-based method ( y IP = I + IP ) requires estimat-ing the investment cost ( I ) alone, without any explicit attention to theprotability of the investment ( ). The output-based method ( y OP =k I + OP ) requires estimating both I and . This fundamental differ-ence leads to a structural difference in the market mispricing of rminvestments.

Returning to the analysis of investment distortion induced by mispricing,substitute the pricing function ( P = a OP +b OP z ( x 1 +k I ) +b OP y (k I + OP )) into the expected mispricing, and we have

10 One way to view the natural relation between 2IP and 2OP is that the accounting system

constructs the output-based measure based on two estimates: an estimate of actual investment made (e.g., y IP = I = I +noise ) and an estimate of protability (e.g., = +noise ). Theoutput-based measure is estimated using the true production function (roughly, y OP =2 I ). Viewed this way, it is natural that the overall noise in y OP is likely to be higher than the noise

in y IP

.


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E x 1x 2 [ P ( x 1 +x 2 +2 I ) | ]

= E x 1x 2 a OP +b OP z ( x 1 +k I ) +b OP y (k I + OP )

( x 1 +x 2 +2 I ) |

= b OP z +b OP y k 2 I + E x 1 x 2 a OP +b OP z x 1 +b OP y OP ( x 1 +x 2) .If (b OP z + b OP y ) k 2 = 0, any investment will affect the market pricing,giving the rm an incentive to over- or underinvest. The marginal effect of

investment I on the mispricing depends on true investment protability (because the derivative equals (b

OP z +b OP y )k 2

2I ). This leads to a dampening

effect: The marginal benet is concave in I , providing a diminishing returnto investment deviations. Intuitively, the rms ability to use real investment

tochangethemarketperception of its investmentprotabilityismitigatedby the independent protability estimate built into the output-based measure.

With input-based accounting, the expected mispricing is

E x 1 x 2 [ P ( x 1 +x 2 +2 I ) | ]

= E x 1 x 2 a IP +b IP z ( x 1 +k I ) +b IP y ( I + IP ) ( x 1 +x 2 +2 I ) |

= b IP z k 2 I +b IP y I + E x 1 x 2 a IP +b IP z x 1 +b IP y IP ( x 1 +x 2) .The mispricing will only affect the equilibrium investment through the

rst two terms: ( b IP z k 2) I +b

IP y I . The marginal effect of investment onthe rst term ( b IP z k 2) I depends on the investment protability (the

derivative is b IP z k 2

2I ) while the marginal effect on the second term b

IP y I

depends only on an equilibrium constant b IP y . The dampening effect is activeonly on the rst term, not the second term. Notice, from Theorem 3, that weknow in equilibrium, the market reactions to the cash ow and accountingmeasures ( b IP z and b IP y , respectively) are such that

b IP z k 2 =[(1 + )k 2] 2 2IP +2IP 2 2

2IP 2 2

+ 2 2IP

+k 2IP 2IP

2

,

b IP y =[2 (1 + )k ]

32IP

2 22IP

2 2 + 2 2IP +k 2IP 2IP 2.

When k = k , the sign of b IP z k 2 is always the opposite of the sign of b IP y , which indicates that the marginal effects on the two terms are in theopposite direction. The total effects on mispricing depend on which itemoutweighs the other.

For example, if k > k , b IP z k 2 is positive, which indicates that ahigher investment would increase market mispricing. However, the damp-ening effect provides a diminishing return to overinvestment, which makes


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T A B L E 1The Performance of Three Accounting Regimes

Parameter regions

=0 0< k k < k Cash ow =1 =1 > 1 < 1

OP =1 if 2OP =0 OP =1 if 2OP =0Output-based OP =1 OP =1 OP > 1 if 2OP > 0 OP < 1 if 2OP > 0 IP =1 if 2IP = 2 IP =1 if 2IP = 2Input-based IP =1 IP =1 IP 1 if 2IP 2 IP 1 if IP 1

overinvestment less attractive. On the other hand, the rm is also motivatedto underinvest because b IP y is negative. Notice here that the marginal effect

is a constant and is independent of the private information and invest-ment level I ; no dampening is in effect. This hurts the economy when theaccounting report is too precise. If 2IP is too small, the absolute value of b IP y is too large, which motivates the rm to underinvest by a large amount (Corollary 1). This motivation outweighs the overinvestment motivation by the rst item because a small 2IP reduces the market response to the aggre-gate cash ows report ( b IP z ).

Consider the limiting case, when 2IP =0. Unlike thecase for output-basedaccounting, the rst-best investment is not achieved when the input-basedmeasure is noiseless. Suppose the rm invests the rst-best amount (e.g.,

I = ), then the markets best responses are b IP z = 1 + and b

IP y = 2 (1 + )k < 0. These responses invite the rm to underinvest because at I =

, themarginalbenet of additional investment is b IP z k 2

2 +b IP y =1 (1 + )k 2 ,

which is less than the marginal cost of additional investment ( =1). 11In another knife-edge case, the two opposite effects exactly offset eachother where the rst-best is achieved. That is, if 2IP = 2 and we proposethat IP = 1, then we nd that

b IP z k 22 +b IP y =0 (i.e., the marginal effect of any investment deviation is zero). In equilibrium, there is no incentive

to distort investment. Thus, with the input-based measure, the rst-best isachieved when measurement noise is small but not zero. Table 1 summarizesthe performance of three accounting regimes.

Figure2 illustrates theeffectof accounting reports on theefciency. Here,the expected net project return represents the efciency of the investment.The FB line stands for the net project return when the investment level isthe rst-best ( I FB = ). The basic setting results ( I SB = ) are denoted by

11 Further, in this limiting case of perfect knowledge of the actual investment made ( I ), theinduced investment efciency is worse than that of the basic setup (item iv of Theorem 4). Alternatively, if the protability ( ) is perfectly revealed and z is reported, it can be shown that the induced efciency is notrst-best. This is consistent with Kanodia, Singh,andSpero [2005], where some imprecision in the accounting measurement is preferred. What is different in thecurrent model is that the conclusion on accounting imprecision depends on the accountingmeasurement basis. With an output-based measure, the ideal accounting is noiseless.


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SB-CF

Net project return

2

FB

0

OP

IP

2

m

'

FIG. 2.The effect of accounting quality on the net project return.

the SB-CF line. The performance of input-based accounting ( I IP = IP )and output-based accounting ( I OP = OP ) is described by the IP and OPsolid curves respectively. From the gure, it is easy to see that output-basedaccounting is dominated by input-based accounting in the region when thecommon noise is between and 2 .12

5. Extension I: Accounting Manipulation In this section we expand themodel to consider managerial manipulation

of the accounting measurement. A robust feature of any accrual measure-ment is that rms have varying degrees of inuence (or discretion) on howaccruals are prepared. However, other economic factors (e.g., auditing ormanagerial reputation) prevent the use of complete discretion. We capturethis partial discretion by considering a simple model of cost-benet calculuson the part of the rm.

5.1 EQUILIBRIUM UNDER ACCOUNTING MANIPULATIONSuppose that the accounting signal y m (m {OP , I P }) is subjected tomanagerial manipulation. A rm can prepare its accounting report w m differently from the unmanipulated y m , at a cost. We assume the cost asc (w m ) = c

m

2 (wm y m m )2 , where m is independent of all other random variables and follows a normal distribution with mean zero and variance of

12 Figure 2 shows that input-based accounting induces more efcient investment decisionsthan output-based accounting for all noise levels greater than . We note that this result is not general and that there exist sufcient conditions that the output-based accounting dominatesthe input-based accounting when the measurement noise is large. That is, when the noise ishigh, input-based does not necessarily dominate output-based when both share a commonmeasurement noise.


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2m . Variable m captures the random component of manipulation costs. 13From the earlier results, the market response to the accounting report canbe positive or negative. Then, with accounting manipulation, the rm canbenet from adjusting the accounting report upwards or downwards, at themargin. We next introduce the denition of an equilibrium for the setup with accounting manipulation.

DEFINITION 2. An equilibrium relative to ={z 1, w m } consists of an investment function I m w (), a reporting policy w m (), and a market pricing function P (), such that: (i) Given P (), the optimal investment function I m w () and the reporting policy w m () maximize V ( | I m w (), w m ()) = E [I m w + P () + (1 )( z 1 + z 2) c (w m )].(ii) Given I m

w (

) and w m (

), the pricing function P (

) satises P

=E [z 1

+z 2 | , I m w (), w m ()]. We now analyze the equilibrium behavior of the rm under the accrual

basis accounting with manipulation.

THEOREM 5. If y OP =k I + OP and y IP = I + IP , and c (w m ) =c m 2 (w

m y m m )2 (m {OP , I P }), where k (0, 2) and using (6), there exists a unique linear equilibrium relative to ={z 1, w m }. It is given by (i) an equilibrium linear pricing function:

P (z 1, w m )

=a m w

+b m

z 1

+d m

w m , w here

b OP =(1 + )k 2 OP 2 2 +(1 + ) 2 2OP +2k OP 2OP 2

k 2 OP 2 2 + 2 2OP +k 2 OP 2OP 2,

d OP =[2 (1 + )k ]k OP 2 2

k 2 OP 2 2 + 2 2OP +k 2 OP 2OP 2,

a OP w = (2 b OP ) +(2 kb OP kd OP ) OP 0 (b OP )2

c OP ,

b IP = (1 +

) 2

IP 2 2

+(1 +

) 2 2

IP +2k

IP 2

IP 2

2IP

2 2 + 2 2IP +k 2 IP 2IP 2,

d IP =[2 (1 + )k ]

32

IP 2 2

2IP 2 2 + 2 2IP +k 2 IP 2IP 2

,

a IP w = (2 b IP ) +(2 kb IP IP d IP ) IP 0 (d IP )2

c IP ,

2m = 2m +2m . (23)

13 See a similar assumption and more discussions of the cost structure in Dye and Sridhar[2004a].


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(ii) an equilibrium investment function and an equilibrium reporting policy:

I m w ( ) = m , if 0

0 if < 0,where (24)

OP = 1 + k (b OP +d OP )

2

2

, IP =1 +

kb IP

21 d IP

2

, (25)

w m = y m + m + d m

c m . (26)

According to theequilibrium reportingpolicy(the w m

expression in equa-tion 26), the equilibrium accounting report varies from the one without accounting manipulation: The additional noise from a random variable m is added as well as a xed constant d

m

c m .The market can perfectly calculate the xed constant d

m

c m . (Parameters and c m are common knowledge, and d m is the equilibrium market re-sponse to the accounting report.) The intercept of the pricing function a m w is adjusted accordingly to undo the expected manipulation. The randomcomponent of manipulation injects noise m into the accounting report. Asa result, accounting manipulation worsens thequality of accounting reports.

The pricing function is adjusted in a way that the variance of the measure-ment noise ( 2m ) in the previous equilibrium pricing function is replacedby 2m , which equals the sum of 2m and 2m .

5.2 VALUE OF ACCOUNTING MANIPULATIONThe accounting manipulation leads to a more noisy accounting measure

and a dead-weight loss (the manipulation cost to the rm). The latter cost isincorporated into the following cost-benet analysis of accounting manip-ulation.

COROLLARY 2. Under output-based accounting, accounting manipulation always makes the rm worse off; under input-based accounting, a sufcient condition for accounting manipulation to be value enhancing is the combination of (i) 2IP < 2 and (ii) c IP 0 is sufciently high.

As we have shown, the performance of output-based accounting mono-tonically decreases as the quality of the accounting report worsens. There-fore, under output-based accounting, accounting manipulation reduces ef-ciency for two reasons. First, a more noisy accounting report induces lessefcient investment. Second, the rm incurs a manipulation cost c (w m ).

However, with input-based accounting, the nonmonotonicity feature

makes it possible that the incremental noise due to manipulation may ben-et the rm. Recall that in gure 2, making the accruals more noisy canimprove efciency when the accruals are too precise ( 2IP <

2 ). Thus,


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if 2IP < 2 , the expected net project return strictly increases as a result of

accounting manipulation, provided that 2IP is not too large.Since the expected net project return is (2 m m ) 0, the higher 0

can magnify the gain from more efcient investment choices. The cost of manipulation is

c (w IP ) =c IP

2w IP y IP IP

2

= d IP 2

2c IP .

A higher c IP can reduce the dead-weight loss from the rms myopic de-cision. Thus, if c IP 0 is sufciently high, the cost of earnings management is outweighed by the increase in the expected investment returns. 14

Intuitively, when theactual investment made is measured tooprecisely, themarket pricing places too much valuation weight on the accounting report,providing an unmitigated incentive to over- or underinvest. By allowingaccounting manipulation, more noise is injected into accounting reportsand the market reacts by reducing the valuation weight. As a result, lesspressure leads to less inefcient investment choices. 15 Table 2 summarizesthe effect of accounting manipulation.

5.3 REAL VERSUS ACCOUNTING MANIPULATIONThe above analysis points to a link between the so-called real earnings

management and accounting earnings management. Real management typically refers to the rms discretionary choices that affect the rms cash

ow for the sole purpose of inating reported performance. These choicesare not in the best interest of the shareholders. Accounting management typically refers to the rms discretionary choices that affect the rms re-ported performance by altering the accounting measurement process. Inour model, we interpret investment deviations from the rst-best as an ex-ample of real earnings management and accounting manipulation of y intow as accounting earnings management.

14 It might also be interesting to compare the efciency of discretionary accounting re-ports with the basic cash ow setting. This takes the view that allowing either output-based orinput-based accounting measurement implicitly grants the rm the ability to manipulate itsaccounting performance. In other words, accrual accounting and the manipulationoption area bundle.

From Theorem 4, under output-based accounting, the investment is always more efcient than in the cash ow setup, but manipulation always incurs positive costs c (w OP ). Whetheroutput-based accounting with manipulation is preferable to the cash ow setup depends onthecost incurred andon benets derived from the investment improvement.Thesame tensionexists under input-based cost accounting with manipulation. Specically, input-based account-ing with manipulation is more efcient than the cash ow setup when (1) 2IP > and (2)c 0 is sufciently high. On the other hand, input-based accounting with manipulation is lessefcient than the cash ow setup when 2IP < .

15 This intuition does not follow when output-based accounting is used because when the

measurement noise is low, the actual investment only partially affects the accounting signal(recall y OP =k I +OP ).


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T A B L E 2The Effect of Accounting Manipulation for Different Accounting Regimes

Parameter Region

Accounting regime k >

k k 1, 2OP OP > 0 OP < 1, 2OP

OP < 0

Input-based 2IP > 2 IP > 1,

2IP

IP > 0 IP < 1, 2IP IP < 0

2IP < 2 IP < 1,

2IP

IP < 0 IP > 1, 2IP IP > 0

Under the output-based measure, our results indicate that accountingearnings management always leads to more real management. They arecomplements. This is because accounting management leads to more mis-pricing. Under an input-based measure,a similar result is obtained when the

measurement noise is high. However, when noise is low (especially in theregion where investment efciency increases in noise), accounting earn-ings management leads to less mispricing, and more efcient investment choices are made. Here accounting management is a substitute for realmanagement. The intuition is that the accounting management introducesadditional noise, which leads to less market reaction to the accounting re-port. Less pressure on the accounting numbers mitigates over- and under-investment incentives.

6. Extension II: Technology for Future Investments

In this section we modify the model to include a situation where the rmtechnology may be re-used in the future. That is, from date-2 onwards, theowners of the rm may generate future cash ow by investing in the rmtechnology they have acquired on date-1. In other words, the owners havean option to invest and will exercise the option if the technology turns out to be protable. In turn, on date-1, the capital market not only prices thecash ows generated from the date-0 investment, it also prices the value of owning the technology that produces future cash ows. 16

6.1 MODEL MODIFICATIONS

To simplify the problem, we change the model as follows. On date-0, therm chooses an investment level, denoted I 1R +, based on the private

signal , same as before.On date-1, shares of the rm are traded in a competitive capital market.

However, we assume there are no ongoing activities 17 (x 1 and x 2) and only one public signal m (m {OP , I P }), which is produced by the accountinginformation system, observable to outsiders. Similarly, we assume that the

16 We wish to thank the referee for suggesting that we pursue this extension.17 This assumption is made for simplicity only. All results in this section survive if stochastic

ongoing activities are introduced as long as, as before, they are independent of other random variables in the model.


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rm can choose either an input-based approach ( IP ) or an output-basedapproach ( OP ) as the accounting measurement bases. For the output-basedmeasure,

OP

=2 I 1 +OP

, (27) where OP N (0,

2OP ). The output-based accounting reports provide a noisy

measurement of the total investment return. For the input-based measure,

IP = I 1 + IP , (28) where IP N (0,

2IP ). The input-based accounting reports provide a noisy

measurement of the investment costs.Finally, on date-2, the owners observe the return of the initial investment

2 I 1, and choose additional investments into the existing technology. Weassumethe present value of futurecashowsis r , r

R +. A simple interpre-tation of this representation is that the future protability of the technology isthesameas , and thefutureownersmake an optimal investment decision,knowing the true . In this case, in every period this technology is viable,the owners choose optimal I t to maximize 2 I t I t and generate peri-odic prots equal to , assuming a positive . As a result, future prots canbe represented by an annuity (or perpetuity). Parameter r summarizes theimportance of these future cash ows relative to the cash ow generatedby the initial investment. If the technology is long-lived or if the ownersdiscount rate is low, more rm value comes from future investments (orgrowth opportunities), leading to a higher r .

A more complex, perhaps more realistic, interpretation involves a non-stationary technology, or future rm owners subjected to additional market frictions, or that the true is revealed to the owners gradually throughlearning-by-doing. However, under these scenarios, it may be reasonableto assume that the value of this reinvestment option is proportional to thepast protability . As a result, we believe our characterization is a reason-able approximation to capture the idea of this option without bringing inadditional complexity to the model.

6.2 EQUILIBRIUM UNDER MODIFIED MODEL

Now return to the pricing problem on date-1; without the ongoing cashows, the capital market must estimate the value of the date-2 cash owgenerated by past investment and the value of the potential future cashows generated by future investments. As a result, the market price is equalto the expected value of the cash ow from existing projects plus the present value of the reinvestment option, that is

P = E [2 I 1 +r | ] . As before, when making the initial investment on date-0, the rm is

motivated by both the long-term interest and the short-term interest. The

convex combination of these concerns is the same as the basic setup, that is,I 1 + P +(1 )(2 I 1 +r ) .


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We now analyze the equilibrium behavior of the rm with technology forfuture investments.

THEOREM 6. If OP =2 I 1 + OP and IP = I 1 + IP , and using (6), there exists a unique linear equilibrium relative to ={

m }. It is given by

(i) an equilibrium linear pricing function:

P ( m ) = a m +b m m , where

b OP =4 OP +2r OP 2

4 OP 2 + 2OP

,

a OP = 2 OP +r 2 OP b OP 0,

b IP =2 IP +r IP 2( IP )2

2 + 2IP

,

a IP = 2 IP +r IP b IP 0; (29)

(ii) an equilibrium investment function:

I m 1 ( ) = m , if 0

0 if < 0,where

OP = 1 + b OP 2, IP =

1 1 b IP

2

. (30)

With a valuable technology for future investments, the share price in-cludes the market estimate of how much the rm will benet from thetechnology ever after. Higher r indicates the rm is able to generate morefuture cashowsfor a given positive . Thus, themarket response coefcient b m is strictly increasing in r . With a higher market response coefcient, therm has incentive to inate the perceived protability. This can be achieved

by inating initial investment, which increases the mean of either the input-based measure or the output-based measure. Therefore, the investment decision I m 1 is strictly increasing in the parameter r .

Since the accounting measurements may not be perfectly precise, themarket also responds to the measurement noise. As the variance of mea-surement noise 2m gets higher, the accounting reports are less informative,leading to a less responsive market price to the accounting report and therm has less incentive to overinvest. Therefore, the initial investment ( I m 1 ) isstrictlydecreasing in themeasurement noise 2m . Because thetwoexogenousparameters ( r and 2m ) provide opposite incentives of investment choices,

there exists a knife-edge case that the two opposite effects exactly offset eachother, leading to the rst-best initial investment.


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6.3 COMPARING OUTPUT -BASED AND INPUT-BASED ACCOUNTING MEASURESThe following corollary summarizes the properties of the equilibrium.

COROLLARY 3. If OP

=2 I 1

+ OP and IP

=I 1

+ IP ,

(i) if 2OP =2r 2 , investment choice is the rst-best level, and (ii) if 2IP = (r +1) 2 , investment choice is the rst-best level.(iii) If r > 1( r < 1), there exists a that lies in the interval between (r + 1) 2 and 2r 2 , such that a sufcient condition for input-based accounting to be more efcient than output-based accounting is 2IP = 2OP [( r +1) 2 , ]( 2IP = 2OP [ , (r +1) 2 ]). 18

The presence of the reinvestment option changes the market pricingand (thus) initial investment decisions. Compared with the correspond-ing results in section 4 (see Theorem 4), the results are different in two ways. First, even with the output-based measure, investment efciency is nolonger monotonic in measurement noise. In particular, the output-basedaccounting measure does not perform best when the measure is noiseless.Second, the parameter r , a growth potential index so to speak, is important in determining economic efciency, in addition to accounting rules andmeasurement errors. Intuitively, as r increases, the market imposes morepressure on the accounting reports, and the rm is motivated to inate thereports. 19

To see the trade-off precisely, we briey review the mispricing structure.Substituting the pricing function ( P =a OP +b OP OP ) into the mispricingexpression, we have

E OP P (2 I 1 +r )

= E OP a OP +b OP 2 I 1 + OP 2 I 1 +r

= b OP 1 2 I 1 r +a OP + E OP b OP OP .The investment choice is rst-best when the mispricing does not depend

on the investment choice. Under the output-based measure, the mispricingis not a function of the initial investment only if the market response to

the accounting measure is equal to unity (i.e., b OP = 1). However, in thismodied model, a noiseless output-based measure will no longer lead to a

unity market response. This is because, in this modied model, the market ispricing two streams of cash ows. First, for cash ows due to the initial invest-ment, a unity response is needed with a noiseless measure. Second, for cashows due to future investments, a nonzero response is needed. Combined,the total response would be greater than unity in the noiseless case, thusproviding ex ante incentive to deviate from the rst-best investment.

18 When r

=1, we can show that output-based accounting is (weakly) more efcient than

input-based accounting once is small enough.19 When r = 0, the model reverts back to the basic model with the output-based model

performing best when noiseless.


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Net project return

FB

0

OPIP

2

m

* 24 23

FIG. 3.Investment efciency in the modied model ( r = 2).

Figure 3 provides an illustration of the results in Corollary 3. Compared toFigure 2, the main difference is that the efciency under the output-basedmeasure peaks when the variance of the measure is not zero. Furthermore, when r > 1, the peak occurs to the right of the peak under the input-basedmeasure. This is because, relative to the input-based measure regime, a

higher r imposes more market pressure on the accounting measure, thusleading a more distorted investment choice.Finally, the extension leads us to rethink the subtleties of output-based ac-

counting when (reinvestment) option value is important. From an account-ing measurement perspective, OP can be viewed as a measure of value inuse, ignoring the option value of future use (through future investments).These measures do exist in accounting practices, such as the re-valuation ex-ercise in accounting for asset impairment. However, one may argue that theoption value would be impounded in a would-be exchange price of the as-set in question. Fair-value accounting measures, as proposed by the recent

FASB exposure draft, may be close to having this characteristic. One caneven argue that these measures already exist in accounting practices, suchas the use of market value in initial and re-valuation of certain assets and inthe initial recording of goodwill (provided the market prices are reectiveof various option values).

7. Conclusion

In this paper, we explore the trade-offs between two dominant account-ing measurement bases: input-based measures and output-based measures.

We discover that the trade-offs go beyond relevance and reliability issuescommonly mentioned in accounting debates. We show that these two mea-sures affect investment incentives in fundamentally different ways. The


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output-based measures have a natural advantage in aligning the rm andsocial investment incentives through a dampening effect, which limits over-andunderinvestmenttendencies. However, high levelsof noise andaccount-ing manipulation, which are typically associated with output-based mea-sures, may make output-based accounting far from a perfect solution to allaccounting problems.

With an input-based accounting measurement basis, accounting num-bers are less comprehensive, but their advantages are a lower level of noiseand fewer accounting manipulation opportunities. In fact, being less com-prehensive makes small but positive noise and/or manipulation desirable.Based on our analysis, the move toward output-based accounting, such as afair-value principle, may not be benecial and requires more care and moreextensive debates.

Our model is simple. Future works may benet from including operatingand nancial choices and from analyzing more general settings with het-erogeneous rms where accounting standards are central to an economicanalysis of accounting.

APPENDIX

Proof of Limit Properties of the Approximation Assumption.In this part of theappendix, we show that the approximation error, denoted AE , gets smallerand approaches zero as the mean of z increases, for every realized value of

the conditioning variable, y + az . To prove this formally, notice for every W y +az ,AE ( W ) = z > 0 x ( x +z ) f ( x , z | y +az ) dx dz +G (0 | y +az ) x x f ( x | y ) dx

z x ( x +z ) f ( x , z | y +az ) dx dz =G (0 | y +az ) x x f ( x | y ) dx z < 0 x ( x +z ) f ( x , z | y +az ) dx dz ,

(A1)and we need to prove both components of the AE (W ) expression approachzero as E [z ] increases to every realization of W y +az .By assumption, the joint distribution of x and y is

x y

N x y

, 2x xy xy 2 y

and z N [ z , 2z ] is independent of x and y . Hence, the joint distributionof x , z , and y +az is

x z y +az

N

x z y +a z

,

2x 0 xy 0 2z a 2z

xy a 2z 2 y +a 2 2z .


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By the property of normal density function, the conditional distributionof z given any realization of W y +az is

z |W N z +a 2z

2 y +a 2 2z ( W y a z ) , 2z

2 y

2 y +a 2 2z . (A2)To simplify the notation, we denote the above by z |W N [ z , 2z ]. Andthe joint conditional distribution of x and z given any realization of W is

x z

W

N

x + xy

2 y

+a 2 2z

( W y a z )

z +a 2z

2 y +a 2 2z ( W y a z )

,

2x 2xy

2 y

+a 2 2z

xy a 2z 2 y

+a 2 2z

xy a 2z

2 y +a 2 2z 2z

2 y

2 y +a 2 2z .

(A3) Again, to simplify, we denote the above by

x z W N

x z

, 2x xz xz 2z

.

Using (A2), the conditional cumulative density function of z , G ( z

|W )

=( z z z ) , where () is the standard normal cumulative distribution func-tion (cdf). Hence, G (0 |W ) = (

z z

). If z increases, z z goes to nega-tive innity because z

2 y 2 y +a 2 2z

z +a 2z

2 y +a 2 2z ( W y ) for any given W .By the property of standard normal cumulative density function ( ()),( z z ) 0 as z

z . Therefore, the rst component of (A1),G (0 |W ) x x f ( x | y ) dx , goes to zero.Using (A3), the second component of (A1) is

z < 0

x

( x +z ) f ( x , z |W y +az ) dx dz

= z < 0 x ( x +z ) f ( z |W ) f ( x | z , W ) dx dz = z < 0 x xf ( x | z , W ) dx +z x f ( x | z , W ) dx f (z |W ) dz = z < 0 x + xz 2z (z z ) +z f ( z |W ) dz

= x + xy 2 y ( W y ) G (0 |W ) + 1

a xy 2 y z < 0 zf (z |W ) dz .


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Similar to the rst component, the rst part of the second component,[ x +

xy 2 y

( W y )] G (0 |W ), also approaches zero when z increases.Finally, we need to show the second part of the second component,

z < 0

z f (z |W ) dz , goes to zero as

z increases.

z < 0 z f ( z |W ) dz = 1 z 2 0

z exp

(z z )22 2z

dz

=1

z 2 0

z exp

z 2

2 2z

expz z 2z exp

2z 2 2z

dz .

Because z is always negative, exp( z z 2z ) must be positive and alwayssmaller than one. So the absolute value of the above expression must besmaller than the absolute value of 1 z 2

0

z exp( z 2

2 2z ) exp(

2z 2 2z

) dz = z 2 exp(

2z 2 2z

). As the mean of z increases, 2z 2 2z

goes to negative innity.

Hence, the absolute value of 1 z 2 0

z exp( z 2

2 2z ) exp(

2z 2 2z

) dz goes tozero, and thus, the second part of the second component,

z < 0 z f ( z |W ) dz ,also approaches zero.

To summarize, we have shown for every W y + az , as z increases,AE (W ) approaches zero.Proof of Theorem 1. We begin with the linear pricing conjecture:

P (z 1) =a +bz 1.

The managers maximization program becomes:

Choose I ( ) to max V ( | I ()) G ( ) d = E x 1x 2 [I + P +(1 )( z 1 +z 2)] G ( ) d = I + a +b + k I G ( ) d

+(1 ) 2 + 2 I G ( )d .


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The point-wise rst-order condition with respect to I is, for > 0,

0=

1+

1 + bk 2

I I = 1 +

bk 2

2

= 1 +bk 2 1

2

,

and for < 0, I =0. So, it must be the case that = 1 +

bk 2 1

2

.

Given that and k are constants,

x 1 +x 2 +2 x 1 +k

N 2 +2 0 +k 0

,2(1 + ) 2 +4 2 (1 + ) 2 +2k 2(1 + ) 2 +2k 2 2 +k 2 2

.

So, we have the approximate pricing function, using (6):

P = E [ z 1 +z 2 | z 1] = E [ x 1 +x 2 +2 I |x 1 +k I ]= E [ x 1 +x 2 +2 |x 1 +k ]

= 2 +2 0 +(1 + ) 2 +2k 2

2 +k 2 2( z 1 k 0)

=(1 ) 2 +2(k 2 k ) 2

2 +k 2 2 +

(2 (1 + )k ) 2 2 +k 2 2

0

+(1

+ ) 2

+2k 2

2 +k 2 2 z 1.So, it must be the case that

a = (2 b ) +(2 kb ) 0,

b =(1 + ) 2 +2k 2

2 +k 2 2.

To show the existence of and b , substituting b into , and simplifying, wehave

1 k 2 2

2 +1 = k (1 + )2 1 .


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As [0 , 1] , k [0 , 2] , and (1, 1), the range of the right-handside is from 1 to 1. As > 0, and all of the functions are continuous, theleft-hand side covers the range from 1 to +. Therefore, there is at least one positive root of .20Proof of Theorem 2. Suppose the manager chooses the rst-best investment

=1. Correspondingly, using (7), (8), and (9) the market pricing function would beP ( z 1) = a +b =1 z 1, where

b =1 =(1 + ) 2 +2k 2

2 +k 2 2, a = (2 b =1) +(2 kb =1)0.

To sustain the equilibrium, the managers reaction to b =1 =(1 + ) 2 +2k 2

2 +k 2 2must indeed be to set

=1, which is the rst-best investment level. That is,

it must be the case that |b =1 ,k =k =1

Using (11), we have

|b =1 ,k =k = 1 +b =1k

2 12

= 1 +(1 + ) 2 +2

21 +

2

2 +2

1 +2

2

2

1 +2 1

2

=1.Generally, we rewrite bk as the following

bk =(1 + ) 2 +2k 2

2 +k 2 2 k

= 2 +[(1 + )k 2] 2

2 +k 2 2.

So

= 1 +bk 2 1

2

= 1 +

2 [(1 + )k 2] 2

2 +k 2 22

.

Now it is clear that if k > k , > 1, and k < k , < 1.

20 Technically, in some rare cases (when the right-hand side is very close to

1), the function

could have three positive roots of . Then, in our study, we only consider the root that is closest to one. That is, in multiple equilibrium cases, the economy chooses the most efcient one.


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PROPOSITION 1. Comparative statics in the basic setup.

1. and b strictly increase in ;2 . When k > k (k < k ), and b strictly decrease in

2

2 (strictly increase in 2 2 );

and 3 . When k > k (k < k ), strictly increases in (strictly decreases in ). Also

b strictly decreases in for any k =k .Proof of Proposition 1. Using the pricing function (8), substituting b into

the managers investment decision (11), we have the following equation, which has to hold in equilibrium.

1 k 2 2 2 +1 =

k (1 + )2 1 .

The left-hand side increases monotonically in 21, and is independent of . The right-hand side increases monotonical in , and is independent of . It is easy to see that strictly increases in . By (11), b strictly in-creases in , therefore, b strictly increases in . From Theorem 2, > 1 when k > k . Given the equation above, the right-hand side is positive ask > k . As

2

2 increases, has to decrease to let the equation sustain. Sim-ilarly if k < k , increases in

2

2 . By (11), b changes in the same way as . As we show above, the left-hand side increases monotonical in . Whenk > k , the right-hand side increases in , so strictly increases in .Similarly, strictly decreases in when k < k , and is equal to one re-gardless of when k = k . To analyze the change in b , rewrite (8) as thefollowing

b =2k +

(1 + ) 2k

1 +k 2 2 2

.

If k > k , (1 + ) 2k > 0. As increases, increases from the above analy-sis. Therefore, b decreases in . Similarly, b also strictly decreases in whenk < k , and is equal to 1 + regardless of when k =k .

Remarks: The rst result is quite straightforward. As increases, the short-term and long-term cash ows from the ongoing activities are more corre-lated. Therefore, the market response coefcient b increases, and the rmhas an incentive to invest more to inate the market price. That is, in-creases.

Combined with Theorem 2, the second comparative static result showsthat a higher

2

2 can induce a more efcient investment level (i.e., pushing

21 The monotonicity is based on the previous assumption that in the case of multiple roots

for the equilibrium , theeconomy chooses theroot with which the investment is most efcient.See proof of Theorem 1.


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closer to one). As 2 gets lower, the ongoing cash ows are less noisy,making the aggregate cash ow report ( z 1 = x 1 +k I ) more informativeabout the short-term investment return ( k I ). Intuitively, it is easier forthe market to identify the rst-period investment return. (Technically, themarket response coefcient is closer to 2k as

2 2 increases.) As a result, the

short-term investment return is less mispriced while the ongoing cash owis more mispriced. However, given that the mispricing of the ongoing cashow does not have any negative effect on the real investment decision, anincrease in

2

2 improves the investment efciency. In the extreme, when 2

approaches zero, the equilibrium achieves rst-best results. With short-term pressure ( > 0), the rm investment deviates from the

rst-best (by Theorem 2). The magnitude of the deviation increases in mar-ket pressure ( ). As the rm makes less efcient investment under more

market pressure, the market responds less to the cash ow report ( b de-creases in ) because the value lost due to inefcient investment increasesin .

Proof of Theorem 3. Suppose the rm chooses an input-based accountingsystem, based on the linear conjecture

P z 1, y IP = a IP +b IP z z 1 +b IP y y IP .The manager maximizes the expected payoff:

Choose I (

)to

max V

(

|I ())

G (

)d

= E x 1x 2 [I + P +(1 )( z 1 +z 2)] G ( ) d = I + a IP +b IP z + k I G ( ) d +b IP y I

+(1 ) 2 + 2 I G ( )d .The point-wise rst-order condition with respect to I is for > 0, and wehave

0 = 1 +1 +

kb IP z 2

I + b

IP y ,

I IP =1 +

kb IP z 2

1 b IP y

2

,

and for < 0, I IP =0.


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So it must be the case that

IP =1 +

kb IP z 2

1 b IP y

2

.

Using (21) and (22), we have

x 1 +x 2 +2 IP x 1 +k IP

IP +IP is normally distributed with mean

2 +2 IP 0 +k IP 0

IP 0

and variance

2(1 + ) 2

+4IP 2 (1 + )

2

+2k IP 2 2

32

IP 2

(1 + ) 2 +2k IP 2 2 +k 2IP 2 k 32IP

2

232IP

2 k

32IP

2

2IP

2 + 2IP

.

Using the approximation assumption (6), the pricing function becomes

P = E z 1 +z 2 z 1, y IP = E x 1 +x 2 +2 I x 1 +k I , I + IP

= E x 1 +x 2 +2 x 1 +k , +IP

= 2 +2 IP 0 + (1 + ) 2 +2k IP 2 , 232IP

2

2 +k 2IP 2

32IP

2

k 32IP

2

2IP

2 + 2IP

1

z 1 k IP 0

y IP IP 0

= 2 +2 IP 0 +(1 + )2IP 2 2 +(1 + ) 2 2IP +2k IP 2IP 2

2IP 2 2 + 2 2IP +k 2IP 2IP 2

z 1 k IP 0 + [2 (1 + )k ]32

IP 2

2

2IP 2 2 + 2 2IP +k 2IP 2IP 2

y IP IP 0 .

8/3/2019 Accounti

accounting measurement basis market mispricing and firm investment efficiency 2007

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