The economic consequences of discrete recognition and continuous measurement

Journal of Accounting and Economics xxx (xxxx) xxx

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The economic consequences of discrete recognition and continuous measurement* Pingyang Gao a, *, Xu Jiang b a b

Booth School of Business, The University of Chicago, USA Fuqua School of Business, Duke University, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 15 March 2018 Received in revised form 29 July 2019 Accepted 14 August 2019 Available online xxx

Discrete recognition is a long-standing and ubiquitous accounting practice, but it has been widely criticized for suppressing information and inducing accounting-motivated transactions. We study a model to examine the economic consequences of shifting away from discrete recognition to a continuous measurement approach. Without manipulation, discrete recognition is less informative than the continuous approach. However, the continuous regime induces more manipulation. The equilibrium informativeness is determined by both the accounting standard and endogenous manipulation. Discrete recognition is more informative than its continuous counterpart precisely when manipulation is a severe threat. We respond to the recent call in Kothari, Ramanna, and Skinner (2010) for using positive accounting theory to explain certain long-standing accounting practices. We also discuss the model's implications for fair value accounting. © 2019 Elsevier B.V. All rights reserved.

JEL classification: M41 M45 G28 G38 Keywords: Accounting standards Evidence management Recognition Measurement Disclosure Fair value

1. Introduction Recognition, defined as the process of admitting information into the basic financial statements (SFAC 5), is a fundamental accounting concept and long-standing accounting practice. It determines when and how a transaction is included in the financial statements. Depending on recognition criteria, an expenditure may be capitalized as an asset on the balance sheet or expensed on the income statement, the future payment under a lease contract may be recorded as a liability on the balance sheet or left off the balance sheet, and the securitization of a portfolio of mortgages may lead to revenue recognition or secured borrowing. * We are grateful for comments from Rick Antle, Tim Baldenius, Qi Chen, Doug Diamond, Phil Dybvig, Ron Dye, Michelle Hanlon (the editor), Zhiguo He, Jonathan Glover, Bill Mayew, Pierre Liang, Katherine Schipper, Phil Stocken, two anonymous referees and participants of workshops at CUHK, Dartmouth College, Duke University, National University of Singapore, Purdue University, SWUFE, Singapore Management University, University of Chicago, University of Hong Kong Accounting Theory conference and Washington University in St. Louis. We gratefully acknowledges financial support from the University of Chicago Booth School of Business, the PCL Faculty Scholarship, the Duke University Fuqua School of Business, the Center for Financial Excellence at the Fuqua School of Business at Duke University, and National Nature Science Foundation of China (project 71620107005 “Research of Capital Market Trading System and Stability”). * Corresponding author. E-mail addresses: [email protected] (P. Gao), [email protected] (X. Jiang).

https://doi.org/10.1016/j.jacceco.2019.101250 0165-4101/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: Gao, P., Jiang, X., The economic consequences of discrete recognition and continuous measurement, Journal of Accounting and Economics, https://doi.org/10.1016/j.jacceco.2019.101250

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Recognition entails discrete accounting consequences. Consider as an example a long-term asset lease contract in which a firm promises to pay annual installments for a period that accounts for t percent of the useful life of the leased asset. Under the previous standard, the promised future payment is recognized as a lease liability if t exceeds 75 but may remain off the balance sheet if t stays just below 75. When t changes from 74 to 76, the amount of lease liability on the balance sheet jumps disproportionately. In other words, the accounting consequence around t ¼ 75 is discrete. In contrast, the new lease standard (ASC 842) takes a more continuous approach. The recognized lease liability is roughly proportional to t: The same change of t around 75 leads only to a small change in the amount of the lease liability on the balance sheet. Thus, the accounting consequence around t ¼ 75 becomes more continuous. Dye et al. (2014) have argued that recognition suppresses information in two ways. First, it directly excludes information. In the lease example, the promised payment may remain off the balance sheet when t stays just below 75. Second, the discreteness in accounting consequences motivates firms to engage in activities that are designed primarily for financial reporting benefits. We use the broad term evidence management to refer to such activities. Evidence management further degrades the report's informativeness.1 In light of these shortcomings, Dye et al. (2014) recommend that standard setters “adopt a continuous approach” whereby a transaction's expected economic value is recognized without truncation. Consistent with this recommendation, recent accounting standards have been moving towards a more continuous approach, as exemplified in the new lease accounting standard. The deliberation of contingent liabilities is also moving significantly towards this continuous approach. How do we evaluate this shift towards the continuous approach to standard setting? The positive accounting theory (e.g., Watts and Zimmerman (1986, 1990) and Kothari et al. (2010)) cautions us not to abandon long-standing accounting practice without careful economic analysis. In light of the long history and prevalence of discrete recognition in accounting practice, we would like to understand both the discrete and continuous regimes. Is the shift towards a continuous approach more consistent with the stated goal of financial reporting of providing more useful information to investors? How will it affect firms' evidence management? What are the efficiency consequences? We present a model to compare the informativeness and efficiency of a discrete recognition regime and a continuous regime. In the model, the manager engages in evidence management to influence the mapping from the state (a transaction's economics substance) to evidence (a transaction's characteristics), and the accounting system converts evidence to an accounting report. Evidence management reduces the correlation between the state and its accounting evidence, while the reporting process determines how much accounting evidence is communicated in the report. In particular, relative to the continuous regime, discrete recognition excludes some evidence (information) from the report. After receiving the report, the investor makes an investment decision. We define efficiency as a weighted average of the expected payoffs to the manager and to the investor. Our main result is that discrete recognition generates more information and higher efficiency than the continuous regime if and only if evidence management poses a severe threat. If evidence management were absent, then the continuous regime would always be more informative than the discrete recognition regime. If and only if the manager's evidence management cost is sufficiently low, the discrete recognition regime is more efficient than the continuous regime. The intuition is as follows. On one hand, for a given evidence distribution, the discrete recognition regime communicates less information than the continuous regime, as suggested by conventional wisdom. On the other hand, the recognition regime induces less evidence management than the continuous regime. The manager engages in evidence management to influence the report, because the investor's decision is sensitive to the report. All else equal, since the discrete recognition regime reveals less evidence to the investor, it also mitigates the manager's incentive for evidence management. Overall, the report's equilibrium informativeness is jointly determined by the manager's endogenous evidence management and by the accounting regime. Therefore, there is a trade-off between the discrete recognition regime and the continuous regime, leading to our main result. Empirically, our model predicts more evidence management when we move away from the discrete recognition regime and more towards the continuous regime. Note that this prediction is not inconsistent with the compelling evidence on evidence management around recognition thresholds we have discussed in footnote 1. In the discrete recognition regime, evidence management is intensified around the recognition threshold but muted away from the threshold. Its intensity around the threshold makes it more conspicuous and easier to document empirically. However, in the continuous regime, evidence management may arise not only for firms around the threshold but also in other ranges. While individually it might be less noticeable, it could be even more damaging to the report's informativeness in aggregation. Such empirical bias demands careful research design to test our predictions. Our primary contribution is to provide a rationale for the long-standing accounting practice of discrete recognition. Dye et al. (2014) have developed a formal framework to demonstrate that discrete recognition degrades the report's informativeness. Based on this result, their major recommendation is to abandon discrete recognition and “adopt a continuous approach.” Apart from their main argument, they have also pointed out that they might be missing the merits of discrete

1 Evidence management has been empirically documented in the accounting literature, including, among others, Imhoff and Thomas (1988) (for leases), Lys and Vincent (1995) (for merger and acquisition), Engel et al. (1999) (for hybrid securities), and Dechow and Shakespear (2009) (for securitization). It has also long been a major concern for standard setters and regulators. For example, in its report to the Congress on the financial reporting of off-balance sheet arrangements, the first recommendation the Securities and Exchange Commission (SEC) makes is “to discourage transactions and transaction structures primarily motivated by accounting and reporting concerns, rather than economics” (see SEC (2005)).

Please cite this article as: Gao, P., Jiang, X., The economic consequences of discrete recognition and continuous measurement, Journal of Accounting and Economics, https://doi.org/10.1016/j.jacceco.2019.101250

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recognition and made some conjectures. Our paper proposes one merit of discrete recognition and provides a more balanced evaluation of the shift to the continuous approach. Contrary to the main conclusion in Dye et al. (2014), we show that the continuous approach generates more informative financial reporting if and only if the threat of evidence management is not severe.2 Glover (2013) has pointed out another merit of discrete recognition. It facilitates auditing and improves verifiability. Under the discrete recognition approach auditors can focus their attention on the most verifiable features of transactions (around the thresholds). A shift towards the continuous approach can reduce verifiability and increase information asymmetries about verifiability, which can have qualitative impact on the nature of the contracting problem between investors and managers (e.g., Glover et al. (2005)). Our second contribution is to extend the two-step representation of accounting measurement (e.g., Dye (2002) and Gao (2013a)). Motivated by the prevalence of discrete recognition in accounting, Dye (2002) has pioneered a tractable model to study its consequences. While this framework has been adopted by a number of papers to study various issues in standard setting, the literature so far has taken the discrete recognition as given so as to focus on its consequences.3 By specifying the conditions for the optimality of discrete recognition, we lend support to the prior literature that takes the discrete recognition as a starting point. In Section 5 we elaborate the innovations in our model that have made it possible to identify the conditions under which discrete recognition is efficient. Third, our paper also makes a methodological contribution by using integral precision as a criterion of ranking information systems. This powerful criterion was recently developed in Ganuza and Penalva (2010) and first used in accounting by Marinovic (2013). Evidence management does not reduce the report's informativeness in the Blackwell sense, but it does so in the sense of integral precision. Moreover, it is both necessary and sufficient that a more integral precise information system leads to higher decision efficiency. Thus, as far as the decision-making usefulness is concerned, integral precision is a strong enough informativeness criterion. It adds a useful tool to our toolbox to deal with informativeness issues. Finally, our paper is also related to agency theory. The broad economic force in our paper, that controlling a manager's ex ante incentive to engage in evidence management requires the inefficient use of information ex post, is a recurring theme in the agency literature. Discrete recognition, by suppressing information ex post, substitutes for the decision maker's commitment to mute her reaction to accounting reports. As such, this paper could be viewed as an application of the agency theory to accounting standard design. For example, Arya et al. (1997) explore a similar trade-off in studying the optimal design of information system. In a model with both control and decision problems, they show that when the principal cannot commit to the use of information ex post, the optimal information system suppresses some information. The suppression of ex post information serves as a substitute for the principal's lack of commitment. For another example, Chen et al. (2007) study the role of conservatism in dampening the manager's incentive to manipulate earnings. They argue that conservatism mutes the market's reaction to bad news, reduces the manager's benefit from earnings management, and, hence, results in less earnings management. Yet another example is Mittendorf (2010) who studies the role of audit thresholds in the misreporting of private information. The commitment to tolerating misreporting within the materiality threshold makes the threat of punishing egregious misreporting above the threshold more effective. The rest of the paper proceeds as follows. Section 2 describes the model. Section 3 solves the model and characterizes the equilibrium under each regime. Section 4 compares the efficiency of the two regimes and presents our main result. Section 5 explains our model's relation to the prior literature, Section 6 discusses our model's implications and Section 7 concludes. 2. The model The model consists of two components. The first component creates a demand for information in the context of investors' decision making. The second component articulates an accounting measurement process that produces the information. In the baseline model, there are two players, a manager and an investor. For ease of reference, we refer to the manager as “he” and the investor as “she”. When we refer to a random variable, we use “~” to denote the random variable itself and the same notation without “~” to denote its realization. There is also a passive player, labeled as the accountant, who administers the measurement process but does not play any strategic role.4 Each moves once in the following sequences. 1. At date 0; an accounting system is installed to report transactions. We specify the accounting systems in more detail below. 2. At date 1; the manager engages in unobservable evidence management m that influences accounting evidence t.

2 Since our model endogenizes accounting evidence that accounting standards classify, our approach responds to Demski's calls in his presidential address to American Accounting Association (Demski (2004)) for providing micro-foundations of equilibrium expectations in evaluating accounting policies. He argues that “the FASB has […] a penchant for focusing on a type of transaction and then determining the proper accounting treatment of that transaction. It does not […] overly concern itself with the supply of transactions if it proscribes one particular accounting treatment. […] The FASB's Conceptual Framework strikes me as […] being built upon a foundation that sidesteps micro foundations of the underlying choices, and largely inadequate for scholarly purpose.” 3 See, e.g., Fan and Zhang (2012), Gao (2013a, b, 2015), Dye et al. (2014), Laux and Stocken (2018). 4 For a theory about the externality of accounting standards and disclosure regulation, see Dye (1990), Admati and Pfleiderer (2000) and Dye and Sridhar (2008). We also abstract away from the political economy issues in accounting standards setting and refer readers to, among other, Dye and Sunder (2001), Bertomeu and Magee (2011); Bertomeu and Magee (2015a, b), Bertomeu and Cheynel (2013), and Friedman and Heinle (2016).

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3. At date 2, state u is realized but not observable to anyone. The accountant collects accounting evidence t and converts it to report s according to the prevailing accounting system. 4. At date 3; the investor observes report s and makes a decision I: I and u jointly determine the payoffs of the investor and the manager, vðu; IÞ and uðu; IÞ; respectively. ~ ; is either high ðHÞ with probability qH or low ðLÞ with probability qL ≡1  qH ; i:e:; u2 The firm's viability, denoted as u ~ is not observable but measured by an accounting system. Thus, fH; Lg: We normalize H ¼ 1 and L ¼ 0 for simplicity. Viability u ~ the economic substance of a representative transaction or simply the state.5 we also call u The investor makes an investment decision I. The future net cash flow to the investor is

l

v ¼ uI  I 2 : 2

(1)

She benefits from an investment decision better matched to the firm's true viability, as reflected in the interaction term uI. The investment costs 12 lI 2 , with l being a parameter of the adjustment cost. In contrast, the manager's payoff is

u ¼ bI  kKðmÞ:

(2)

bI is the manager's gross payoff from investment, while kKðmÞ is the private cost of evidence management that will be explained later. Since the investor prefers to match the investment to the firm's viability while the manager prefers larger investment, their interest is not fully aligned. The misalignment is captured by the parameter. b > 0: So far we have created a setting in which the investor demands information about state u: The second component of our model articulates a two-step accounting process that generates a report about state u; which can be characterized as follows: ~ Þ    evidence ð~tÞ     report ð~sÞ stateðu |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} evidence management

accounting standards

ðtÞ ~ manifests itself in accounting evidence ~t; with density f u ðtÞ over ½t u ;t u . f u ðtÞ satisfies MLRP, i:e:; ff L ðtÞ State u is increasing in 6 t: Thus, evidence t is informative about the state in the sense of Milgrom (1981). Accounting evidence t is converted into a report, denoted as ~s; according to the prevailing accounting regime. We consider two accounting regimes. In a discrete recognition regime, accounting evidence t is partitioned into two sets by a threshold T; resulting in two reports: H

 s¼

h if t  T : l if t < T

(3)

Threshold T could be arbitrary in our model as long as it is in the common support of f H ðtÞ and f L ðtÞ, i.e. T2 ½t H ; t L  Other cases are trivial. This representation of accounting recognition is simplistic. In practice, recognition is followed by measurement and thus the report ~s is less discrete. We consider an alternative form of the recognition regime in Section 4.3. We also consider a continuous regime in which the accounting evidence is directly communicated to the investor without truncation. Thus, the resulting report is

s ¼ t; for any t:

(4)

Our main research question is to compare the discrete recognition regime with the continuous regime. We denote the regime using a subscript i2fC;Dg, where C denotes the continuous regime and D denotes the discrete recognition regime. We omit the subscript i whenever no confusion arises. The key friction in the accounting process is that the manager can take costly actions to influence the evidence distribution. Specifically, before observing u; the manager can choose manipulation intensity m2½0; 1 that changes the evidence distribution from f u ðtÞ to H u fu m ðtÞ ¼ mf ðtÞ þ ð1  mÞf ðtÞ:

(5)

5 Note that binary states do not imply that the optimal accounting rule generates binary signals because the decision is continuous. As we show later, continuous signals dominate binary ones when there is no manipulation. 6 ~ Dye (2002) standardizes the accounting evidence by introducing the probability of the state 4; which can be recovered from the distribution of state u fG and accounting evidence ~t by the Bayes rule, that is, 4≡Prðu ¼ HjtÞ ¼ q fqGHþq . fB H

L

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u If m ¼ 0; the evidence distribution is not affected by evidence management, i:e:; f u m¼0 ðtÞ ¼ f ðtÞ: If m ¼ 1; then the H manager always receives accounting evidence that is not distinguishable from a high state, i:e:; f u m¼1 ðtÞ ¼ f ðtÞ: If m2 ð0; 1Þ; evidence management improves the evidence distribution in the sense of first-order stochastic dominance.7 This ex ante evidence management technology enables us to compare the two regimes in a tractable manner. We postpone the discussions and justification of this assumption to Section 5 after we have explained how our model works. Given an accounting regime i; evidence management affects the final report ~s: We denote its probability density function 8 (when ~s is continuous) or probability function (when ~s is discrete) as g u m ðsÞ for realization s. Evidence management m costs the manager kKðmÞ privately. KðmÞ has the standard properties of a cost function. It is 0

increasing and convex with Kð0Þ ¼ K ð0Þ ¼ 0. In addition, we assume that equivalently, that K 0 ðmÞ is weakly log-concave and that

lim KðmÞ 0 m/0 K ðmÞ

0

K ðmÞ K 00 ðmÞ

is weakly increasing in m on ð0; 1Þ, or,

¼ 0. The conditions are satisfied for common convex

functions used in the literature (see Bagnoli and Bergstrom (2005)), e.g. KðmÞ ¼ mn for n  2. k > 0 is a cost parameter that will be used for comparative statics later. In sum, report s is a function of evidence t whose distribution is determined by state u and evidence management m: Thus, the report's equilibrium informativeness with respect to the state is affected by both the manager's evidence management and the accounting regime. Evidence management influences the mapping from state u to evidence t; and the accounting regime shapes the mapping from evidence t to report s: One example that fits our model is revenue recognition. Consider a sales transaction in which all criteria for revenue recognition have been satisfied except the uncertainty about collectability. Suppose the customer pays either the whole amount $1 or nothing. Thus, the state is the collectability, i:e: u2f1; 0g: The firm has various information t about the collectability u; such as the customer's payment history, credit scores and general business prospect. The manager may take some costly actions to influence these data. Conditional on the evidence, the firm's belief is pðtÞ ¼ E½ujt: Because of MLRP, pðtÞ is informationally equivalent to t. In a discrete recognition regime, the firm recognizes revenue if and only if the expected collectability exceeds a threshold. In other words, it reports r ¼ 1 (or r ¼ p) if t  T and r ¼ 0 if t < T: In a continuous regime, the firm recognizes revenue r ¼ pðtÞ regardless of its magnitude. Note that investors know the existence of the transaction through the change in inventory, regardless of whether revenue is recognized or not. The recent change in revenue recognition standards from ASC 605 to ASC 606 resembles the movement from the discrete recognition regime to the continuous regime. The solution concept is Perfect Bayesian Equilibrium (PBE). A PBE consists of the manager's evidence management m and the investor's investment strategy I  ðsÞ such that, for any given regime, 1. the manager chooses m to maximize U ¼ E½uðmÞ; 2. the investor chooses I  ðsÞ to maximize V ¼ E½vðIÞjs; 3. the investor has rational expectations about the manager's evidence management.

3. The equilibrium For a given regime, we solve the model by backward induction. We first solve for the investor's investment response given her conjecture about the manager's manipulation. We then solve for the manager's manipulation decision and finally impose the rational expectations requirement to obtain the entire equilibrium. We first analyze the investor's decision at date 3 after she receives report s. Since she does not observe the manager's b Based on this conjecture, she chooses investment I to manipulation, she conjectures that the manager has chosen m: b Denote I BR as her best investment response to report s: The first-order condition maximize her expected payoff Eu ½vðIÞjs; m results in

  1 b ¼ E½ujs; m b ¼ arg max Eu vðIÞjs; m b : I BR ðs; mÞ I

l

(6)

The second-order condition is satisfied due to the concavity of the investment return function vðIÞ. The investor's best investment response characterized in equation (6) is intuitive. The investor invests more if the investment adjustment cost l is lower or if her posterior belief about the state is higher.

7 The technology implies that evidence management does not improve the evidence distribution in the good state, that is, there is no “good evidence management.” If evidence management improves the evidence distributions in both states, our main results are qualitatively the same provided that the improvement is larger in the bad state than in the good state. 8 The accounting system measures a firm's existing transactions and past events, while the report in our model is useful for making future investment decisions. What is implicit in our model is that the firm's profitability (state) in the past and in the future are correlated. Therefore, the report from the accounting system that measures the firm's past transactions is also useful for investors to make new investment. For example, investors use income statement to make new investment decisions. To economize on notation, we have assumed that they are perfectly correlated.

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b at date 3 and chooses m to maximize his At date 1 the manager anticipates the investor's investment response I BR ðs; mÞ expected payoff E½u: With manipulation m; the manager expects to receive evidence distribution f u m ðtÞ in equation (5) and its BR b corresponding report distribution g u m ðsÞ:The investor then responds to report s with investment I ðs; mÞ: Therefore, the b is manager's expected payoff with manipulation m when the investor conjectures m

b ≡ bEu Uðm; mÞ

Z

 b u I BR ðs; mÞg m ðsÞds  kKðmÞ:

(7)

Since the investor does not observe the manager's actual choice of m, her investment decision does not directly respond to b is independent of m: Instead, evidence management affects the manager's expected payoff only m: In other words, I BR ðs; mÞ through its effect on the report distribution. g u m ðsÞ: e We now explain how g u m ðsÞ is determined. In the continuous regime, sC ¼ t and thus report sC has the same distribution as the evidence. This results in

  u H u ; for any s : gu C m ðsC Þ ¼ f þ m f  f

(8)

Under the discrete recognition regime, report seD has a binary distribution, i:e:; sD 2 fh; lg: We can write it out as:

n o  8Z > f u þ m f H  f u dt if sD ¼ l > > < Zt < T n  o  : gu m ðsD Þ ¼ > f u þ m f H  f u dt if sD ¼ h > > tT : 0 otherwise

(9)

To highlight the effects of evidence management on the manager's expected payoff, we define the manager's expected b as payoff in state u when he does not manipulate (but the investor still conjectures m)

Z

b ≡ Uð0; mÞ b ¼ b I BR ðs; mÞg b u ðsÞds: Pu ð mÞ

(10)

b in equation (7) as We can rewrite Uðm; mÞ

  b ¼ qH PH ð mÞ b þ qL PL ð mÞ b þ qL m PH ð mÞ b  PL ð mÞ b  kKðmÞ: Uðm; mÞ

(11)

b Equation (11) is intuitive. In the absence of evidence management, the manager's base expected payoff is qH PH ð mÞþ b Evidence management m improves the distribution of evidence from g L ðsÞ to g H ðsÞ with probability m in the low qL PL ð mÞ. b PL ð mÞÞ b at the cost of. kKðmÞ: state, increasing the manager's expected payoff by qL mðPH ð mÞ Substituting the expressions of the distribution of report ~s; g u m ðsÞ from expression (8) and (9), into the manager's objective b in equation (11) and taking the first-order condition of m, we obtain the manager's best evidence manfunction Uðm; mÞ b mBR ð mÞ; b as agement response (to the investor's conjecture of m);

    b  PL ð mÞ b  kK 0 mBR ð mÞ b ¼ 0: qL PH ð mÞ

(12)

The second-order condition is satisfied because of the convexity of the cost function KðmÞ: Equation (12) explains the b the marginal benefit of manipulation is manager's evidence management decision. Given the investor's conjecture m; b PL ð mÞÞ; b qL ðPH ð mÞ the manager's expected payoff difference with and without a favorable evidence distribution in the low b state. The manager trades off this marginal benefit against the marginal cost of kK 0 to determine the best response mBR ð mÞ: Finally, the investor has rational expectations about the manager's evidence management. Even though she does not observe evidence management m; the investor's conjecture is consistent with the manager's actual choice in equilibrium. b ¼ m: b As we show in the proof, Thus, we obtain the equilibrium by solving the fixed point problem (equation (12)) of mBR ð mÞ this equilibrium has a unique solution and we denote it as m . Evaluating the investor's investment best response in equation (6) at the equilibrium evidence management level m ; we b b  : The equilibrium ðI  ; m Þ is summarized below. obtain the equilibrium investment function I  ðs; m Þ ¼ I BR ðs; mÞj m ¼m Proposition 1. The equilibrium consists of the manager's evidence management m and the investor's investment decision I  ðsÞ. The equilibrium investment decision I  ðsÞ is characterized by

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1 I  ðsÞ ¼ E½ujs; m :

l

7

(13)

The unique interior equilibrium evidence management m is characterized by the first-order condition

  qL PH ðm Þ  PL ðm Þ  kK 0 ðm Þ ¼ 0:

(14)

m increases in b and decreases in k and l.We can now verify the impossibility theorem in Dye et al. (2014), that is, m ¼ 0 cannot be an equilibrium. If the investor believes that the manager does not manipulate and thus takes the report at its face value, then the marginal benefit of manipulation is strictly positive. Since the marginal cost of manipulation is 0 at m ¼ 0; the manager will deviate by engaging in some positive amount of evidence management.9 The determinants of the interior m are intuitive. b measures the misalignment of interest between the manager and the investor. An increase in b exacerbates the manager's conflict of interest with the investor and thus induces the manager to engage in more evidence management. A higher k indicates a higher marginal cost of evidence management and thus leads to lower evidence management. Finally, l is the adjustment cost of the investment. It reduces the investor's response to the report. As we can see from the equilibrium investment decision (equation (13)), l scales down the investment level I  for any given report s: Because the manager's manipulation is driven by the investor's response to the report, the equilibrium manipulation is decreasing in adjustment cost l: 4. The comparison of the two accounting systems Having solved the equilibria, we can compare them under the two regimes. For a given regime, we have obtained a unique equilibrium characterized by m and I  ðsÞ. We first examine the report's equilibrium informational properties in the two regimes. These properties are the key building blocks to prove the subsequent results. We then present the main results about the efficiencies of the two regimes. We finish this section with an alternative characterization of the recognition regime. 4.1. The informational properties of the reports To compare the informativeness of reports from each regime, we need criteria for informativeness. The classic criterion is Blackwell informativeness (Blackwell (1953)). Definition 1. X and Y are two signals about state u with respective p.d.f. (or p.f.) fX ðxjuÞ and fY ðyjuÞ: X is more Blackwell informative than Y if and only if there exists a conditional density nðyjxÞ such that

fY ðyjuÞ ¼

Z

nðyjxÞfX ðxjuÞdx:

suppðXÞ

Definition 1 provides a formal expression to capture the underlying intuition that X is more Blackwell informative than Y if Y is a garbling of X with respect to the underlying state u. Blackwell informativeness is a strong criterion and cannot be used to rank all information systems (e.g., Demski (1973)). In our model, an intuitive indicator of the report's informativeness is evidence management. As m increases, the report becomes less informative under each regime. However, the report with lower equilibrium evidence management is not Blackwell more informative than its counterpart with higher equilibrium evidence management. Fortunately, the economics literature has developed an alternative criterion of informativeness, known as integral precision. It was developed by Ganuza and Penalva (2010) in the context of ranking information systems the seller provides to buyers in an auction setting. Marinovic (2013) applies the notion to study accounting quality. Definition 2. X and Y are two signals about state u: X is more integral precise than Y if and only if E½ujY second-order stochastic dominates E½ujX: Integral precision has an intuitive appeal. Information is useful because it moves the posterior belief away from the prior. The more informative a report is, the more dispersed the posterior belief distribution is. Second-order stochastic dominance is a natural measure of the dispersion of a distribution. Our definition of integral precision is equivalent to Definition 2.(ii) in Ganuza and Penalva (2010) as their notion of “convex order” is equivalent to our notion of second-order stochastic dominance.

9 All four conditions for the impossibility theorem in Dye et al. (2014) are satisfied in our model. In particular, the manager has reasonable preference and the standards are typical because the equilibrium investment I ðsÞ is increasing in the report. The continuous evidence t satisfies condition 3 and the continuous cost function with K 0 ð0Þ ¼ 0 satisfies condition 4. Thus, evidence management arises in equilibrium, that is, m > 0.

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Integral precision is a weaker criterion than Blackwell informativeness and thus can rank more information systems. If X is more Blackwell informative than Y; then X is also more integral precise than Y; but the converse is not true. For example, suppose X ¼ q þ ε1 and Y ¼ q þ ε2 where qeNðm; s2q Þ, ε1 eNð0; s21 Þ and ε2 eNð0; s22 Þ, q, ε1 and ε2 independent of each other and s21 < s22 . It can be shown that X is more integral precise than Y but X is not more Blackwell informative than Y. Equipped with these definitions, we can study the equilibrium report's properties. The equilibrium report s is a function of both the regime i2fC; Dg and the equilibrium evidence management m : Lemma 1. 1. The investor's average posterior belief about state u is equal to the prior: E½E½ujsi ; m  ¼ qH ; for any m and i 2 fC; Dg: 2. Given an evidence distribution, report seC is Blackwell more informative than seD : 3. Given a reporting regime i 2 fC; Dg, report sei ðm Þ is more integral precise as manipulation m decreases.Part 1 of Lemma 1 is a reminder of the investor's rational expectations requirement. Even though the investor does not observe manipulation m; her conjecture is consistent with the manager's actual choice in equilibrium. Conditional on the realizations of the reports, the investor's posterior belief about the state will be updated to E½ujsi ; m  However, on average E½ujsi ; m  is expected to be equal to qH : In other words, the rational investor is not systematically misled by evidence management. Part 2 of Lemma 1 confirms the conventional wisdom that recognition suppresses evidence. Given an evidence distribution, the report under the continuous regime reveals all available evidence through sC ¼ t: In contrast, the report under the recognition regime garbles the available evidence through the recognition rule characterized in equation (3). Thus, the discrete recognition regime suppresses evidence and is less informative even under the strongest criterion of Blackwell informativeness. Part 3 of Lemma 1 is an intuitive but important result. It states that evidence management reduces the report's informativeness (in the sense of integral precision) under both regimes. The purpose of financial reporting is to help the investor to differentiate the high state from the low, but evidence management makes it more likely that different states may generate the same reports. Since the investor rationally anticipates this contamination, she responds less to the report when the equilibrium manipulation is higher. 4.2. The main result Now we are ready to present our main result. In each equilibrium, we can measure the ex ante payoffs to the investor and the manager. At date 0; the investor's ex ante expected payoff is

h  i l h 2  V ≡ E v I  ¼ Es I  2

(15)

V is also the investment efficiency as the investor relies on accounting reports to make investment decisions. The higher the investment efficiency, the higher the investor's expected payoff. The manager's ex ante expected payoff is

U ≡ Uðm ; m Þ ¼ E½uðu; I  Þ ¼ bEs ½I  ðsÞ  kKðm Þ

(16)

U is increasing in the expected level of equilibrium investment I  and decreasing in manipulation cost. We define the efficiency in terms of the weighted average of both the investor's and the manager's ex ante expected payoffs:

W ¼ aV þ ð1  aÞU; with a 2 ð0; 1 By allowing a ¼ 1; we include the extreme case that focuses only on the investor's ex ante expected payoff. For reasons we will explain later, both the investor's expected payoff and the efficiency go in the same direction as far as the comparison of the two regimes are concerned. Therefore, it is inconsequential which outcome variable we focus on. For completeness, we focus on the efficiency W: Our main result is the following theorem. Theorem 1. There exists a unique k such that the discrete recognition regime is more efficient than the continuous regime (i:e:; WD > WC ) if and only if k < k. k decreases in l and a and increases in b. Theorem 1 states that the efficiency could be higher in the discrete recognition regime than in the continuous regime. Moreover, the discrete recognition regime is more efficient if and only if the threat of evidence management is sufficiently high, i.e. k is sufficiently low. In other words, it is precisely the threat of evidence management that makes the discrete recognition regime more efficient than the continuous regime. For the rest of this subsection, we explain the proof of the theorem in four steps. First, given the same evidence distribution, the continuous regime is more efficient than the discrete recognition regime. Second, evidence management reduces the efficiency in both regimes. Third, the equilibrium evidence management is higher in the continuous regime than in the Please cite this article as: Gao, P., Jiang, X., The economic consequences of discrete recognition and continuous measurement, Journal of Accounting and Economics, https://doi.org/10.1016/j.jacceco.2019.101250

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discrete recognition regime. As a result of these three steps, there is a trade-off between the two regimes. The discrete recognition regime suppresses evidence but induces less evidence management. The last step is to show that this trade-off is well-behaved and generates the partition by k presented in Theorem 1. Proposition 2. For the same evidence distribution, the efficiency is higher in the continuous regime than in the discrete recognition regime. That is, for any m 2 ½0; 1Þ, WC ðm Þ > WD ðm Þ: This is an out-of-equilibrium comparison. The implication is that if we were to impose the same m across regimes, then the efficiency would be higher in the continuous regime. We first demonstrate that the manager's expected payoff U are the same in the two regimes when evidence management levels are equal. When the evidence management levels are the same across the two regimes, the evidence distributions are the same. The manager benefits from the expected level of investment Es ½I  ðsÞ and incurs the cost of evidence management kKðm Þ: When the evidence management levels are the same, the cost of evidence management is the same across the two regimes. Moreover, since investment I  is linear in the investor's conditional expectation of the state (equation (13) from Proposition 1), part 1 of Lemma 1 implies that the expected level of investment is the same across the two regimes as well. Since evidence management does not systematically mislead the investor's belief about the state, it does not systematically mislead the investor's investment decision either. Therefore, the manager's expected payoff is the same across the two regimes when m is the same. We then compare the investment efficiency V across the two regimes. The expression of V in equation (15) suggests that the investment efficiency is convex in the equilibrium investment level I  and thus convex in the investor's posterior belief E½uj~s (due to equation (13) from Proposition 1). This is intuitive. More efficient investment decision implies matching the investment decision more accurately to the state, that is, to invest more (less) in the high (low) state. Thus, the convexity simply means that the investment decision benefits from a more informative report (which induces a more volatile posterior belief distribution). By part 2 of Lemma 1, we know that, for the same evidence distribution, the continuous regime generates a more informative report and thus higher investment efficiency than the discrete recognition regime. Therefore, if evidence management is the same in the two regimes (which implies the same evidence distribution), the efficiency is higher in the continuous regime than in the discrete recognition regime. Specifically, the manager's expected payoffs are the same while the investment efficiency is higher in the continuous regime. Proposition 3.

The efficiency is decreasing in evidence management in both regimes. That is,

vWi vm

< 0; for i2fC; Dg.

This again is an out-of-equilibrium comparison. We again start with the manager's expected payoff U. We have shown earlier that the expected level of investment is independent of evidence management. Moreover, the cost of evidence management is increasing in evidence management. Thus, the manager's expected payoff is decreasing in evidence management. Even though the manager finds it optimal to engage in evidence management, the investor rationally anticipates the manipulation and makes proper adjustment when interpreting the reports. As a result, the manager is stuck in this “bad” equilibrium, in which he engages in costly evidence management without systematically misleading the investor. Now we turn to the investment efficiency. We have shown that V is convex in the posterior belief E½ujs (combining equation (13) and equation (15)). Part 3 of Lemma 1 shows that evidence management reduces the report's informativeness (in the sense of integral precision) in both regimes. Thus, evidence management reduces the investment efficiency. Therefore, the efficiency in each regime is decreasing in evidence management. Specifically, both the manager's and the investors' expected payoffs are decreasing in evidence management. Proposition 4. The equilibrium evidence management is higher in the continuous regime than in the discrete recognition regime. That is, mC > mD . Proposition 4 is perhaps surprising at first glance, but it is a natural consequence of Proposition 2. Recall that the manager's equilibrium evidence management is determined by equation (14). The manager engages in evidence management precisely because the investor is sensitive to the report, as captured by the term PH ðm Þ  PL ðm Þ. The manager manipulates more when the investor is more responsive to the report. For given evidence distribution, part 2 of Lemma 1 shows that the report from the continuous regime is more informative than that in the discrete recognition regime, and Proposition 2 further shows that the investor's investment decision is more responsive to the report from the continuous regime. Therefore, the manager's incentive to manipulate is stronger as well. In other words, evidence management is more rampant in the continuous regime precisely because the continuous regime is more informative for a given evidence distribution. Propositions 2, 3 and 4 together create a trade-off between the two regimes. Fixing the evidence distribution, the continuous regime generates a more informative report (Proposition 2). This induces the manager to manipulate more, which in turn reduces the report's informativeness and the investor's response to the report (Proposition 3). In equilibrium, the continuous regime induces more evidence management (Proposition 4). Theorem 1 demonstrates that this trade-off generates an interior threshold k that partitions the efficiency of the two regimes. To see the intuition, we could consider the two extremes. At one extreme when evidence management is absent, the continuous regime is more efficient, i:e:; WC ð0Þ > WD ð0Þ: At the other extreme, suppose mC ¼ 1 > mD (which occurs with k properly chosen). The report is completely uninformative under the continuous regime while still informative under the discrete recognition regime. Then we have WC ð1Þ < WD ðmD Þ: Therefore, there exist situations in which the continuous regime could be less efficient than the discrete recognition regime. The last step to prove Theorem 1 is to show that the trade-off is

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well-behaved, that is, WC ðkÞ ¼ WD ðkÞ has a unique solution k. This part is technical and thus delegated to the appendix. We therefore have proved Theorem 1. Theorem 1 holds for any non-trivial threshold T. This robustness highlights our main economic force that the relative efficiency of the discrete recognition regime is driven by the threat of manipulation. It does not rely on the optimal design of the threshold in the discrete recognition regime. Nonetheless, it would be interesting to endogenize the threshold and recharacterize the comparison of the two regimes. Such an extension is unfortunately intractable in our setting mainly because the optimal threshold would be determined only by an implicit function (as do the manipulation choices in both regimes). The optimal design of the threshold in the recognition regime has been studied in different settings in the literature (e.g., Dye (2002), Laux and Stocken (2018) and Gao (2015)). The determinants of the threshold k in Theorem 1 further corroborate the main message that discrete recognition is more efficient only when evidence management is a severe threat. A larger k indicates that the recognition regime is more likely to dominate the continuous regime in efficiency. As we have discussed following Proposition 1, when the interest misalignment b is larger or when the adjustment cost l is lower, the equilibrium evidence management is more severe. As a result, k increases. 4.3. Semi-discrete recognition In our baseline model we have compared the continuous regime with the regime where the recognition is modelled as a discrete classification. While discrete classifications have been used in the literature (e.g. Dye (2002); Gao (2013a); Laux and Stocken (2018)), the accounting practice of recognition is often more complicated in reality. For example, in a typical revenue recognition transaction, if the revenue is not recognized, the firm reports 0. However, if the revenue is recognized, then the measurement process follows and the expected revenue is reported. Thus, the report is partially binary and partially continuous. We study this alternative characterization. In particular, we modify equation (3) as follows

 s¼

t if t  T : l if t < T

(17)

We show that our main results about the comparison of the two regimes are qualitatively the same. Theorem 2. There exists a unique kalt such that the semi-discrete recognition regime is more efficient than the continuous regime (i:e:; WD > WC ) if and only if k < kalt . kalt decreases in l and increases with b. The reason that Theorem 2 is qualitatively the same as Theorem 1 is because the driving force of our main result, that suppression of evidence also mitigates the manager's evidence management, still survives in this alternative characterization of semi-discrete recognition. Since the proofs are qualitatively the same, they are omitted here but available upon request. 5. The ex ante manipulation technology Having explained the tractable trade-off between the discrete recognition and continuous regimes, we now further discuss our model's relation to the prior literature. We model evidence management as an ex ante choice to apply the informativeness criterion of integral precision. The manager chooses m without knowing u or initial evidence t. Conceptually, evidence management is consequential both because it consumes resources and because it may degrade the report's informativeness and reduce the efficiency of decisions made on the basis of the report. The ex ante manipulation technology enables us to model both consequences in a tractable manner. It has four benefits. First, the ex ante manipulation technology has an intuitive property that, ceteris paribus, the report's informativeness is monotonically decreasing in the equilibrium level of evidence management (Lemma 1). Second, it negates the need for aggregating ex post evidence management since the manipulation choice is a scaler to start with. Third, it accommodates general distributions of both the signal and the impact of the manipulation. Finally, it has an additional benefit of avoiding the tricky issues associated with ex post signaling. Dye et al. (2014) have discussed the major difficulty of evaluating the efficiency in the prior literature that uses ex post evidence management: the manager chooses the amount of evidence management after observing the initial realizations of accounting evidence. To evaluate the standards ex ante, one has to aggregate the consequences of the ex post evidence management. This task is complicated and creates tractability issues. Specifically, there are two problems. First, the aggregation of ex post evidence management in general is not technically tractable. Mathematically, the aggregate evidence R management, m ðtÞdFðtÞ; depends on the functional forms of both m ðtÞ and FðtÞ, and a tractable form is only available for special functions. For example, Dye (2002) focuses on uniform distributions for FðtÞ: Second, even if a tractable aggregator of ex post evidence management levels is available, it is not necessarily a proper measure of the informational consequences of evidence management. Specifically, evidence management induces accounting evidence distributions that are typically not Blackwell comparable. In other words, an increase in evidence management does not make the accounting evidence distribution less informative in the Blackwell sense. In the absence of a proper aggregator for the consequences of evidence management, the comparison across the two regimes becomes intractable. Evidence management in practice can occur in all stages of a transaction, both before and after the transaction's economic substance becomes clear to the manager. Thus, both ex ante and ex post evidence management are probably descriptive. Moreover, while it is technically convenient to assume that the manager chooses evidence management before receiving private Please cite this article as: Gao, P., Jiang, X., The economic consequences of discrete recognition and continuous measurement, Journal of Accounting and Economics, https://doi.org/10.1016/j.jacceco.2019.101250

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information, what is essential for the ex ante evidence management modeling choice is that the manager does not have full control over the impact of evidence management on the final report. This feature implies that receivers cannot fully back out the impact of evidence management from only observing the report, leading to the result that evidence management reduces the report's informativeness. The manager's residual uncertainty about the impact of his evidence management choices on the final report can result from various channels. For example, after evidence management, the auditor may challenge the manager's accounting choices and force the manager to make adjustments to the final report. For another example, some evidence management choices are made before the transaction is finalized and thus their effects may interact with the realization of the transaction's other characteristics, which may be outside the manager's control. Bird et al. (2016) provide strong empirical evidence that managers face inherent uncertainty about how their accounting choices affect the final earnings. Our paper is also related to Dye and Sridhar (2004) who study the consequences of aggregation. While aggregation reduces the amount of information communicated to investors, it also mitigates the manager's manipulation and saves the manipulation cost. Our discrete recognition regime may be interpreted as aggregation to the extent that discrete recognition in our model suppresses information. There are several differences between the two papers. First, we study a very specific form of suppressing information (i.e. information is suppressed because of discrete recognition), which can be related to specific accounting measurement rules (e.g. revenue recognition and lease accounting examples as discussed before). In contrast, in Dye and Sridhar (2004), the suppression comes from summing a hard signal and a soft signal that are not related to specific accounting measurements. Second, Dye and Sridhar (2004) use an ex post manipulation technology whereas we adopt an ex ante manipulation technology. Their comparison has been elaborated earlier in this section. Finally, Dye and Sridhar (2004) study a capital market setting where the manager chooses investment to maximize short-term prices whereas we study a decision-making setting to focus on the informational consequences of manipulation. 6. The empirical and policy implications The model generates straightforward empirical implications. Accounting standards that adopt a more continuous approach always generate more evidence management. In addition, the standards generate less informative reports and less efficient investment decisions if and only if the transaction characteristics are sufficiently vulnerable to managerial manipulation. Moreover, the standards are more likely to result in these adverse effects if the manager's manipulation incentive is higher or if the investment decisions are more sensitive to accounting reports (after controlling for the reports' informativeness). Our results provide several policy implications. First, our model formalizes one benefit of recognition (above and beyond measurement) and provides a counter argument to the tidal move towards a more continuous approach. This benefit is consistent with the prevalent use of recognition in extant accounting standards. We warn that a more continuous approach is desirable only for transactions that are robust to managerial manipulation. Second, our comparative statics about the determinants of the relative efficiency of the two regimes provide direct implications. We expect to see that discrete recognition is more likely to prevail when the investment adjustment cost is lower or when the interest misalignment between the manager and the investor is more severe. Finally, our results also have policy implications for the debate on fair value accounting.10 While the debate is multidimensional, one aspect of the fair value accounting resembles the continuous regime in our model. For example, a fair value approach to contingent liability moves us towards the continuous regime in our model. Proponents of fair value accounting often hold the view that fair value accounting, by providing more information, is more value relevant and helps investors to make better decisions (e.g. Barth (1994), Barth et al. (1996), and Barth et al. (2008). See Barth et al. (2001) for a

Fig. 1. A graphical illustration of the proof of Theorem 1.

10 We use the term “fair value” accounting to be consistent with the literature. However, the term is not appropriate for the debate, as explained in Sunder (2008). “Current value” accounting or “exit value” accounting might be more appropriate.

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review.) Opponents, however, argue that there is plenty of managerial discretion embedded in fair value estimates and managers will exploit it for their own benefits (e.g. Dietrich et al. (2000), Dechow et al. (2010). See Kothari et al. (2010) for a review). We show that the equilibrium informativeness of fair value accounting is conditional: it provides more information if and only if there is little room for managers to exploit the discretion. 7. Conclusions The paper aims to explain a puzzle about recognition. Discrete recognition is ubiquitous in accounting practice, even though it seems to degrade the report's informativeness. Not only does recognition directly suppress information, it also induces evidence management that further degrades the report's informativeness. We develop a model to compare a discrete recognition regime with a continuous regime. We show that the discrete recognition regime is more efficient than the continuous regime if and only if the threat of evidence management is sufficiently severe. As such, we provide a rationale for the observed prevalence of discrete recognition in accounting practice and offer a balanced view on the shift towards a more continuous approach in standard setting. Our model introduces an ex ante evidence management technology into the two-step representation of accounting measurement, whose implications for studying standard setting are elaborated in Gao (2013b). The ex ante evidence management technology provides additional desirable features for modeling accounting standard setting. It overcomes the issue of aggregating ex post manipulation from an ex ante perspective and provides a single measure of managerial influence on the accounting process. It delivers the intuitive property that manipulation decreases the report's informativeness. This technology could be exploited in other settings in future research. Our model has a number of limitations that future research can address. First, we have modeled the accounting process in a restrictive manner. In practice, recognition is followed by measurement. Recognition determines whether or not to admit a transaction or event into the accounting system. Conditional on recognition, measurement determines the amount of the elements by choosing a measurement attribute (such as historical cost, net realizable value and fair value (SFAC 5)). In our model, recognition and measurement are combined as a joint process that converts evidence to accounting reports. In a richer model, recognition and measurement could be separated and their different combinations could be examined. Second, the model has been confined to the comparison of two regimes with and without binary recognition. It does not address the issue of the optimal design of recognition. While binary recognition seems consistent with accounting practice, there is no obvious reason to believe that it is optimal among other possible forms. We leave the design of the optimal recognition regime to future research. The novel approach of modeling information design in the Bayesian persuasion literature (e.g., Kamenica and Gentzkow (2011); Gentzkow and Kamenica (2016); Bergemann and Morris (2016)) can be very useful in studying the optimal shape of the recognition rules. The literature has so far focused on the design problem from the sender's perspective, but the receiver (investors) has the right to design the measurement rules in our setting. Thus the design problem has to be shifted from the sender's perspective to the receiver's perspective. Third, we have assumed in the model that accounting reports are unambiguously valuable for the investor's decisions (the demand side) and have instead focused on how the production of accounting reports is constrained by evidence management (the supply side). Recognition arises as an equilibrium response to evidence management. In contrast, there has been a large literature on biased performance measure and earnings management (see our literature review in Introduction). Most of this literature focuses on how the use of an accounting report is affected by earnings management. It would be desirable to better integrate the demand for and the supply of accounting information in the same model, which might generate new insights into the institutional features of accounting measurement. Finally, discrete classifications are observed in many areas outside accounting. Professors often give students discrete letter grades instead of continuous scores (e.g., Dubey and Geanakoplos (2010); Harbaugh and Rasmusen (2018)), and rating agencies issue discrete ratings instead of continuous default probabilities (e.g., Goel and Thakor (2015)). Future research may compare and contrast these phenomenon with accounting recognition to better understand them. Appendix For convenience of exposition in the proof we will refer to E½ujs; m as gðs; mÞ, i.e.

gðs; mÞ ≡ E½ujs; m ¼ Prðu ¼ 1js; mÞ ¼ ¼

g H ðsÞqH gm ðsÞ

(18) gH ðsÞqH H

ðqH þ qL mÞg ðsÞ þ qL ð1  mÞg L ðsÞ

where gm ðsÞ is the unconditional distribution of the report s when the manager chooses m; i:e: gm ðsÞ ¼ ðqH þqL mÞg H ðsÞþ qL ð1  mÞg L ðsÞ: Please cite this article as: Gao, P., Jiang, X., The economic consequences of discrete recognition and continuous measurement, Journal of Accounting and Economics, https://doi.org/10.1016/j.jacceco.2019.101250

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Proof of Proposition 1: Proof. The existence of the equilibrium has been proved in the text. Here we prove that the pair ðI  ; m Þ is unique. Denote the left hand side of equation (14) as





Uðm Þ ≡ qL PH ðm Þ  PL ðm Þ  kK 0 ðm Þ:

(19)

If m is interior, then its uniqueness is proved by the monotonicity of Uðm Þ:

vUðm Þ vm

b ¼ qL l

Z

v

h

i

gðs; m Þ gH ðsÞ  gL ðsÞ ds vm

00

 kK ðm Þ

h i2 H L Z H qL b g ðsÞ g ðsÞ  g ðsÞ qH qL 00 ds  kK ðm Þ ¼  2 l gðs; m Þ < 0: Now we have proved the uniqueness of m ; we can use the implicit function theorem to conduct the comparative statics.

vm vU=vb vU ¼ f vb vU=vm vb Z h i 1 ¼ gðs; m Þ gH ðsÞ  gL ðsÞ ds

l

> 0: Similarly,

vm vU f ¼ K 0 ðm Þ < 0; vk vk Z h i vm vU b f ¼  2 gðs; m Þ g H ðsÞ  g L ðsÞ ds < 0: vl vl l Thus, we have proved that the interior m is increasing in b and decreasing in k and. l: Proof of Lemma 1: Proof. We prove the three informational properties one by one. Part 1 is obtained immediately by the law of iterated expectations. We next verify Part 2 that s C Blackwell dominates sD according  to RDefinition 1. R   Given any gðsC juÞ, gðsD ¼ 1uÞ ¼ sC T gðsC juÞdsC and gðsD ¼ 0uÞ ¼ sC < T gðsC juÞdsC . We can define a new distribution n :



nðsD ¼ 1jsC Þ ¼

1 if sC  T 0 if sC < T

and

 nðsD ¼ 0jsC Þ ¼

1 if sC < T : 0 if sC  T

R Then we have gðsD juÞ ¼ nðsD jsC ÞgðsC juÞdsC . Thus by Definition 1, sC Blackwell dominates sD . Finally we prove Part 3 that manipulation reduces informativeness in the sense of integral precision as defined in Definition 2. For ease of exposition, we drop the index of regime i2C; D as the comparison is within a given regime. We also replace equilibrium manipulation mi with a generic m as it is a parameter of the report distribution ~s: Moreover, we use GX to denote the CDF of the random variable of. X: The methodology developed in Ganuza and Penalva (2010) to compare two signals in terms of integral precision consists of two steps. First, apply the probability integral transform to the original report ~sðmÞ to generate the transformed signal YðmÞ:

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Second, Ganuza and Penalva (2010) proves in Proposition 3.(i) (page 1014) that ~sðm1 Þ is more integral precise than ~sðm2 Þ if and only if

GYjH ðy; m1 Þ  GYjH ðy; m2 Þ:

(20)

We use this methodology to prove Part 3. We start with the continuous regime. The original report is ~sðmÞ. We define a new random variable Y as

YðmÞ ¼ G~s ð~s; mÞ:

(21)

Y is the transformed signal after applying the probability integral transform to the original report ~s. Since the distribution of ~s is affected by m; so is Y: To prove equation (20), we write out GYjH as

GYjH ðy; mÞ≡PrðYðmÞ  yju ¼ HÞ ¼ PrðG~s ð~s; mÞ  yju ¼ HÞ     ¼ Pr ~s < G~1 ðy; mÞ u ¼ H  s   ðy; mÞ ¼ G~sjH G~1 s   ðy; mÞ : ¼ F H G~1 s The first line and the fourth line use the definition of CDF. The second line uses the definition of Y in equation (21). The third line uses the monotonicity of G in s, which will be proved below. The last step uses the fact that ~s has the same distribution as evidence ~t in the continuous regime. Recall that F H is the CDF of the evidence distribution when u ¼ H. Before we prove equation (20), we first establish two properties of G~s ðs; mÞ: it is increasing in s and decreasing in m: G~s ðs; mÞ is increasing in s because of the properties of CDF. G~s ðs; mÞ is increasing in m as more manipulation shifts the distribution of s to the right in the sense of first order stochastic dominance (as less weight is put on F L and more weight is put on vG ðs;mÞ

F H ), implying that ~svm < 0. With these two properties, we show that F H ðG1 ðy; mÞÞ is increasing in m.

  vF H G~s1 ðy; mÞ vm

¼

1 vF H ðxÞ vG~s ðy; mÞ vx vm

vG1 ðy; mÞ f ~s vm vG ðy; mÞ=vm ¼  ~s vG~s ðy; mÞ=vy > 0: The second line uses the fact that F H is a CDF and thus an increasing function. The third line utilizes the implicit function theorem. The last line is due to properties of G~s proved above. Therefore, we have proved equation (20). By Proposition 3(i) in Ganuza and Penalva (2010), we have shown that if m1 < m2 ; then ~sðm1 Þ is more integral precise than ~sðm2 Þ. We now prove the case for the discrete regime. The only difference from the proof in the continuous case lies in the first step of applying the probability integral transform to the original report ~s; which is now discrete. Ganuza and Penalva (2010) develop a separate procedure to apply the transformation for discrete random variables in their footnote 5 (page 1011), which we follow here. The trick is to transform the discrete random variable ~s to a continuous one, which we denote as ~s0 : More specifically, denote

s0 ¼



~s þ Prð~s ¼ lÞ~se1 if ~s ¼ l ~s þ Prð~s ¼ lÞ þ Prð~s ¼ hÞ~se2 if ~s ¼ h

where ~se1 , ~se2 are random variables uniformly distributed on ½0; 1 and are independent with each other as well as independent with ~s.

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15

Starting with this continuos random variable ~s0 ; we are able to use the same procedure as in the continuous measurement regime above to prove that GYjH ðy; mÞ is weakly increasing in m and the increasing is strict in a non-zero set of y, where YðmÞ ¼ G~s0 ð~s0 ; mÞ.11 Therefore sD ðm1 Þ is more integral precise than sD ðm2 Þ and the proof is complete. Proof of Proposition 2: Proof. We first calculate the ex ante payoffs to the investor V.

V ≡ E½vðu; I  Þ

(22)

i 1 h 2 E g ð~s; m Þ 2l Z 1 g2 ðs; m Þgm ðsÞds ¼ 2l Z q ¼ H gðs; m Þg H ðsÞds 2l ¼

¼

qH H  P ðm Þ: 2b

(23)

(24)

where the second line (i.e. equation (23)) comes from equations (13) and (15), the third line comes from the definition of expectation, the fourth line follows from equation (18) and the fifth line (i.e. equation (24)) follows from the definition of PH , i.e. equation (10). The ex ante payoffs to the manager U can be similarly calculated:

U ≡ E½uðu; I  Þ ¼ bEs ½I  ð~sÞ  kKðm Þ ¼

qH b

l

 kKðm Þ:

The third line uses Part 1 of Lemma 1. Therefore, the efficiency W can be calculated as

W¼

q b  a h 2~ i E g ðs; m Þ þ ð1  aÞ H  kKðm Þ : 2l l

(25)

Now we compare the efficiency across the two regimes in the off-equilibrium case in which the evidence distributions (or equivalently the evidence management levels) are the same across the two regimes. In this case,

DjmC ¼mD ¼m ≡ðWC  WD ÞjmC ¼mD ¼m i h io an h 2~ E g ðsC ; m Þ  E g2 ð~sD ; m Þ 2l i h i h fE g2 ð~sC ; m Þ  E g2 ð~sD ; m Þ ¼

> 0: The last inequality holds because, By Lemma 1, ~sC strictly Blackwell dominates ~sD when m 2½0; 1Þ. In other words, in the off-equilibrium case with the same evidence distribution, the report from the continuous regime is more informative and thus leads to higher investment efficiency. Proof of Proposition 3: Proof. Recall the expression of W in equation (25). The first component is strictly decreasing in m because report ~s is more integral precise as m decreases (by Part 3 of Lemma 1). By the definition of integral precision (i.e. Definition 2), we have that EðE½uj~s; m Þ2 ¼ E½g2 ð~s; m Þ decreases in m . In addition, Kðm Þ increase in m . Therefore, Wðm Þ in equation (25) decrease in m for any interior m : Proof of Proposition 4: Proof. We prove mC > mD by contradiction. Suppose mC  mD : Recall that mC and mD are characterized by the first-order conditions, i.e. equation (14). We thus have

11

The reason for weak monotonicity in m stems from the discreteness of s, resulting in Y being a constant of m over certain regions of y.

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mC  mD f h

H

¼ PC

k 0



K mC  K 0 mD

qL i



i h 

L 

mC  PLC mC  PH D mD  PD mD

(26)

i h

i h

L 

L 

   PH  PH C mD  PC mD D mD  PD mD i h

h

i H 

  PLC mD  PLD mD ¼ PH C mD  PD mD > 0;

(27)

resulting in contradiction. Therefore, we have proved that mC > mD : The third line is from the proof of Proposition 1 that PH ðm Þ  PL ðm Þ is decreasing in m for both regimes. The fifth line is proved in two steps. First, we show that

















PHC mD  PHD mD f VC mD  VD mD > 0:

(28)

The equality comes from equation (24) and the inequality comes from Proposition 2. Second, we use the rational expectations (but off-equilibrium) condition to prove that PLC ðmD Þ  PLD ðmD Þ < 0. In any regime i 2 fC; Dg; if the manager's actual choice of manipulation is consistent with the investor's conjecture (i.e. expectations are rational), then the expected investment level is qlH by the law of iterated expectations. Therefore, the manager's expected payoff is

Ui ðm; mÞ ¼ b

qH

l

 kKðmÞ:

(29)

Note that m can be any choice of manipulation as long as it is consistent with the investor's conjecture. Therefore, we have





q UC mD ; mD ¼ b H  kK mD ;

l

UD





q mD ; mD ¼ b H  kK mD :

l

b ¼ mD and rearranging the terms, we obtain Writing out both equations by using equation (11) with m ¼ m





PLC mD  PLD mD ¼



i qH þ qL mD h H 

H 

< 0: P P  m  m C D D D qL 1  mD

The inequality comes from equation (28). Therefore, we have proved the Proposition. The intuition of the proof is as follows. At mD ; the marginal benefit of further manipulation for the manager in the discrete L   0  regime is PH D ðmD Þ  PD ðmD Þ; which is equal to the marginal cost of kK ðmD Þ: However, suppose the manager in the continuous regime starts at mD : The key step of our proof that expression (26) is positive means that the manager in the  continuous regime enjoys a larger marginal benefit than the manager in the discrete regime. The reason is that PH C ðmD Þ > PHD ðmD Þ > PLD ðmD Þ > PLC ðmD Þ, i.e. continuous regime generates a larger change in beliefs because the signals are more informative (conditional on the same level of manipulation). Therefore, the manager in the continuous regime will manipulate more in equilibrium, that is, mC > mD : Proof of Theorem 1: Proof. The efficiency difference across the two regime is defined as

DðkÞ ≡ WC  WD : We now prove that there exists a unique k such that DðkÞ > 0 if and only if k > k: The proof consists of four parts. In part 1, we express DðkÞ in three equivalent ways for subsequent use. In part 2, we use the intermediate value theorem to prove the existence of k: In part 3, we prove its uniqueness. Finally, we prove the comparative statics of k in part 4. Part 1: DðkÞ can be written as follows.

DðkÞ ≡ WC  WD ¼a

i



 qH h H 

PC mC ðkÞ  PHD mC ðkÞ  ð1  aÞk K mC ðkÞ  K mD ðkÞ 2b

(30)

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17









PHC mC ðkÞ  PHD mC ðkÞ  A K mC ðkÞ  K mD ðkÞ k

f

(31)











f 1  mC K 0 mC  1  mD K 0 mD  A K mC  K mD ; with A ≡

(32)

2bð1  aÞ : qH a

Expression (30) follows directly from taking the difference of the expression of W as in equation (25). Expression (31) follows directly from the second line. We now derive equation (32). Comparing equation (31) and equation (32), we need to show that







PHC mC ðkÞ  PHD mC ðkÞ k

(33)







¼ 1  mC K 0 mC  1  mD K 0 mD b ¼ m , we can write out equation (29) as, for i ¼ C; D, Using the definition of U from equation (11) with m

 b 



L 

H L 

  ¼ qH qH PH i mi þ qL Pi mi þ qL mi Pi mi  Pi mi

l

(34)

Rearranging terms of equation (14) results in



kK PLi mi ¼ PHi mi 

0



mi



qL

(35)

Insert equation (35) into equation (34) and rearranging terms results in

b



PHi mi ¼ qH þ 1  mi kK 0 mi l

(36)

Taking the difference of equation (36) with i ¼ C;D, divide by k and rearranging terms result in equation (33). The proof of step 1 is thus complete. Part 2: use the intermediate value theorem to prove the existence of. k: In step 1 we show that when k is sufficiently small, DðkÞ < 0. We then show in step 2 that when k/ þ ∞, DðkÞ > 0. The intermediate value theorem then implies that there is at least one value of k, denoted as k, such that DðkÞ ¼ 0. We proceed to prove step 1. When k is sufficiently small, mC ¼ 1 > mD (which we show later), resulting in, from equation (32), that

DðkÞ





f  1  mD K 0 mD  A Kð1Þ  K mD < 0:

We now prove that mC ¼ 1 > mD when k is sufficiently small. Previous results have focused on interior m. We now consider the possibility of corner solutions. First, we show that mC ¼ 1 is an equilibrium (i:e: the first-order condition UC ðmC ¼ 1Þ  0) with a sufficiently small k. Recall from equation (19) that

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UC ð1Þ

  L 0 ¼ qL PH C ð1Þ  PC ð1Þ  kK ð1Þ

¼ qL

b l

¼ qL

b l

Z

tL

Z

h

tH

tH tH

i

gðsC ; 1Þ gH ðsC Þ  gL ðsC Þ dsC  kK 0 ð1Þ

(37)

h i qH gH ðsC Þ  gL ðsC Þ dsC  kK 0 ð1Þ

b l

¼ qL qH GL ðt H Þ  kK 0 ð1Þ:

The third line is due to gðsC ; 1Þ ¼ qH when sC ¼ t2½t H ; t H  and the belief that gðsC ; 1Þ ¼ 0 when sC ¼ t 2½t L ; t H Þ since the support of F H is ½t H ; t H . Thus, in the continuous regime, if k is sufficiently small so that kK 0 ð1Þ  qL qH bl ðt H  t L Þ, then UC ð1Þ  0 from equation (37) and mC ¼ 1. Second, we show that mD < 1, i.e. mD ¼ 1 is not an equilibrium by contradiction. Suppose mD ¼ 1, then in the recognition regime, gðsD ; 1Þ ¼ qH when sD ¼ h; l as T2½t H ; t L . Therefore from equation (19),

UD ð1Þ L 0 ¼ PH D ð1Þ  PD ð1Þ  kK ð1Þ h i X gðsD ; 1Þ gH ðsD Þ  gL ðsD Þ  kK 0 ð1Þ ¼ sD

i X h qH g H ðsD Þ  g L ðsD Þ  kK 0 ð1Þ ¼ sD

¼ kK 0 ð1Þ < 0; resulting in mD < 1, which is a contradiction. Therefore mD < 1. We now prove step 2 that lim DðkÞ > 0. When k/ þ ∞, mC and mD /0. We therefore have, from equation (30) that k/þ∞

lim DðkÞ

k/þ∞

¼ lim

qH h

k/þ∞ 2b







i

PHC mC ðkÞ  PHD mC ðkÞ 



1a  

k K mC ðkÞ  K mD ðkÞ

a

i 1a



 q h H ¼ H PH lim k K mC ðkÞ  K mD ðkÞ C ð0Þ  PD ð0Þ  k /þ∞ a 2b i qH h H ¼ P ð0Þ  PHD ð0Þ  0 2b C > 0:

 H The fifth line is because PH lim k KðmC ðkÞÞ ¼ C ð0Þ  PD ð0Þ > 0, which follows from Proposition 2. The fourth line is because k/þ∞   lim k KðmD ðkÞÞ ¼ 0, which we now prove. Note that

k/þ∞

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19



lim kK mC ðkÞ

k/þ∞



K m ðkÞ ¼ lim kK 0 mC ðkÞ 0 C

k/þ∞ K mC ðkÞ

h i K mC ðkÞ L

¼ qL PH ð0Þ  P ð0Þ lim C C k/þ∞ K 0 mC ðkÞ h i KðmÞ L ¼ qL PH C ð0Þ  PC ð0Þ lim 0 m/0 K ðmÞ ¼ 0: 0

The third line comes from equation (14) and the fifth line follows from L'Hospital's rule, i.e. lim KKðmÞ ¼ lim KK00ðmÞ ¼ 0. The 0 m/0 ðmÞ m/0 ðmÞ  proof for lim kKðmD ðkÞÞ ¼ 0 is essentially the same and thus omitted. k/þ∞

The second part is thus complete. Part 3: prove that k is unique. We prove uniqueness by showing that DðkÞ crosses zero only once. One sufficient condition for this is that vDvðkkÞjk¼k > 0 for any k such that DðkÞ ¼ 0. We use the quadratic cost specification, i.e. KðmÞ ¼ 12m2 , to illustrate the intuition underlying the proof of vDvðkkÞjk¼k > 0. The proof for the general case has essentially the same intuition but is technically more involved, which we delegate to the very end. Note that DðkÞ ¼ 0 implies, from equation (32), that

DðkÞ







¼ 1  mC ðkÞ K 0 mC ðkÞ  AK mC ðkÞ





 1  mD ðkÞ K 0 mD ðkÞ  AK mD ðkÞ ¼0



⇔L mC ðkÞ  L mD ðkÞ ¼ 0

(38)

where we introduce the notation that

LðmÞ ≡ ð1  mÞK 0 ðmÞ  AKðmÞ:

(39)

In the binary case, LðmÞ ¼ ð1 mÞm  12 Am2 and equation (38) can be simplified as





L mC ðkÞ  L mD ðkÞ 



A  mC ðkÞ þ mD ðkÞ mC ðkÞ  mD ðkÞ ¼ mC ðkÞ  mD ðkÞ  1 þ 2  





A  mC ðkÞ þ mD ðkÞ ¼ mC ðkÞ  mD ðkÞ 1  1 þ 2 ¼ 0; resulting in

mC ðkÞ þ mD ðkÞ ¼

2 2þA

(40)

as mC > mD from Proposition 4. 1 > m ðkÞ, where 1 is the unique value of m that maximizes Given that mC > mD , equation (40) implies that mC ðkÞ > 2þA D 2þA 1 , L0 ðm ðkÞÞ < 0 < L0 ðm ðkÞÞ, resulting in LðmÞ. Since LðmÞ is decreasing (increasing) in m when m > ð < Þ 2þA D C

vDðkÞ j vk k¼k

vmC ðkÞ

vmD ðkÞ ¼ L0 mC ðkÞ jk¼k  L0 mD ðkÞ j >0 vk k¼k vk as

vmC ðkÞ vk

and

vmD ðkÞ vk

are both negative from Proposition 1.

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We now provide a graphical illustration for the proof for general specifications of KðmÞ in Fig. 1. The assumptions on KðmÞ implies that LðmÞ first increases with m, reaches a unique maximum when m ¼ m for some m2ð0; 1Þ, and then decreases with m, as illustrated by the purple unbroken line in the figure (i.e. similar to the case when KðmÞ is a quadratic function of m).12 This implies that L0 ðmÞ < ð > Þ0 if m > ð < Þm. Since mC > mD , it must be that mC ðkÞ > m > mD ðkÞ, as illustrated by the three vertical dashed lines. Therefore L0 ðmC ðkÞÞ < 0 < L0 ðmD ðkÞÞ, resulting in vDvðkkÞjk¼k > 0. Therefore there is a unique k that satisfies equation (38) and WC > WD if and only if k < k. The rest of the proof is merely showing the properties of LðmÞ, as discussed above, in rigorous algebra. Key step: show that LðmÞ first increases with m, reaches the maximum at m and then decreases with m. We divide the proof of the key step into three substeps. In the first substep, we show that LðmÞ has at least one argmax when m is interior. In the second step, we show that LðmÞ has only one such argmax, denoted as m. In the third step, we show that L0 ðmÞ > 0 when m < m and that L0 ðmÞ < 0 when m > m. Substep 1: LðmÞ will reach at least one maximum when m2ð0; 1Þ, i.e. when m is interior. 00 First, Lð0Þ ¼ 0 and Lð1Þ  0. In addition, L0 ðmÞ ¼ K ðmÞ  AK 0 ðmÞ > 0 when m is sufficiently close to 0. To see this, note 0 0 00 K 0 ðmÞ ¼ 0 implies that cε > 0, dmðεÞ s.t. KK00ðmÞ < ε, or, equivalently, K ðmÞ > K ðmÞ 00 ε . ðmÞ m/0 K ðmÞ 00 K 0 ðmÞ mðεÞ, K ðmÞ > ε > AK 0 ðmÞ, resulting in L0 ðmÞ > 0 when 0 < m < mðεÞ.

that lim

Pick 0 < ε <

1 . 1þð1a aÞ

Then when 0 < m <

Thus, LðmÞ will reach (possibly more than one) maximum when m2ð0; 1Þ. Substep 2: LðmÞ will reach one and only one maximum, denoted as m, when m2ð0; 1Þ, i.e. when m is interior. A necessary condition for an interior maximum is the satisfaction of the first-order condition, i.e. L0 ðmÞ ¼ 0, or, equivalently,

1  m ¼ ð1 þ AÞ

K 0 ðmÞ 00 K ðmÞ

(41) 0

will have at least one solution on ð0; 1Þ. Since 1  m is strictly decreasing in m and by assumption KK00ðmÞ is weakly increasing in ðmÞ m, there is a unique m2ð0; 1Þ that satisfies equation (41), i.e. L0 ðmÞ ¼ 0. Since m is the only solution of the first-order condition, LðmÞ reaches its maximum at m. Substep 3: L0 ðmÞ > 0 when m2ð0; mÞ and that L0 ðmÞ < 0 when m2ðm; 1Þ. !  !  !  We prove by contradiction. Suppose that L0 ðmÞ  0 for some m 2ð0;mÞ. Clearly L0 ð m Þ < 0 as L0 ð m Þ ¼ 0 would violate  0 ! the condition that m is the unique solution that satisfies equation (41). When L ð m Þ < 0, there must exist at least one m2 !  !  !  ð m ; mÞ such that LðmÞ > 0. The reason is that if this is not true, then L0 ðmÞ < 0 for all m2ð m ; mÞ, resulting in Lð m Þ >  0 ! LðmÞ and thus violating the assumption that LðmÞ reaches its maximum at m. Now suppose that LðmÞ > 0 and L ð m Þ < 0, !  then by the intermediate value theorem there must exist a m↔ 2ð m ; mÞ such that L0 ðm↔ Þ ¼ 0. Since m↔ < m, it is in contradiction with m being the unique solution that satisfies equation (41). We thus have proved that L0 ðmÞ > 0 when m2 ð0; mÞ. The proof that L0 ðmÞ < 0 when m2ðm; 1Þ is essentially the same and thus omitted. Part 4: Comparative statics of k Recall that k is defined by









DðkÞ ≡ L mC ðkÞ  L mD ðkÞ ¼ 0

(42)

where LðmÞ≡ð1 mÞK 0 ðmÞ  AKðmÞ from equation (39). By the implicit function theorem, vDðkÞ

jk¼k vk ¼  vDvðlkÞ vl vk jk¼k We already show that vDvðkkÞjk¼k > 0. In addition,

vDðkÞ ¼ L0 j vl k¼k

 

  vmC ðkÞ vmD ðkÞ mC k mD k  L0 vl vl vm ðkÞ

vm ðkÞ

As discussed in the proof of part 3, L0 ððmC ðkÞÞ < 0 and L0 ððmD ðkÞÞ > 0. In addition, from Proposition 1, vCl < 0 and vDl < 0. This results in vDvðlkÞjk¼k > 0. Therefore vvkl < 0. Similarly, using the implicit function theorem and take the derivative of both sides of equation (42) with respect to b results in

12 Note that this does not mean that the second order derivative of LðmÞ is always negative, i.e. the first order derivative of LðmÞ decreases from positive to zero to negative. It implies a weaker condition that the first order derivative of LðmÞ crosses zero only once.

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21

vDðkÞ

j vk vb k¼k ¼  vDðkÞ vb vk jk¼k We already know that vDvðkkÞjk¼k > 0 as shown above. In addition,

vDðkÞ j ¼ L0 vb k¼k

 

  vmC ðkÞ vmD ðkÞ mC k mD k  L0 vb vb

Again, it has already been shown in step 3 that L0 ððmC ðkÞÞ < 0 and L0 ððmD ðkÞÞ > 0. In addition, from Proposition 1, and

vmD ð vb

kÞ

vmC ðkÞ > vb

0

> 0. This results in vDvðbkÞjk¼k < 0. Therefore vvbk > 0.

References Admati, A.R., Pfleiderer, P., 2000. Forcing firms to talk: financial disclosure regulation and externalities. Rev. Financ. Stud. 13, 479e519. Arya, A., Glover, J., Sivaramakrishnan, K., 1997. The interaction between decision and control problems and the value of information. Account. Rev. 72, 561e574. Bagnoli, M., Bergstrom, T., 2005. Log-concave probability and its applications. Econ. Theor. 26, 445e469. Barth, M.E., 1994. Fair value accounting: evidence from investment securities and the market valuation of banks. Account. Rev. 1e25. Barth, M.E., Beaver, W.H., Landsman, W.R., 1996. Value-relevance of banks' fair value disclosures under SFAS No. 107. Account. Rev. 513e537. Barth, M.E., Beaver, W.H., Landsman, W.R., 2001. The relevance of the value relevance literature for financial accounting standard setting: another view. J. Account. Econ. 31, 77e104. Barth, M.E., Hodder, L.D., Stubben, S.R., 2008. Fair value accounting for liabilities and own credit risk. Account. Rev. 83, 629e664. Bergemann, D., Morris, S., 2016. Information design, Bayesian persuasion, and Bayes correlated equilibrium. Am. Econ. Rev. 106, 586e591. Bertomeu, J., Cheynel, E., 2013. Toward a positive theory of disclosure regulation: in search of institutional foundations. Account. Rev. 88, 789e824. Bertomeu, J., Magee, R.P., 2011. From low-quality reporting to financial crises: politics of disclosure regulation along the economic cycle. J. Account. Econ. 52, 209e227. Bertomeu, J., Magee, R.P., 2015a. Mandatory disclosure and asymmetry in financial reporting. J. Account. Econ. 59, 284e299. Bertomeu, J., Magee, R.P., 2015b. Political pressures and the evolution of disclosure regulation. Rev. Account. Stud. 20, 775e802. Bird, A., Karolyi, S., Ruchti, T., 2016. Understanding the ”numbers game” (CMU working paper). Blackwell, D., 1953. Equivalent comparisons of experiments. Ann. Math. Stat. 24, 265e272. Chen, Q., Hemmer, T., Zhang, Y., 2007. On the relation between conservatism in accounting standards and incentives for earnings management. J. Account. Res. 45, 541e565. Dechow, P.M., Myers, L.A., Shakespeare, C., 2010. Fair value accounting and gains from asset securitizations: a convenient earnings management tool with compensation side-benefits. J. Account. Econ. 49, 2e25. Dechow, P.M., Shakespear, C., 2009. Do managers time securitization transactions to obtain accounting benefits? Account. Rev. 84, 99e132. Demski, J.S., 1973. The general impossibility of normative accounting standards. Account. Rev. 48, 718e723. Demski, J.S., 2004. Endogenous expectations. Account. Rev. 79, 519e539. Dietrich, J.R., Harris, M.S., Muller, K.A., 2000. The reliability of investment property fair value estimates. J. Account. Econ. 30, 125e158. Dubey, P., Geanakoplos, J., 2010. Grading exams: 100, 99, 98,… or a, b, c? Games Econ. Behav. 69, 72e94. Dye, R., 1990. Mandatory versus voluntary disclosures: the cases of financial and real externalities. Account. Rev. 65, 1e24. Dye, R., 2002. Classifications manipulation and Nash accounting standards. J. Account. Res. 40, 1125e1162. Dye, R., Glover, J., Sunder, S., 2014. Financial engineering and the arms race between accounting standard setters and preparers. Account. Horiz. 29, 265e295. Dye, R., Sridhar, S., 2004. Reliability-relevance trade-offs and the efficiency of aggregation. J. Account. Res. 42, 51e88. Dye, R., Sridhar, S., 2008. A positive theory of flexibility in accounting standards. J. Account. Econ. 46, 312e333. Dye, R., Sunder, S., 2001. Why not allow FASB and IASB standards to compete in the US? Account. Horiz. 15, 257e271. Engel, E., Erickson, M., Maydew, E.L., 1999. Debt-equity hybrid securities. J. Account. Res. 37, 249e274. Fan, Q., Zhang, X.J., 2012. Accounting conservatism, aggregation, and information quality. Contemp. Account. Res. 29, 38e56. Friedman, H.L., Heinle, M.S., 2016. Lobbying and uniform disclosure regulation. J. Account. Res. 54, 863e893. Ganuza, J., Penalva, J.S., 2010. Signal orderings based on dispersion and the supply of private information in auctions. Econometrica 78, 1007e1030. Gao, P., 2013a. A measurement approach to conservatism and earnings management. J. Account. Econ. 55, 251e268. Gao, P., 2013b. A two-step representation of accounting measurement. Account. Horiz. 27, 861e866. Gao, P., 2015. Optimal Thresholds in Accounting Recognition Standards. Working Paper. Gentzkow, M., Kamenica, E., 2016. A Rothschild-Stiglitz approach to Bayesian persuasion. Am. Econ. Rev. 106, 597e601. Glover, J., 2013. Can Financial Accounting Regulators and Standard Setters Get (And Stay) Ahead of the Financial Engineers? Emanuel Saxe Lecture at Baruch College. Glover, J., Ijiri, Y., Levine, C., Liang, P., 2005. Verifiability, Manipulability, and Information Asymmetries about Verifiability and Manipulability. Working Paper. Carnegie Mellon University. Goel, A.M., Thakor, A.V., 2015. Information reliability and welfare: a theory of coarse credit ratings. J. Financ. Econ. 115, 541e557. Harbaugh, R., Rasmusen, E., 2018. Coarse grades: informing the public by withholding information. Am. Econ. J. Microecon. 10, 210e235. Imhoff, E., Thomas, J.K., 1988. Economic consequences of accounting standards: the lease disclosure rule change. J. Account. Econ. 10, 277e310. Kamenica, E., Gentzkow, M., 2011. Bayesian persuasion. Am. Econ. Rev. 101, 2590e2615. Kothari, S.P., Ramanna, K., Skinner, D.J., 2010. Implications for GAAP from an analysis of positive research in accounting. J. Account. Econ. 50, 246e286. Laux, V., Stocken, P., 2018. Accounting standards, regulatory enforcement, and investment decisions. J. Account. Econ. 65, 221e236. Lys, T.Z., Vincent, L., 1995. An analysis of value destruction in ATandT's acquisition of NCR. J. Financ. Econ. 39, 353e378. Marinovic, I., 2013. Internal control system, earnings quality, and the dynamics of financial reporting. RAND J. Econ. 44, 145e167. Milgrom, P.R., 1981. Good news and bad news: representation theorems and applications. Bell J. Econ. 12, 380e391. Mittendorf, B., 2010. The role of audit thresholds in the misreporting of private information. Rev. Account. Stud. 15, 243e263.

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SEC, 2005. Report and Recommendations Pursuant to Section 401(c) of the Sarbanes-Oxley Act of 2002 on Arrangements with Off-Balance Sheet Implications, Special Purpose Entities, and Transparency of Filings by Issuers. SEC Staff Report. Sunder, S., 2008. Econometrics of fair values. Account. Horiz. 22, 111e125. Watts, R.L., Zimmerman, J.L., 1986. Positive Accounting Theory. Prentice Hall, Englewood Cliffs, NJ. Watts, R.L., Zimmerman, J.L., 1990. Positive accounting theory: a ten year perspective. Account. Rev. 65, 131e156.

Please cite this article as: Gao, P., Jiang, X., The economic consequences of discrete recognition and continuous measurement, Journal of Accounting and Economics, https://doi.org/10.1016/j.jacceco.2019.101250