on the relation between conservatism in accounting ...qc2/bio/research/conservatism.pdf ·...

DOI: 10.1111/j.1475-679X.2007.00243.x

Journal of Accounting ResearchVol. 45 No. 3 June 2007

Printed in U.S.A.

On the Relation betweenConservatism in AccountingStandards and Incentives for

Earnings Management

Q I C H E N , ∗ T H O M A S H E M M E R , † A N D Y U N Z H A N G ‡

Received 13 March 2006; accepted 13 November 2006

ABSTRACT

This paper studies the role of conservative accounting standards in alle-

viating rational yet dysfunctional unobservable earnings manipulation. We

show that when accounting numbers serve both the valuation role (in which

potential investors use accounting reports to assess a firm’s expected future

payoff) and the stewardship role (in which current shareholders rely on the

same reports to monitor their risk-averse manager), current firm owners have

incentives to engage in earnings management. Such manipulation reduces

accounting numbers’ stewardship value and leads to inferior risk sharing. We

then show that risk sharing, and hence contract efficiency, can be improved un-

der a conservative accounting standard where, absent earnings management,

accounting earnings represent true economic earnings with a downward bias,

compared with under an unbiased standard where, absent earnings manage-

ment, accounting earnings represent true economic earnings without bias.

∗Duke University; †University of Houston; ‡Duke University. We appreciate comments from

seminar participants at Duke University, 2004 Duke/UNC Fall Camp, University of Houston,

University of Maryland, the 15th FEA conference at University of Southern California, an

anonymous referee, and Doug Skinner (the editor). We acknowledge the financial support of

the Fuqua School of Business at Duke University (for Qi Chen and Yun Zhang) and the Bauer

College of Business at University of Houston (for Thomas Hemmer).

541

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

542 Q. CHEN, T. HEMMER, AND Y. ZHANG

1. Introduction

In this paper, we study the role of standard-specific and thus observableaccounting biases in alleviating rational yet dysfunctional unobservable bi-ases introduced by earnings manipulation. Specifically, we study the relativeperformance of two alternative financial reporting regimes: one where ac-counting standards are designed to be bias free and one where accountingstandards are designed deliberately to introduce a conservative bias in finan-cial reports. We identify conditions under which conservatism in accountingstandards is effective in reducing incentives to manage earnings upwards. Inaddition, we demonstrate that doing so can reduce contracting costs, thusproviding insights into the source of the seemingly universal popularity ofconservatism in financial accounting standards.1

A variety of alternate perspectives on the demand for conservatism cur-rently exist in the accounting literature. Watts [2003a] provides a numberof intuitive explanations for the beneficial implications of conservative ac-counting, in particular from a contracting perspective. His main argumentis that conservatism constrains opportunistic behavior and offsets biases in-troduced by self-interested parties. Our paper is somewhat related, as wealso focus on the role of conservative accounting in alleviating biases in fi-nancial reports introduced by opportunistic reporting behavior. Our mainpoint, however, differs significantly from his. By allowing for both systemicconservative accounting biases and opportunistic (liberal) reporting biasesin an analytical model with endogenous demand for financial reports, weare able to show that conservatism need not offset opportunistic biases byimposing explicit constraints and reducing firm insiders’ opportunities tointroduce biases. Rather, conservatism can have such an effect simply bydampening firm insiders’ incentives to manage earnings.

Our study of accounting conservatism is based on a specific model inwhich we seek to integrate a number of general observations about the fea-tures of standard financial reporting regimes. First, we note that financialreports are produced by jointly confronting auditable historical data andan array of subjective estimates with specific predefined accounting princi-ples, procedures, and methods. Any biases in financial reports are thereforebound to be the net result of biases introduced by specific accounting prin-ciples, procedures, and methods, and biases introduced by estimates. Whilethe former are largely precommitted publicly through the choice of specificaccounting principles, the latter, by their subjective nature, are harder tocredibly commit to and much more difficult for outsiders to identify ex post.

Second, while some biases are deterministic in the sense that the finalreport always over- or underestimates the truth, biases are often introduced

1 Examples of accounting conservatism abound: Generally accepted accounting principles

(GAAP) require most research and development expenditures to be expensed as incurred

rather than capitalized. Similarly, many assets are valued at the “lower of the market or historical

value” and recognition of appreciations in value is not permitted under most circumstances.

CONSERVATISM IN ACCOUNTING STANDARDS 543

only in expectation. This is certainly true for biases introduced by particularaccounting methods and principles, as these have to be used consistentlyafter their adoption. Accordingly, while straight-line depreciations may beused with the intent of creating a conservative bias, in the absence of aliquid market for the asset being depreciated, the realized bias introducedinto accounting income may well end up being liberal. And while there maybe more degrees of freedom available when it comes to biasing estimates,the very fact that estimates are made ex ante allows for the possibility thatwhat seems biased in one direction ex ante actually leads to realized biasesin the opposite direction ex post.

Third, although biases introduced by manipulative actions such as the useof biased estimates could potentially be offset exactly “on average” by usingaccounting methods biased in the opposite direction, two different sourcesof bias are not likely to be perfect substitutes. Stated differently, two wrongsare not likely to make a right: While the different biases introduced by biasedestimates and biased accounting methods may offset in a way to produce anoverall unbiased report, the information content of that report is unlikely tobe identical to one produced without the introduction of any bias at all. Forexample, “lower of cost or market” may well provide an appropriate offsetfor optimistic estimates used for bad debt expense. Yet the final report thatincorporates such offsetting biases still differs substantively from a reportwhere all assets are marked-to-market in an unbiased way.

Finally, we note that accounting numbers in financial reports generallyplay a multitude of roles that inevitably create diverging incentives for at-tempting to endow them with certain biases. For example, while from astewardship perspective, (current) shareholders may favor a cautious ap-proach to financial reporting, from a valuation perspective they may bemore concerned about avoiding undervaluation of their shares.

Our model captures these features in the following ways. First, for speci-ficity we concentrate on diverging reporting incentives related to the twoaforementioned key uses of financial reports: “valuation,” where potentialinvestors use accounting information to assess a firm’s expected future pay-off, and “stewardship,” where accounting information is used by the currentowners to evaluate and incentivize employees. We do this by studying a set-ting where an innovator/entrepreneur contracts with a professional man-ager to help manage the firm in a way to increase the value of the firm priorto selling (all or part of) it to a second generation of shareholders, for exam-ple, through an initial public offering (IPO).2 Our analysis takes accountingnumbers’ dual roles as given and focuses instead on their consequences.3

2 Another interpretation of our model is that it describes a venture capitalist contracting

with an entrepreneur (agent) and subsequently bringing the firm to IPO. The venture capitalist

is likely to assume a dominant role in the financial reporting involved in the IPO process.3 For an in-depth discussion on accounting numbers’ multiple roles, see Watts and Zimmer-

man [1986]. Bushman, Engel, and Smith [2006] provide empirical evidence for the relation

between the valuation role and the stewardship role of accounting reports. Also, if investors


Second, to capture the point that biases are introduced ex ante and thustypically are “in expectation,” all biases in our model are introduced bystochastically altering the representation of the underlying facts. Concretely,biases are introduced through reducing the probability that an event is rep-resented accurately in the financial report while at the same time increasingthe probability that the event is either over- or underrepresented dependingon the particular bias being introduced.

This stochastic representation also allows us to capture the property thattwo wrongs do not make a right in a particularly parsimonious way. Wedo this by assuming that a liberal bias can only be introduced by actionsthat increase from zero (decrease from one) the probability of a favorable(unfavorable) earnings report when the underlying true earnings are unfa-vorable. A conservative bias, on the other hand, can only be introduced byactions that increase from zero (decrease from one) the probability of anunfavorable (favorable) earnings report when the underlying true earningsare favorable. Accordingly, a liberal bias that makes bad outcomes appeargood can always be offset in expectation by a conservative bias. The offsetis not a “one for one” though, as the tools available for engineering thedifferent biases are different themselves.

In addition to capturing the notion that biases may be offset mainlythrough dissimilar actions, there are several other advantages to the par-ticular representation we adopt here. For one, the representation capturesthe notion that actions intended to distort financial reports are not innocu-ous in the sense that a user of financial statements anticipating a particularbias is able to undo the damage simply by deducting the anticipated biasfrom the published earnings. With the stochastic structure, the user can takeout only the expected bias but cannot restore the information content of thereport lost due to the introduction of biased noise. Another advantage of ourrepresentation is that the model itself is biased against finding any benefitsof conservatism in financial reporting. This is true regardless of the pres-ence of a liberal bias since introducing conservatism here always reducesthe information content of the financial report, thus making it less useful.

The analysis of our model yields a number of insights. Starting in a bench-mark setting where the accounting regime is unbiased, we first establish thatwhen there is uncertainty about the future payoff of the firm, the currentowner has an incentive to manage the accounting earnings upward (in ex-pectation) to induce more favorable (potential) investors’ belief about thefirm’s prospects. Because we allow for the fact that there is a fair amountof discretion in the estimation part of preparing the financial report, thisincentive leads, in equilibrium, to the introduction of a liberal bias in thefinancial report.

rationally rely on the accounting numbers in pricing a firm, it has to be the case that the

accounting numbers are related to the true underlying output. If the true output depends

on the firm manager’s productive effort, then accounting numbers, by their relation with the

true output, are informative about these productive efforts. The valuation role of accounting

numbers then almost surely implies their stewardship value.


The equilibrium we identify implies inefficiency, however. While the cur-rent owner in fact does choose to manipulate earnings upwards in equi-librium, he does not benefit from the manipulation for two reasons. First,although potential investors do not directly observe the earnings manage-ment, they rationally expect and price protect against it. Accordingly, theycorrectly incorporate the expected bias in the financial report when valuingthe firm. Second, such manipulation actually reduces the current owner’swelfare relative to a (here unattainable) no-manipulation equilibrium. Thisis because the introduction of a nondeterministic bias also reduces the in-formativeness of accounting numbers for stewardship purposes, which inturn leads to inefficiencies in the incentive arrangement provided by thecurrent owner to the manager.

Again, the central inefficiency arises in our setting essentially becauseof the need for making estimates in producing financial reports. Becausesuch estimates are subjective, unverifiable, and likely unobservable, it is im-possible for the current owner to credibly commit not to bias his estimatesdeliberately with the intent to distort accounting earnings.4 It is in this con-text that we study the potential consequences of using publicly observableaccounting principles with publicly observable (expected) bias. We do so bycomparing the efficiency loss (which is captured by the inefficiency in risksharing in our model) under the two accounting regimes we contrast: anunbiased regime where, absent earnings management, accounting earningsrepresent the underlying true output without bias, and a conservative onewhere, again in the absence of earnings management, accounting earningsmay underrepresent the true underlying output.

We show that the degree of earnings manipulation is lower in the conser-vative regime than in the unbiased accounting regime. The intuition behindthis result is the following. Under the conservative system, observing a lowaccounting earnings number does not necessarily mean that the true eco-nomic earnings are as low. This in turn reduces the benefit of earningsmanipulations, as the reason for the manipulation is exactly to make the in-ference drawn from observed earnings numbers more favorable. Moreover,we show that conservatism increases the current owner’s marginal cost tomotivate the agent, further dampening his incentive to manage earnings.

We also identify situations where the conservative accounting regime im-proves the contracting efficiency for the current owner. This result is some-what counterintuitive at first: after all, conservatism introduces additionalnoise in the accounting report, which in and of itself reduces the steward-ship value of accounting data and therefore increases the cost of motivat-ing the manager. However, we show that under reasonable conditions, theconservative standard actually reduces earnings management so much sothat the noise introduced by the conservative standard is more than offsetby the reduction in earnings management. This overall net reduction in

4 Firms can, and are often required to, disclose the detailed estimates. But it may still be

difficult for firms to convince investors that these estimates are completely bias free.


(equilibrium) noise makes the accounting reports more informative aboutmanagerial action.

In terms of empirical implications, our analysis provides a theoreticalframework for interpreting existing evidence regarding both the time-serieschanges and the cross-country differences in the degree of accounting con-servatism. Specifically, our model implies that conservative principles aremore likely to arise in situations where accounting numbers play dual rolesand where the self-interested parties involved in the financial reportingprocess (e.g., controlling shareholders, top management teams) have sig-nificant equity stakes in their firms. Casual examination of existing evidenceseems to support this prediction. For example, Watts [2003b] notes that con-servatism in the United States has increased after the Financial AccountingStandards Board (FASB) began managing the accounting standards, a pe-riod in which equity incentives for top management in U.S. companies havealso increased significantly (Milliron [2000]).

Ball, Kothari, and Robin [2000] provide evidence that accounting mea-surement is more conservative in common law countries (e.g., the UnitedStates and United Kingdom) than in code law countries (e.g., France andGermany). To the extent that management in the United States and UnitedKingdom has higher equity stakes, and investors in the United States andUnited Kingdom are more diverse and rely more on accounting earningsfor valuation than those in France and Germany, where ownership tends tobe concentrated and direct monitoring is more prevalent (thus accountingis less likely to play dual roles), their findings are consistent with our model’sprediction.

Our paper also relates to two strands of theoretical research in account-ing: earnings manipulation and accounting conservatism. Prior studies onearnings manipulation focus on identifying situations where earnings ma-nipulations arise in equilibrium (Stein [1989] and Dye [2002]).5 In thesepapers, the cost of earnings manipulation is assumed to be exogenous. Ourpaper differs from theirs in that the cost of earnings manipulation is endo-genized as a contract cost and is tied to the stewardship role of accountingnumbers. What is significant about this difference is that conservatism here,unlike in prior studies, plays a crucial role in reducing overall earningsmanagement.

Kwon, Newman, and Suh [2001] also study the need for conservatismin a pure agency setting. In their model, accounting numbers have onlythe stewardship value and conservatism arises because the agent has lim-ited liability. There, conservatism per se enhances accounting numbers’stewardship value by loosening the limited liability constraint. Our paperinstead analyzes the implications of the dual roles of accounting numbers.6

5 Recent studies by Dye and Sridhar [2004] and Stocken and Verrecchia [2004] analyze the

impact of aggregation on earnings manipulation.6 Narayanan and Davila [1998] study a similar trade-off between the performance evaluation

and belief-revision uses of accounting signals. Their focus is on delegation, while ours is on

accounting conservatism.


More importantly, in our model, conservatism per se is “bad,” as it makesaccounting numbers less valuable for stewardship purposes. The benefitof adopting a conservative principle stems from its effect on the currentowner’s incentive to manipulate earnings. The net effect of conservatism inequilibrium is to increase both accounting numbers’ stewardship value andtheir usefulness in assessing firms’ future payoff. Accordingly, our paper alsodiffers from Gigler and Hemmer [2001] and Venugopalan [2004], whichtake conservatism as given and focus on its implications for firms’ voluntarydisclosure behavior and investment efficiency.

In what follows, we first set up the model in section 2 and solve the modelunder the unbiased accounting standard in section 3. We then introduceconservatism in section 4. Conclusions are offered in section 5.

2. Basic Model Setup

Our basic model commences at date 1 when a firm’s current owner hiresa professional manager (the agent) to manage the operations of his firm.We assume (entirely for simplicity) that the current owner (being a venturecapitalist) is risk neutral while the agent is risk and effort averse. The agentcan take an (unobservable) productive action a ∈ {ah, al } (the subscriptsh and l represent “high” and “low,” respectively) where ah more positively,albeit stochastically, influences the firm’s payoff than does al . To ensure thatthe problem at hand is not trivial, we assume that the expected net valuegenerated by ah is so much larger than that generated by al that the currentowner always finds it worthwhile to motivate the agent to work hard.

To simplify the relation between productive effort and (expected) ter-minal output, we assume that there are only two possible terminal payoffs:xh > xl , where xl is normalized to zero for simplicity and without loss of gen-erality. Production is represented by assuming that Pr(xh | ah) ≡ p ≥ 1

2>

Pr(xh | al ) = 0, and Pr(xl | ah) = 1 − p < Pr(xl | al ) = 1. That is, if the agentchooses al (i.e., shirks), the payoff is low (xl ) for sure; if he chooses ah (i.e.,works), the real output is more likely to be high. As only the marginal costof effort is relevant, we normalize the agent’s cost of al to zero and let D >

0 denote the agent’s incremental disutility for taking the high rather thanthe low effort. His (additively separable) utility from consuming a paymentof s is U (s) where U ′ > 0 and, due to risk aversion, U ′′ < 0. Without loss ofgenerality, we normalize the agent’s reservation utility to zero.

We note that the binary production structure assumed above is withoutloss of generality for our purpose. This is because our goal is not to iden-tify the optimal effort level for the principal per se; rather, it is to identifyconditions under which conservatism helps reduce earnings managementas well as reduce the cost of implementing any given effort level. A standardanalysis of principal-agent models with multiple or continuous effort levelsinvolves invoking assumptions to ensure only the local incentive compatible(IC) constraint is binding (relevant) in equilibrium. Typically, the monotonelikelihood ratio and the convexity of the distribution function conditionsare assumed, but even if left implicit (equilibrium), concavity of the agent’s


expected utility in effort is central to most, if not all, principal-agent studies(e.g., Grossman and Hart [1983]).

From a technical vantage point, this means that, to establish whether aninformation system is of value, one simply needs to show that the optimaleffort absent the information system (regardless of what that effort levelis) can be implemented over the immediate adjacent (lower) effort morecost-effectively with the information system present. We show that, betweenany two effort levels where, absent conservatism, the principal wants to im-plement the higher effort (which excludes the degenerate case where thelowest effort level is preferred), conservatism can help the principal im-plement the higher effort level at a lower cost. Thus, the beneficial role ofconservatism is not unique and special to a binary setting. It is an insight thatcan be applied to general settings with multiple effort levels or continuouseffort choice.

At date 2, the current owner needs to sell the firm to a second genera-tion of shareholders (whom we refer to as investors or future sharehold-ers) for reasons (such as liquidity needs) exogenous to our model. We as-sume that the second generation of shareholders bids for the firm’s sharescompetitively. To ensure a meaningful role for accounting information inthis transaction, we assume that the sale takes place before x is realized(x accrues to the future shareholders) and that the accounting reports e(statements of earnings) are the only publicly observable and contractiblesignals about x available at date 2. Thus, the firm’s selling price depends onboth parties’ beliefs about x based on accounting earnings. To capture theidea that (some) firms last longer than their managers, we assume that theagent turns over when ownership changes hand at date 2. Accordingly, theagency problem ends at the time of transfer and the current owner contractswith the agent by specifying compensation contingent upon accountingearnings.

Accounting earnings are generated by the firm’s accounting system. Howwell they represent the true underlying output depends on both the prevail-ing accounting standards/principles adopted by the jurisdiction in whichthe firm operates (e.g., U.S. GAAP or international accounting standards)and the earnings manipulation chosen by the firm’s owner. Our focus hereis on how conservatism in accounting standards affects earnings manipula-tion. As such, we confine our analysis to comparing two accounting standardsthat differ fundamentally with respect to conservatism.

The first accounting standard we consider is an unbiased standard that,without earnings management, simply reveals the underlying economic con-ditions. Accordingly, under this standard, Pr(e h | xh) = 1 and Pr(e l | xl ) = 1.The second standard is conservative and differs from the unbiased standardin that, absent earnings management, accounting earnings are stochasti-cally biased downward via Pr(e h | xh) = 1 − � and Pr(e l | xh) = �, with � ∈(0, 1). As accounting standards are publicly disclosed and can be enforcedby auditors and regulators, � is assumed to be publicly observable. Withthis representation, � reflects the degree of conservatism in the accounting


FIG. 1.—Information structure—unbiased accounting standard with � earnings

manipulation.

FIG. 2.—Information structure—conservative standard with �� earnings manipulation.

standard, as a higher value of � implies that true high output is more likelyto be reported as low earnings numbers by the accounting system.

As discussed earlier, accounting earnings may also be affected by thecurrent owner’s preference for (opportunistic) earnings management. Toparsimoniously capture the notion that earnings manipulation always re-duces the informativeness of the accounting system, in our setting the cur-rent owner manages earnings by introducing some unobservable noise (�)into the system such that Pr(e l | xl ) is lowered from its default value of 1 to1 − �, and Pr(e h | xl ) = �. Accordingly, a higher � means that a true lowlevel of payoff is more likely to be reported as high accounting earnings.7

Figures 1 and 2 provide an illustration of the information structures underthe two standards and earnings management.

While not crucial to our results, we assume that there is some upper boundfor �, that is, � ∈ [0, �max], with �max < 1. The purpose of this assumptionis simply to eliminate the potential for a “pathetic” equilibrium where �

is set equal to one and the current owner therefore is unable to motivate

7 Earnings manipulation may also result in a lower value for Pr(e h | xh), i.e., when the cur-

rent owner manages earnings downward by choosing � such that Pr(e h | xh) = 1 − �, and

Pr(e l | xh) = �. It can be easily shown that the current owner optimally sets � to zero. The

intuition is fairly straightforward, as the purpose of earnings management in our model is to

boost earnings and selling price.


FIG. 3.—Timeline.

the agent to take ah. Imposing an upper bound on � also makes intuitivesense as accounting standards and auditors constrain the current owner’searnings manipulation behavior.

We assume that the earnings manipulation, �, is observed (only) by thecurrent owner and the agent, but not by the future shareholders/investors.However, investors are not naive about the reported accounting earnings.Although they do not observe � directly, they take into account the currentowner’s incentives to manipulate earnings and use their (rational) conjec-ture of � (denoted �) when pricing the firm. The current owner, in choos-ing his optimal �, takes � as given. In equilibrium, investors’ conjecture iscorrect. A formal definition for an equilibrium is given below:

DEFINITION. An equilibrium in this model is characterized by (�, �, Pek , sk),where k = l or h, such that

1) investors’ conjecture, �, is correct in equilibrium, i.e., � = �;2) given the compensation contract sk and �, the agent chooses ah;3) given �, � and sk maximize the current owner’s expected payoff from selling

the firm to investors, net of the expected employment compensation for the agent;4) given �, the market price, Pek , equals the investors’ expected future payoff from

purchasing the firm.

Figure 3 summarizes the timeline and main events of the model.

3. Unbiased Accounting Standard

In this section, we solve the model under the unbiased accounting stan-dard following the standard approach of backward induction. Accordingly,we first determine how much investors are willing to pay for the firm at date 2as a function of both the realized earnings and investors’ conjecture about�. We then use this pricing function to identify the current owner’s choiceof earnings manipulation, �, and compensation contract for the agent, s,at date 1.

Investors purchase the firm for the expected output x. Given their con-jecture of the current owner’s earnings manipulation, investors’ expectedbenefit when the realized earnings is eh and el can be respectively written


as:

Peh = Pr(xh | e h, �)xh + Pr(xl | e h, �)xl

= p

p + (1 − p)�xh > 0,

Pe l = xl = 0.

That is, investors pay a higher price upon observing eh than upon observingel . Ceteris paribus, this implies that the current owner would like to makeeh appear more frequently, which can be achieved by increasing �.

In addition to the earnings management �, at date 1, the current owneralso needs to choose the compensation contract for the agent. Specifically,the contract will pay the agent sh upon observing the high earnings realiza-tion eh (or equivalently upon obtaining the high price of Peh ) and sl uponobserving el (or equivalently upon obtaining the low price of Pe l ).8 Thetriplet (sh, sl , �) is jointly chosen to maximize the current owner’s expectedpayoff as follows:

maxsh ,sl ,�

(p + (1 − p)�)( Peh − sh) + (1 − (p + (1 − p)�))( Pe l − sl )

s .t. (p + (1 − p)�)U (sh) + (1 − (p + (1 − p)�))U (sl )

−D ≥ �U (sh) + (1 − �)U (sl ), (IC)

(p + (1 − p)�)U (sh) + (1 − (p + (1 − p)�))U (sl ) − D ≥ 0. (IR)

To solve this maximization problem, we can simplify the IC constraint to

U (sh) − U (sl ) ≥ Dp(1 − �)

.

The solution has both the IR and IC constraints binding. Thus, the optimals∗

h and s∗l can be obtained directly from these constraints as:

U(s∗

h

) = Dp

,

U(s∗

l

) = Dp

(1 − 1

(1 − �)

).

Because of the dual uses of accounting information, the contract with theagent also depends on the earnings reports and is thus, in turn, affected bythe current owner’s � choice. Recall when 1 − � approaches its maximumof one, the resulting accounting reports are the most informative. Thus, as1 − � increases, the risk imposed on the agent, and hence the differencebetween s∗

h and s∗l , should become smaller. This is easily verified by noting

that

∂(s∗

h − s∗l

)∂(1 − �)

= ∂s∗h

∂(1 − �)− ∂s∗

l

∂(1 − �)= − D

p1

U ′(sl )(1 − �)2< 0.

8 Conditioning the agent’s payments on earnings is equivalent to conditioning on price

because price here is uniquely determined by a given earnings realization.


Stated differently, earnings management is costly to the current owner be-cause a higher � introduces more noise into the system and leads to inferiorrisk sharing between the current owner and the agent. As a result, earningsmanagement increases the current owner’s expected compensation to theagent, as shown formally by the following derivative:

∂E (s∗)

∂�= − ∂E (s∗)

∂(1 − �)= (1 − p)

[(s∗

h − s∗l

) − U(s∗

h

) − U(s∗

l

)U ′(s∗

l

) ]> 0 (1)

The last inequality follows as U (·) is a strictly concave function.To solve for the optimal �, substitute in the optimal s∗

h and s∗l and simplify

the current owner’s problem to:

max�

(p + (1 − p)�)(Peh − s∗

h

) + (1 − (p + (1 − p)�))(Pe l − s∗

l

).

Here, we only consider interior solutions. For convenience, to obtain thefirst-order condition, we take the derivative with respect to (1 − �) insteadof �:

FOC = −(1 − p)( Peh − Pe l ) − ∂E (s∗)

∂(1 − �).

Substituting in Peh − Pe l yields9

− ∂E (s∗)

∂(1 − �)= p(1 − p)

1 − (1 − p)(1 − �)xh . (2)

The equilibrium �∗ is obtained by setting � = � in the above equation. Asplotted in figure 4, the left-hand side of equation (2) is decreasing in 1 −�, while the right-hand side is concavely increasing in 1 − �, starting fromthe y axis at p(1 − p)xh with the two curves intersecting at most once.

The following proposition summarizes our results so far.

PROPOSITION 1. Suppose the parameter values are such that an interior solutionto equation (2) exists. Then, there exists an equilibrium where the current owner

9 Notice that the objective function is everywhere concave in 1 − �, as − ∂2 E (s∗)

∂(1 − �)2 < 0 and

hence any point satisfying equation (2) is the maximum. To see this,

− ∂2 E (s∗)

∂(1 − �)2= (1 − p)

⎧⎪⎪⎨⎪⎪⎩U ′′ (s∗

l

) ∂s∗l

∂(1 − �)

(U

(s∗

h) − U

(s∗l))

(U ′ (s∗

l

))2

−[

∂(U

(s∗

h

) − U(s∗l

))∂(1 − �)

1

U ′ (s∗l

) − ∂(s∗

h − s∗l

)∂(1 − �)

]⎫⎪⎪⎬⎪⎪⎭ .

Note that∂(U (s∗

h )−U (s∗l ))

∂(1 − �)1

U ′(s∗l )

= − ∂U (s∗l )

∂(1 − �)1

U ′(s∗l )

= − ∂s∗l

∂(1 − �) and∂(s∗

h − s∗l )

∂(1 − �) = − ∂s∗l

∂(1 − �) .

Thus, the two terms in the square brackets cancel each other out and the first term is strictly

negative.


FIG. 4.—Optimal �—unbiased standard.

introduces upward earnings manipulation to the accounting information system,i.e., �∗ > 0.

While Proposition 1 establishes that the current owner may optimallychoose to manipulate earnings, in equilibrium outside investors are notfooled by the earnings manipulation. Investors price protect themselves bytaking � into account when bidding for the firm. However, with the ma-nipulation, the current owner ends up paying more to motivate the agent(recall E(s) is an increasing function of �). Thus, the manipulation is in-efficient for the current owner. This inefficiency arises from the currentowner’s inability to commit not to manipulate earnings and the fact that ac-counting numbers are used in both external and internal reporting. In thenext section, we show that under the conservative standard the equilibriummanipulation can be reduced, and the current owner can achieve betterrisk sharing with the agent and lower his expected payment to the agent.

Before moving on to section 4, we detour for a moment to look atthe impact of a liberal accounting standard on the equilibrium. A liberalaccounting system is characterized by Pr(e h | xh) = 1, Pr(e h | xl ) = q , andPr(e l | xl ) = 1 − q . q here measures the degree of overestimation of true


output by the accounting system. Similar derivation yields the followingcondition for the equilibrium earnings manipulation �∗:

− ∂E (s∗)

∂(q − �)= p(1 − p)

1 − (1 − p)(q − �)xh .

Notice that the above expression implies d�∗dq = 1, thus d(q − �∗)

dq = 0, be-cause both sides are independent of q when holding q − � constant. Thisimplies that a better equilibrium risk sharing is not achieved by increasingq, that is, any attempt to make the accounting principles more liberal isperfectly offset by the current owner’s manipulation in equilibrium.

4. Conservative Accounting Standard

We now turn to solve the model under the conservative accounting stan-dard. Let �� represent the amount of earnings manipulation introduced bythe current owner at date 1 under the conservative accounting standard.10

For notational ease and for reasons that will be clear soon, define

Z ≡ 1 − �� − �.

Intuitively, since both �� and � are biases, Z roughly measures the infor-mativeness of the accounting system that eventually generates the earningsnumber.

Let �� be investors’ conjecture of �� and P �e h

( P �e l

) be their bidding pricefor the firm when the earnings report is eh(el ). Then

P �e h

≡ Pr(xh | e h, ��

)xh + Pr

(xl | e h, ��

)xl = p(1 − �)xh

η,

P �e l

≡ Pr(xh | e l , ��

)xh + Pr

(xl | e l , ��

)xl = p�xh

1 − η,

where

η ≡ Pr(e h

∣∣ ��, �) = p(1 − �) + (1 − p)��

= 1 − � − (1 − p)Z

is the probability of observing eh under the conservative standard when the

high effort is taken and η is investors’ conjecture of η under �� .Since

P �e h

− P �e l

= p(1 − p)Z xh

η(1 − η)> 0,

the current owner has an incentive to make eh appear more frequently bychoosing a positive �� . However, �� is now a function of �, the degree ofconservatism in the accounting system. Next, we identify conditions under

10 An implicit assumption in this section is that the current owner cannot reduce �. This

can be justified because accounting standards are enforced by regulators and monitored by

auditors. It is therefore unlikely that firms can completely undo the effect of these standards.


which a positive degree of conservatism is desirable in that it reduces ��

and improves contracting efficiency.The current owner’s problem at date 1 is very similar to that in section 3.

Let s�h and s�

l be the agent’s compensation if e = e h and e = el , respectively.The current owner chooses s�

h , s�l , and �� to

maxs�

h ,s�l ,��

η(P �

e h− s�

h

) + (1 − η)(P �

e l− s�

l

)s .t. ηU

(s�

h

) + (1 − η)U(s�

l

) − D ≥ ��U(s�

h

) + (1 − ��)U(s�

l

),

ηU(s�

h

) + (1 − η)U(s�

l

) − D ≥ 0.

Since the solution binds at both constraints, the optimal s�∗h and s�∗

l solvethe following equations:

U(s�∗

h

) =(

1 + �

Z

)Dp

, (3)

U(s�∗

l

) =(

1 + �

Z− 1

Z

)Dp

. (4)

Let E(s�∗) denote the expected payment to the agent. The following lemmaestablishes that E(s�∗) is decreasing in Z . All remaining proofs are relegatedto the appendix.

LEMMA. ∂E (s�∗)∂Z < 0.

Intuitively, since Z ≡ 1 − �� − � measures the informativeness of ac-counting earnings, a higher Z reduces the agency cost.

Substituting s�∗h and s�∗

l into the current owner’s problem yields:

max��

η(P �

e h− s�∗

h

) + (1 − η)(P �

e l− s�∗

l

).

Using a similar approach to that in section 3, it can be shown that thisobjective function is concave in Z . Since the objective function does notdepend on �� once Z is held constant, we can reparameterize the problemusing Z instead of �� . Again, assuming an interior solution, we have thefollowing first-order condition with respect to Z :

(1 − p)(P �

e h− P �

e l

) = −∂E (s�)

∂Z. (5)

The optimal Z ∗ is obtained by setting Z = Z in the above equation.Our main focus is to analyze the effect of conservatism on the current

owner’s earnings manipulation choice, d��

d�, which is equivalent to analyzing

d Z∗d�

with Z ∗ defined by equation (5). The left-hand side of equation (5),

(1 − p)( P �e h

− P �e l

), represents the current owner’s marginal benefit fromearnings manipulation. The marginal effect of conservatism on this term is

∂( P �e h

− P �e l

)

∂�

∣∣∣∣∣�=0

= (1 − 2η)p(1 − p)2 Zη2(1 − η)2

xh .


Since p ≥ 12, η|�=0 = p + (1 − p)�� > 1

2, the above term is strictly negative,

implying that a higher � decreases the marginal benefit of manipulatingearnings.

The right-hand side of equation (5) represents the current owner’smarginal cost of earnings manipulation in terms of the agent’s incentivecompensation. The impact of � on this term is complicated and dependson the agent’s utility function. The following proposition, which is a key re-sult of our paper, identifies a sufficient condition under which conservatismhelps reduce the equilibrium level of earnings manipulation by the currentowner.

PROPOSITION 2. When � is relatively small, a sufficient condition for Z ∗ to beincreasing in � is that −U ′′

lU ′

l

U ′h

U ′l

Uh − UlU ′

l − U ′h

≤ 11 − p .

Proposition 2 outlines a sufficient condition for d Z∗d�

> 0. Since Z =1 − �� − �, d Z∗

d�> 0 implies that d�∗�

d�< −1, that is, one unit of conserva-

tive noise reduces the noise introduced by earnings manipulation by morethan one unit. The sufficient condition in Proposition 2 guarantees that

− ∂2 E (s∗)∂Z∂�

|�=0 > 0, which ensures that accounting conservatism increases the

marginal cost of earnings management (captured by − ∂E (s∗)∂Z ). Because we

are not confining our attention to a specific class of preferences/utilityfunctions, this condition appears somewhat technical. The appearance isdeceiving since the condition is satisfied for a wide range of standard utilityfunctions, such as the negative exponential, the logarithmic, the quadratic,and the power class with relative risk aversion less than or equal to one-half.In the case of higher levels of relative risk aversion, the condition remainssatisfied as long as the difference in the agent’s payments is not too high.11

Thus, Proposition 2 shows that conservatism can reduce the equilibriumlevel of earnings manipulation under fairly general conditions.

The intuition behind Proposition 2 is the following. Observable conserva-tive noise reduces the current owner’s desire to manipulate earnings for two

reasons. First, given investors’ conjecture �� , such noise makes good newsless good and bad news less bad. That is, adding some known (stochastic)conservative bias reduces the impact of news on share prices and conse-quently reduces the benefit of earnings management, given potential in-vestors’ belief. Second, at the same time, conservative noise makes it morecostly for the current owner to motivate the agent to work, thus increasingthe cost of earnings manipulation. As a result, the current owner finds itdesirable to engage in less earnings management, and investors rationallyincorporate this into the price, resulting in a lower level of earnings man-agement in equilibrium.

Figure 5 provides a graphical representation of the optimal � under theunbiased standard and under the conservative standard. It shows that when� increases from zero to positive, the optimal 1 − �∗ − � increases.

11 Details available from the authors upon request.


FIG. 5.—Optimal ��—unbiased and conservative standards.

The effect of conservative accounting on contracting efficiency is cap-tured by �’s net impact on the current owner’s expected payment to theagent, that is,

dE(s�∗)

d�= ∂E (s∗(Z , �))

∂Zd Zd�

+ ∂E (s∗(Z , �))

∂�.

The first term is the indirect effect of conservatism on the expected paymentto the agent: It reduces the noise introduced by earnings management (i.e.,

�) and thus reduces the expected payment (as d Zd�

> 0 and ∂E (s∗(Z ,�))∂Z < 0).

The second term is the direct effect of conservatism on the expected pay-ment. Expanding the second term, we have

∂E (s(Z , �))

∂�= −(sh − sl ) +

[η∂sh

∂�+ (1 − η)

∂sh

∂�

].

The −(sh − sl ) term captures the fact that a higher � makes it more likely thatthe agent is paid the lower salary, sl , hence reducing the expected payment.The last two terms capture the fact that conservatism itself introduces noiseinto the accounting system and hence reduces efficient risk sharing, whichleads to higher expected compensation. The following proposition outlinesa sufficient condition under which the net of these effects is in favor ofconservatism.

PROPOSITION 3. Suppose the conditions of Proposition 2 are satisfied. Then,when � is relatively small, a sufficient condition for the expected payment to the agentto be decreasing in � is 1

U ′hη + 1

U ′l(1 − η) ≤ sh − sl

Uh − Ul.


Certainly the additional (sufficient) condition in Proposition 3 is notentirely without teeth in the sense that Proposition 3 does not apply in allcases where Proposition 2 applies. However, it is not overly restrictive either.For example, it can easily be verified that this condition can be met forvarious standard preference representations. For example, it is met when theagent’s utility function belongs to the CARA family (constant absolute riskaversion), provided that the absolute risk aversion is not too low. Similarly,for utility functions in the CRRA family (constant relative risk aversion), thecondition is satisfied as long as the risk in the agent’s contract is not toohigh.

We know from common intuition that it is more costly to impose risk onagents with higher risk aversion. This suggests that when the agent is rela-tively risk averse, a conservative accounting standard achieves a higher effi-ciency than an unbiased standard. Intuitively, when the agent is risk averse,more informative accounting reports become more valuable because theyimprove risk sharing. To the extent that conservatism reduces the equilib-rium earnings management and improves the informativeness of accountingreports (as established in Proposition 2), it has a more pronounced cost sav-ing effect for the current owner when the contract inefficiency introducedby earnings management is large.

Proposition 3 shows that conservatism can help firms escape the ineffi-cient equilibrium. This does not, of course, imply that conservatism is theonly way this can be done. It is, for example, possible that, if the inefficiencyidentified in the model is sufficiently severe, firms may have incentives tocreate separate internal performance measures for the purpose of evaluat-ing managerial performance. However, it is unlikely that such alternativeswould completely substitute for the beneficial effects of conservatism wehave identified. Rather, we conjecture that conservatism would always bepart of the mixture of solutions. This conjecture is based on the followingtwo observations.

The first derives from the relative credibility of conservatism vis-a -vis alter-native mechanisms. One key reason for conservative accounting standardsto be beneficial in our model is that they reduce the perceived benefits ofearnings management; they can do so because accounting standards arepublicly observable, and therefore credible. This contrasts with many otheralternatives, such as using separate internal performance measures, whichdue to being private in nature are unlikely to be credible to outsiders. Theinability of firms to credibly commit to unobservable actions is precisely thereason they are trapped in the inefficient equilibrium to begin with in ourmodel.12

Second, and equally if not more importantly, as long as the alternativeperformance measures are not sufficient statistics for the external financial

12 Otherwise firms can always announce that they will commit not to managing earnings. If

the credibility of such commitment is not an issue, then there is indeed no role for conservatism

in accounting standards.


reports with respect to the agent’s effort, it is always optimal for firms to usethe external financial reports for the purpose of evaluating the agent. Asdiscussed in footnote 3, in general, the fact that accounting numbers areuseful for valuation purposes suggests that they also have stewardship value.Accordingly, it seems hard to imagine circumstances under which the exter-nal financial reports would not carry the dual role underlying our analysis.This implies that the value of conservatism as established in Proposition 3would persist even if we were to allow explicitly in our model that the currentowner can rely on other measures.

Separately, we acknowledge that our model is a static one and thus doesnot explicitly consider the dynamic aspect of accrual accounting. While afull-scale analysis of the effects of accrual reversal lies outside the scope of thispaper, we offer some preliminary intuitions here. If the second-generationshareholders need to hire an agent to produce so they can sell to a thirdgeneration before the new output is realized, there is no fundamental dif-ference between the second generation’s problem and the first generation’sproblem. That is, the second-generation shareholders would still be temptedto manage earnings on their own and hence be trapped in the inefficientequilibrium. As a result, conservative accounting standards for the secondgeneration still help reduce the inefficiency, which is the main point of thispaper. This does not depend on the source of the baseline properties of theaccounting system, including reversals of prior periods’ accruals.13

If the second-generation shareholders do not need to sell to a third gener-ation but still need to hire an agent to produce, they would take into accountthe effects of the reversal and pay a lower price to the current owner (be-cause reversal introduces noise and makes it costly to motivate the agent towork). While this may reduce the current owner’s incentive to manage earn-ings, it does not necessarily eliminate the incentive. Thus, as long as thereis a positive amount of earnings management in equilibrium, conservatismhas the potential for helping to reduce such earnings management.

5. Conclusion

In this paper, we explore the role of conservative accounting standards incombating (unobservable) earnings management. When accounting infor-mation serves dual purposes (i.e., valuation and stewardship purposes), ourmodel shows that the current owner of a firm has an incentive to engagein earnings manipulation activities in hope of boosting the market price ofhis firm. However, potential investors rationally expect and price protectagainst the earnings manipulation. Lacking the tools for committing not tomanage earnings, the current owner is trapped by such expectations and

13 Also note that because conservatism in our setting reduces the equilibrium (liberal) earn-

ings management, it is not clear that the net outcome of a conservative system is conservative or

liberal. Therefore, it is not clear that the reversal of total accruals under a conservative system

leads to a liberal accounting report in the next period either.


therefore has to manage earnings in order to fulfill potential investors’ ratio-nal conjecture. As a result, inefficient earnings manipulation exists, addingnot just bias but also noise to the performance measure used to motivatethe firm’s employees.

More significantly, however, our analysis also shows that employing con-servative accounting standards can actually reduce the equilibrium amountof both noise and bias and thus help achieve a more efficient equilibriumoutcome. This occurs despite the fact that conservative accounting itself in-troduces undesirable biases/noises and that the current owner maintainsall manipulation options. As shown, the reason it works is that adding someknown (stochastic) conservative bias reduces the impact of news on shareprices and consequently reduces the benefits of earnings management,given potential investors’ belief. In addition, conservative bias also increasesthe marginal cost of earnings manipulation. Surprisingly, adding one typeof bias can reduce the other so much that the total equilibrium amount ofnoise in the accounting system is diminished.

Although accounting conservatism is an active area of research in empiri-cal accounting, theories explaining its existence and prominence have beensparse. Our paper formally establishes the link between accounting conser-vatism and earnings management. In particular, by demonstrating the abilityof conservative distortions to reduce incentives for earnings managementto the point where the noise introduced by conservatism is more than off-set by the reduction in the noise introduced by earnings management, wehighlight the potential for economic benefits of observable conservative re-porting distortions, even if such distortions are themselves inefficient whenviewed in isolation.

APPENDIX

Proof of Lemma.

∂E (s)

∂Z= −(1 − p)(sh − sl ) + η

∂sh

∂Z+ (1 − η)

∂sl

∂Z

= −(1 − p)(sh − sl ) + η

∂U −1

(Dp

(1 + �

Z

))∂Z

+ (1 − η)

∂U −1

(Dp

(1 + �

Z− 1

Z

))∂Z

= −(1 − p)(sh − sl ) + ηDp

1

U ′h

−�

Z2+ (1 − η)

Dp

1

U ′l

1 − �

Z2

≤ −(1 − p)(sh − sl ) + ηDp

1

U ′l

−�

Z2+ (1 − η)

Dp

1

U ′l

1 − �

Z2.


The inequality uses the fact that U is an increasing, concave function, i.e.,U ′

l > U ′h > 0. Further simplifying the above expression yields

∂E (s)

∂Z≤ −(1 − p)(sh − sl ) + 1

U ′l

Dp

1

Z2[(1 − η)(1 − �) − η�]

= −(1 − p)(sh − sl ) + 1

U ′l

DpZ

(1 − p).

Substitute in Uh − Ul = DpZ , we have

∂E (s)

∂Z≤ (1 − p)

[Uh − Ul

U ′l

− (sh − sl )

]< 0,

where the last inequality obtains because U ′′ < 0. �Proof of Proposition 2. The optimal Z under the conservative standard is

obtained by setting FOC (below) to zero and substituting Z (in P �e h

and P �e l

)with Z .

FOC = −∂E (s�)

∂Z− (1 − p)( P �

e h− P �

e l).

Hence, using implicit function theorem, we have

d Z∗

d�= −

∂FOC∂�

∂FOC∂Z

∣∣∣∣∣∣∣Z=Z

=∂FOC∂�

∂2 E (s�)

∂Z2+ (1 − p)

∂(P �e h

− P �e l

)

∂Z

.

Under the assumption that an interior optimal solution exists, the

second order condition is SOC = − ∂2 E (s�)∂Z2 < 0, hence ∂2 E (s�)

∂Z2 > 0. Since∂(P �

eh−P �

el)

∂Z |�=0 = 1η2 p(1 − p)xh > 0, the denominator is positive. Then, if we

can show ∂FOC∂�

> 0 under the condition in Proposition 2, we are done.

∂FOC∂�

= −∂2 E (s)

∂Z∂�+ p(1 − p)2 Zxh

[η(1 − η)]2(2η − 1).

When p ≥ 1/2, η = p + (1 − p)� ≥ 1/2, therefore, the second termabove is positive. In the following, we identify the condition under which

− ∂2 E (s)∂Z∂�

|�=0 > 0. Recall

E (s) = ηsh + (1 − η)sl , where η = 1 − � − (1 − p)Z .

Hence

∂E (s)

∂�= η

∂sh

∂�+ (1 − η)

∂sl

∂�+ ∂η

∂�(sh − sl ),


and

∂2 E (s)

∂Z∂�= ∂η

∂Z∂sh

∂�+ η

∂2sh

∂�∂Z+ ∂(1 − η)

∂Z∂sl

∂�+ (1 − η)

∂2sl

∂�∂Z− ∂(sh − sl )

∂Z

= −(1 − p)∂(sh − sl )

∂�− ∂(sh − sl )

∂Z+ η

∂2sh

∂�∂Z+ (1 − η)

∂2sl

∂�∂Z.

(6)

From equations (3) and (4), we derive the detailed expressions for thepartials listed above, evaluated at � = 0. We have

∂sh

∂�= D

pZ1

U ′h

;∂sh

∂Z= ∂sh

∂�

−�

Z

∣∣∣∣�=0

= 0;

∂2sh

∂�∂Z=

(−1

Z∂sh

∂�+ −�

Z∂2sh

∂�2

)∣∣∣∣�=0

= −1

Z∂sh

∂�;

∂sl

∂�= D

pZ1

U ′l;

∂2sl

∂�2= D

pZ−U ′′

l

U ′2l

∂sl

∂�= −U ′′

l

U ′l

(∂sl

∂�

)2

;

∂sl

∂Z=

(DpZ

1

U ′l

1 − �

Z

)∣∣∣∣�=0

=(

∂sl

∂�

1 − �

Z

)∣∣∣∣�=0

= 1

Z∂sl

∂�;

∂2sl

∂�∂Z=

(−1

Z∂sl

∂�+ 1 − �

Z∂2sl

∂�2

)∣∣∣∣�=0

= −1

Z∂sl

∂�+ 1

Z−U ′′

l

U ′l

(∂sl

∂�

)2

.

Substituting these expressions into equation (6) and collecting termsyields

∂2 E (s)

∂Z∂�

∣∣∣∣�=0

= −(1 − p)∂(sh − sl )

∂�− ∂(sh − sl )

∂Z+ η

(−1

Z∂sh

∂�

)

+ (1 − η)

[−1

Z∂sl

∂�+ 1

Z−U ′′

l

U ′l

(∂sl

∂�

)2]

= DpZ

(1

U ′h

− 1

U ′l

)︸︷︷︸

>0

⎡⎢⎢⎢⎣−1

Z+ (1 − p)

−U ′′l

U ′l

DpZ

(1

U ′l

)2 1(1

U ′h

− 1

U ′l

)︸︷︷︸

(∗)

⎤⎥⎥⎥⎦ .

Because Z < 1, a sufficient condition for the above expression to be negativeis (∗) ≤ 1

1 − p . The condition shown in Proposition 2 is obtained by replacingDpZ with Uh − Ul and rearranging terms. �

Proof of Proposition 3. Taking total derivative of E(s�∗) with respect to �,we have

dE(s�∗(Z , �))

d�= ∂E (s∗(Z , �))

∂Zd Zd�

+ ∂E (s∗(Z , �))

∂�.


Under the condition for Proposition 2, we have shown that d Zd�

= K > 0.

Also, the lemma proves that ∂E (s∗(�,�))∂Z < 0. So the first term in the above

expression is negative. Writing out the detailed expressions, we have

∂E (s∗(Z , �))

∂Z

∣∣∣∣�=0

=[−(1 − p)(sh − sl ) + η

∂sh

∂Z+ (1 − η)

∂sl

∂Z

]∣∣∣∣�=0

= −(1 − p)(sh − sl ) + (1 − η)DpZ

1

U ′l

1

Z,

and∂E (s∗(Z , �))

∂�

∣∣∣∣�=0

=[−(sh − sl ) + η

∂sh

∂�+ (1 − η)

∂sl

∂�

]∣∣∣∣�=0

= −(sh − sl ) + ηDpZ

1

U ′h

+ (1 − η)DpZ

1

U ′l.

Substituting them in dE(s�∗)d�

yields

dE(s�∗)

d�

∣∣∣∣�=0

= −(1 − p)K (sh − sl ) + (1 − η)KDpZ

1

U ′l

1

Z

+[−(sh − sl ) + η

DpZ

1

U ′h

+ (1 − η)DpZ

1

U ′l

]= −[(1 − p)K + 1](sh − sl ) + D

pZ

[η

1

U ′h

+ (1 − η)1

U ′l

(KZ

+ 1

)]= − [(1 − p)K + 1] (sh − sl )

+ (Uh − Ul )

[1

U ′hη + 1

U ′l(1 − η) + 1

U ′l(1 − p)K

]︸︷︷︸

A

,

where the last equality is obtained by substituting in (Uh − Ul ) = DpZ and

(1 − η)|�=0 = (1 − p)Z . Thus, to show dE(s�∗)d�

< 0, we need to show

Ash − sl

< 1 + (1 − p)K , (7)

which is equivalent to

1

U ′hη + 1

U ′l(1 − η) + 1

U ′l(1 − p)K < ((1 − p)K + 1)

sh − sl

Uh − Ul.

Collecting terms, we have

1

U ′hη + 1

U ′l(1 − η) <

sh − sl

Uh − Ul+

(sh − sl

Uh − Ul− 1

U ′l

)(1 − p)K .

Since K > 0 (from Proposition 2) and sh − slUh − Ul

− 1U ′

l> 0 (by the concavity of

the utility function), a sufficient condition for the above inequality is


1

U ′hη + 1

U ′l(1 − η) ≤ sh − sl

Uh − Ul. (8)

Note that equation (8) can be further simplified by using the equilibrium

equation (2) and substituting in the expression for − ∂E (s∗)∂(1 − �)

. Specifically,equation (2) implies

sh − sl − Uh − Ul

U ′l

= p1 − (1 − p)(1 − �)

xh > pxh

sh − sl

Uh − Ul>

1

U ′l

+ pxh

Uh − Ul.

Thus, a sufficient condition for equation (8) is

1

U ′hη + 1

U ′l(1 − η) ≤ 1

U ′l

+ pxh

Uh − Ul,

which can be simplified as

(Uh − Ul )

(1

U ′h

− 1

U ′l

)≤ pxh .

Since both terms at the left-hand side of the above expression are increasingin sh − sl , the above condition says that for a given p and xh, as long as thedifference between sh and sl is not too large, equation (8) is satisfied.

In the case of CARA utility functions, note that the condition in Proposi-tion 2 implies that

(Uh − Ul ) ≤ 1

(1 − p)

(U ′l )

2

(1

U ′h

− 1

U ′l

)(−U ′′

l

U ′l

) =(U ′

l )2

(1

U ′h

− 1

U ′l

)(1 − p)R

,

where R = −U ′′/U ′ is the absolute risk aversion parameter. Substitute thisin (Uh − Ul )( 1

U ′h

− 1U ′

l) ≤ pxh and simplify, we have(

U ′l

U ′h

− 1

)2

≤ (1 − p)pxh R.

Thus, for a given p and xh, the above condition is satisfied as long asU ′

lU ′

his

not too far away, and/or R is not too low. �

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