The Impact of Personal Income Taxation on Executive ...

The Impact of Personal Income Taxation onExecutive Compensation 1

Peter KatuscakDepartment of Economics

University of Michigan

601 Tappan StreetAnn Arbor, MI 48109-1220, USA

Email: [email protected]: (734) 604-6327

(Job Market Paper)

October 27, 2003

1I would like to thank the members of my dissertation committee, namely Joel Slemrod,Jim Hines, John Laitner, and Yan Chen, as well as John Bound, Charlie Brown, MichelleHanlon, Clemens Sialm, Dan Silverman, and seminar participants at the University ofMichigan and the University of Iceland for comments on an earlier draft of this paper.

Abstract

This paper analyzes the effect of personal income taxation on the sensitivity ofexecutive compensation to company performance and the use of stock option andrestricted stock grants underlying this sensitivity. The theoretical model predictsthat, if a single tax rate applies to the entire compensation package, an increasein the tax rate weakly diminishes the equilibrium level of managerial effort andthe after-tax pay-to-performance sensitivity, with the effect on the pre-tax pay-to-performance sensitivity being ambiguous. When salary and option gains are taxed ata higher rate than stock gains, tax changes also induce shifting among the individualcompensation instruments. Interestingly, this differential taxation leads to a positiveamount of stocks in the compensation contract even in the absence of any desire toincentivize the manager. In addition, an increase in the tax rate applied to salaryand option gains may increase the level of managerial effort in equilibrium. On theother hand, an increase in the tax rate applied to stock gains weakly decreases theequilibrium level of managerial effort and strictly decreases the amount of after-tax pay-to-performance sensitivity generated by stock grants. The second part ofthe paper uses the increase in the U.S. federal personal income tax rate legislatedby the Omnibus Budget Reconciliation Act of 1993 as a “natural experiment” toempirically evaluate the impact of personal income taxation on the pay-to-stock-price sensitivity generated by stock-option and restricted stock grants. The resultsreveal that the tax increase decreased the pay-to-stock-price sensitivity generatedby option grants, both pre-tax and after-tax, consistently with the prediction ofthe one-tax model. In addition, the pre-tax effect is almost entirely concentratedamong companies with a relatively weak corporate governance structure, suggestingthat executive pay may be driven by different factors in strong and weak corporategovernance companies.

1 Introduction

The use of performance-based pay in corporate executive compensation has sky-rocketed in the last 25 years (Hall and Liebman (1998), Murphy (1999)). Thepresumed objective of this compensation design is to align the interests of com-pany’s executives with the interests of its shareholders by relating the executive payto company performance. However, personal income taxes create a wedge betweenwhat the companies pay their executives and what the executives receive after tax.One important implication of this observation is that it is the after-tax rather thanthe pre-tax pay-to-performance sensitivity that accounts for the strength of incen-tives from the executives’ viewpoint. It is therefore interesting to ask to what extentcompanies perceive this tax wedge and adjust their pre-tax compensation packagesin response to changes in the personal income tax system. Since incentive-basedpay is nowadays used in a majority of publicly-traded companies in the U.S., thisquestion is of significant importance for both positive and normative analysis. Inparticular, understanding how personal income taxation affects the design of ex-ecutive compensation is key to evaluating the distortionary impact of taxation onincentives of an important group of individuals.

There have been several previous studies dealing with the impact of personalincome taxation on executive compensation. Hite and Long (1982), Miller andScholes (1982), and Hall and Liebman (2000) investigate the impact of the taxsystem on the design of executive compensation from the perspective of the optimalcompensation deferral using the “global contracting perspective” of Scholes andWolfson (1992). They argue that salary, restricted stock grants, and stock optiongrants can all be understood as different means of compensation deferral. Then,depending on the relative tax burden of these instruments, a company, holding theafter-tax present value of the payout to the manager constant, chooses an instrumentwith the lowest present value of after-tax cost to its shareholders. Hall and Liebman(2000) also provide an empirical test of this theory. They find little evidence that taxchanges in the 1980’s and early 1990’s had any significant effect on the compositionof executive compensation.1

None of these studies, however, explicitly considers how personal income taxation

1There is also literature dealing with the impact on executive compensation of other aspectsand provisions of the tax law. Hall and Liebman (2000), Perry and Zenner (2001) and Rose andWolfram (2002) investigate the impact of the “million-dollar rule” instituted in 1993, and effectivesince 1994, which limits the corporate income tax deductibility of non-incentive-based executivepay to 1 million dollars. They conclude that companies whose executives’ salaries were close tothe threshold limited their post-1993 salary growth and used more incentive-based pay instead.However, the effect of the legislation in reducing salary growth appears to be only modest, andthere is no significant effect on the level of total compensation. Crocker and Slemrod (2003)theoretically model how, in the hidden information model, the principal optimally designs thecompensation scheme of the manager when the manager makes decisions about corporate taxevasion that benefits the principal. Chen and Chu (2003) theoretically investigate the tradeoffbetween gains from corporate tax evasion and the associated loss in internal control of the firmwhen accounting books are “cooked” to aid tax evasion. In a recent paper, Desai et al. (2003)explore the links between corporate governance and corporate tax collection, arguing that thereare positive spillovers from improving either of them on the other one.

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of executives impacts the design of their compensation in the context of agencytheory. Although stock option and restricted stock grants certainly provide meansfor compensation deferral from a tax standpoint, it is not clear why a companywould resort to using these risky instruments to compensate a risk-averse manager,which increases the cost of compensation by a risk premium. The company couldinstead use non-risky instruments such as deferred salary or more risk-diversifiedand still tax-deferred instruments such as pension contributions. This criticism isespecially true of using options since, upon exercise, they are subject to the sametax rate as cash compensation. Therefore it is likely that the use of stock optionand restricted stock grants is presumably mostly driven by a desire to motivate theexecutives. As a result, this paper uses agency theory as a theoretical backgroundfor studying the impact of personal income taxation on executive compensation.

In the principal-agent model, a company (the principal) provides its executive(the manager) an incentive to expend effort via relating his pay to company perfor-mance, labelled as pay-to-performance sensitivity. In practice, this may be imple-mented by bonuses, stock-option and stock-appreciation rights grants, and restrictedstock grants. In this framework, an increase in the income tax rate affects the de-sign of a compensation contract in two ways. First, it decreases the utility of theoutside option that needs to be met by the contract. This first effect, labelled as“average tax incidence,” is analogous to the standard tax incidence result that anincreased labor income tax leads to a lower after-tax wage rate. Second, it increasesthe wedge between pre-tax and after-tax effort incentives on the margin, which canbe thought of as the slope of the compensation in the performance measure. Asa result, the principal may desire to undo (a part of) this effect by increasing thepre-tax incentives on the margin or, in the case of differential taxation, by reallocat-ing compensation to relatively less heavily taxed compensation instruments. Thissecond effect is labelled as “marginal tax incidence.”

This paper is concerned with the second effect. Modelling the magnitude of thefirst effect requires modelling the labor market for executives, which is beyond thescope of this paper. However, even though the extent of the change in the utility ofthe outside option is not explicitly modelled, the model incorporates the fact thatthis utility level is affected by taxation.

The theoretical model leads to several insights. In the benchmark version of themodel when there is only a single income tax rate that applies to all the instrumentsof compensation, the model predicts that an increase in the tax rate weakly dimin-ishes the equilibrium level of managerial effort and the after-tax pay-to-performancesensitivity faced by the manager after the company has optimally adjusted the con-tract. On the other hand, the effect on the pre-tax pay-to-performance sensitivityis ambiguous.

This one-tax model may not be empirically relevant for many executives in theU.S., though. This is because the U.S. personal income tax system allows stockand option gains to be taxed at different rates. In particular, salary, bonus, grantvalue of restricted stock grants and option gains are all taxed at the personal rate,but an executive may elect to tax the capital appreciation of restricted stocks at the

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capital gains rate.2 This distinction poses a new theoretical challenge, as most of thetraditional principal-agent analysis does not explicitly differentiate between constant(salary), linear (stocks), and convex (options) payout instruments of the contract.3

Theoretical results in this modified setting crucially depend on which of the two taxrates is higher and on the magnitude of expected capital gains at the time restrictedstocks are granted. If the personal rate does not exceed the capital gains rate, thecapital gains rate is irrelevant for the design of the contract, and all the conclusionsparallel the results of the simpler one-tax model. However, if the personal rate isabove the capital gains rate, stocks have a tax advantage over salary and options asan instrument of compensation. The distortionary effect of the differential taxationleads to some interesting results. Despite the fundamental result from the principal-agent theory that when the objective to incentivize the manager is absent, optimalrisk sharing between a risk-neutral principal and a risk-averse manager calls fora fixed compensation regardless of the realized performance, a positive amount ofstocks will be used in the compensation contract. Another counterintuitive result isthat an increase in the tax rate applied to salary and option gains may increase thelevel of managerial effort in equilibrium. As a result, this two-tax model generatesambiguous predictions for the shape of the compensation contract in response to achange in the personal income tax rate. However, an increase in the capital gains taxrate weakly decreases the equilibrium level of managerial effort and strictly decreasesthe amount of after-tax stock incentives used in the compensation contract.

The second part of the paper presents an empirical evidence on the impact of thetax increase legislated by the Omnibus Budget Reconciliation Act of 1993 on thedesign of executive compensation. This paper offers several extensions on the earlierwork of Hall and Liebman (2000). First, apart from stock options, it accounts forrestricted stock grants as means of providing incentives. Separately investigatingthe restricted stock grants can potentially reveal a substitution from options intostocks following an increase in the personal income tax rate.

Second, it features alternative and arguably superior dependent variables to theone used by Hall and Liebman. They use as their dependent variable the shareof annual compensation, defined as salary, bonus, and the Black-Scholes value ofstock-option grants, that is paid in options. The theoretical prediction is that ahigher personal income tax rate lowers the amount of options used in the compen-sation contract. However, agency theory suggests examining more direct measuresof the use of incentive devices in the compensation contract: the pay-to-stock-pricesensitivity generated by annual grants of stock options and the analogous sensitivitygenerated by annual grants of restricted stocks.

Third, Hall and Liebman identify personal income tax effects solely from timevariation in the top personal income tax rate, which makes it difficult to separateeffects of tax changes from pure time effects. This is because, in the dataset thatthey use, almost all of the executives in all years are in the top tax bracket. Inorder to evaluate the extent of this identification problem, this study uses a morecomprehensive dataset from Compustat’s Execucomp for the years 1992 through

2See the discussion in the next section.3See Hemmer et al. (2000) for an exception.

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1996. The advantage of using this dataset is that it contains executives who canbe classified as being below the top tax bracket in 1993-96. As a result, the 1993personal income tax increase creates different tax increases for executives in the toptax bracket and executives in the second tax bracket. The presence of this cross-sectional variation in the tax increase makes it possible to check on the robustnessof empirical results that rely solely on time variation in the top tax rate. Moreprecisely, it allows identification of the tax effect from time variation in both of thetax brackets. In addition, it also makes feasible the use of difference-in-differencesidentification.

The empirical analysis reveals that the tax increase decreased the pay-to-perfor-mance sensitivity generated by option grants. There is, however, no statistically sig-nificant effect on the sensitivity generated by restricted stock grants, and the resultsfor the effect of the share of compensation paid in options (the variable used by Halland Liebman (2000)) are inconclusive. I also investigate whether there is any het-erogeneity in the effect of the tax increase depending on the strength of a company’scorporate governance. This is motivated by the hypothesis that while the executivepay design may reflect the considerations of agency theory in strong corporate gov-ernance companies, it may reflect managers paying themselves and tax optimizingin the process without much regard to incentivization among weak corporate gov-ernance companies. When the sample is split into companies with strong and weakcorporate governance, as defined by the corporate governance index of Gompers etal. (2003), the results show that the tax-induced decrease in the sensitivity of optiongrants is only statistically significant among the weak corporate governance com-panies, and this effect is statistically significantly different between the two groups,consistently with the hypothesis outlined above. Furthermore, the share of compen-sation paid in options decreases among the weak corporate governance companies,and is unaffected or increases among the strong corporate governance companies,with the difference being statistically significant. On the other hand, there are nosignificant differences in the tax effect on the sensitivity generated by restrictedstock grants.

This paper proceeds as follows. Section 2 outlines the U.S. federal tax rulesthat apply to various instruments of executive compensation. Section 3 providesa principal-agent analysis of the impact of personal income taxation on executivecompensation. It first looks at the case when all the compensation instruments aretaxed at the same rate. It then proceeds to analyze a more complex case whensalary and option gains are taxed at a different rate than stock gains. Section 4describes the identification strategy and data used in the empirical analysis. Section5 displays the results of the empirical investigation. Section 6 concludes. All thetechnical proofs are relegated to the Appendix.

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TABLE 1: U.S. Federal Taxation Rules

Compensation Item Tax Rate When taxable?Salary Personal PayoutBonus Personal PayoutRestricted stocks, no 83(b) election Personal Restrictions lapseRestricted stock grant value, with 83(b) election Personal Grant dateRestricted stock gains, with 83(b) election C. Gains Sale of the stocksUnrestricted stock gains C. Gains Sale of the stocksNon-qualified stock option gains Personal ExerciseIncentive stock option and acquired stock gains C. Gains Sale of the stocksDividends Personal Payout

2 The Tax Treatment of Executive Compensation

in the U.S.

Executive compensation contracts usually consist of the following parts: salary,bonus, restricted stock grants, stock-option grants, and other compensation, suchas perquisites, signing bonuses, and severance payments (“golden parachutes”).4

Salary and bonus are taxable at the personal rate at the time of payout. In thedefault case, the market value of restricted stock grants is taxed at the personalrate when the restrictions lapse.5 Any subsequent capital gains or losses of (thenunrestricted) stocks are taxed at the capital gains rate. However, Section 83(b) ofthe Internal Revenue Code allows the executives to elect to tax the grant value of arestricted stock grant at the personal rate at the time of the grant, and tax any sub-sequent gains as long-term capital gains when the stock is eventually sold. Becausethe top statutory capital gains tax rate has been below the top statutory personalincome tax rate since 1991, it is possible that executives frequently make the 83(b)election.6,7 Dividend earnings resulting from ownership of restricted or unrestrictedstocks are taxable at the personal rate at the time of payout. Stock-option grantsmay be of two types: non-qualified stock options (NQSOs) and incentive stock op-

4Instead of restricted stock grants, companies sometimes award “restricted stock units,” or“phantom shares,” which mimic the payoff of restricted stock grants. Similarly, instead of stock-option grants, companies sometimes award “stock appreciation rights,” or “phantom stock op-tions,” which mimic the payoff of stock-option grants.

5Restricted stock units are also taxed at the ordinary rate at the time of payout.6Unfortunately, the Internal Revenue Service does not collect data on the amount of restricted

stock grants that utilize the 83(b) election.7To be more precise, choosing the 83(b) election allows an executive to save on taxes in the

future by having the appreciation in his restricted stocks after the grant date taxed at a lower taxrate. However, it also triggers an immediate tax liability on the grant date value of the restrictedstocks, which could otherwise be deferred. Another disadvantage of making this election is thatany taxes paid at the grant date are nonrefundable if the restricted stock grant is later forfeitedwhen, for example, the executive leaves the company before the restrictions lapse. See McDonald(2003) for a more detailed account of the optimality of the 83(b) election.

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tions (ISOs). NQSO are taxed at the personal rate when exercised and the tax baseis the difference between the stock price and the exercise price on the day of theexercise.8 ISOs are taxed neither at the grant date, not at the time when they areexercised. Subject to satisfying a holding period restriction, all option profits andsubsequent capital gains on the acquired stocks are taxed at the capital gains ratewhen these stocks are eventually sold. However, ISOs are subject to an annual capof $100,000 on the amount that vests,9 and as a result they account for a relativelysmall portion of executive compensation (Hall and Liebman (2000)). I am thereforegoing to assume in this paper that all options granted to executives are NQSOs.The details of the tax treatment of these individual pay instruments are summa-rized in Table 1. Other compensation includes nontaxable items such as perquisitesand other personal benefits, 401k contributions, and life insurance premiums, butalso taxable items such as signing bonuses and severance payments.10

3 Model

A principal (shareholders) hires a manager (executive) to run the company. Themanager privately chooses a level of effort a ∈ A ⊂ R unobservable to the principal,where the set A is closed and bounded from below with a ≡ min(A). Conditionalon the effort level a, the distribution of firm’s profits x is given by the cumula-tive distribution function F (◦|a) over the support X ⊆ R. A higher level of effortleads to a “better” distribution of profits in the sense of first-order stochastic dom-inance. Throughout the model, x will be associated with the terminal price ofthe company stock. The manager is risk-averse and effort-averse. His utility whenconsuming m and exerting a level of effort a is given by the exponential utilityfunction U(m, a) = −e−r[m−c(a)], where r > 0 is the coefficient of (absolute) riskaversion, and c(·) is a strictly increasing and strictly convex cost of effort func-tion with c(a) = c′(a) = 0. Although somewhat restrictive, the exponential utilityfunction is used frequently in the analysis of the principal-agent problem.11 Theprincipal is risk-neutral and chooses the manager’s compensation contract w so asto maximize the expected terminal value of the company less the gross compensa-tion of the manager. In doing so, the principal faces two constraints. First, theincentive compatibility constraint (ICC) requires that the effort level is chosen so asto maximize the manager’s expected utility given the design of the contract, takinginto account any taxes that need to be paid on the compensation. Second, theparticipation constraint (PC) requires that the expected utility derived from theafter-tax contract payments and the optimally chosen effort level give the managerat least a reservation level of utility available from the next best employment option.

8Stock appreciation rights are taxed in a similar fashion.9The $100,000 limit applies to the fair market value of the stocks that can be acquired for these

options at the time of the grant. All ISOs above this limit are treated as NQSOs.10The information in this section draws on Hall and Liebman (2000) and the author’s own

investigation. Information contained at http://fairmark.com/execcomp/ is particularly useful.11See, for example, Holmstrom and Milgrom (1987), Gibbons and Murphy (1992), Aggarwal and

Samwick (1999a,1999b), Murphy (1999), Prendergast (1999), and Waldman (1999).

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The following two subsections add personal income taxation of managerial com-pensation to this model. Subsection 3.1 investigates the model in which there is asingle tax rate applying to all instruments of compensation. Subsection 3.2 thenextends the analysis to a more complicated case when constant and convex partsof the contract (salary, grant value of restricted stocks, and options) are taxed atthe personal rate, while the linear part of the contract (capital gains or losses fromrestricted stocks) is taxed at the capital gains rate.

3.1 One-tax System

Consider first the case when all income is taxed at a flat income tax rate t ∈ [0, 1).In addition, assume that the corporate profits net of managerial compensation aretaxed at a flat rate T ∈ [0, 1). Then the principal’s problem is given by

maxa∈A,w(◦)

(1− T )

∫[x− w(x)] dF (x|a) (1)

s.t. −∫

e−r[(1−t)w(x)−c(a)]dF (x|a) ≥ −e−ru(t) (2)

s.t. a ∈ arg maxba −∫

e−r[(1−t)w(x)−c(ba)]dF (x|a). (3)

In this formulation, (2) is the PC, and (3) is the ICC, and u(t) is the after-taxcertainty equivalent compensation net of the cost of the required effort level availableat the managerial job market. Because the income tax is economy-wide, u(t) is likelyto be decreasing in t. This follows from the standard intuition of tax incidence thatboth sides of the market, i.e., both companies and managers, will bear a part of theburden of an increase in the tax rate.12

Note that in this model the corporate income tax is a non-distortionary pureprofits tax that does not play any role in the design of an optimal contract, andtherefore it will subsequently be ignored. The analysis of the problem (1)-(3) furthersimplifies when the profit maximization problem is decomposed into two steps. First,the principal minimizes the expected cost of motivating any given level of effort. Forsimplicity, it is assumed that any a ∈ A is implementable. This leads to an expectedcost as a function of the induced level of effort and the tax rate. Second, the principalmaximizes the expected profit using the cost function derived in the first step.13 Toformalize this intuition, define the (expected) revenue function associated with aneffort level a by

R(a) ≡∫

xdF (x|a). (4)

Note that since F (·|a2) first-order stochastically dominates F (·, a1) whenever a2 >a1, R(a) is strictly increasing in a. Similarly, define the (expected) cost function

12Note that the formulation of the principal’s problem in (1)-(3) also allows for the manager tohave an exogenous stochastic income, as long as the distribution of this income is independent ofF for any given level of effort.

13This approach, without the presence of taxes, was pioneered by Grossman and Hart (1983).

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associated with an effort level a and the income tax rate t by

C(a, t) ≡ minw(◦)

∫w(x)dF (x|a) s.t. (2) and (3). (5)

Assuming that this cost-minimization problem has a unique solution s(·, a, t) for any(a, t), the cost minimization problem can equivalently be restated with the after-taxwage schedule being the choice variable as

z(·, a, u(t)) = arg minw(◦)

∫w(x)dF (x|a) (6)

s.t. −∫

e−r[w(x)−c(a)]dF (x|a) ≥ −e−ru(t) (7)

s.t. a ∈ arg maxba −∫

e−r[w(x)−c(ba)]dF (x|a). (8)

In this formulation, z(·, a, u(t)) ≡ (1 − t)s(·, a, t) is the cost-minimizing after-taxcompensation schedule, and

C(a, t) ≡∫

s(x, a, t)dF (x|a) ≡ 1

1− t

∫z(x, a, u(t))dF (x|a). (9)

Note that the only way in which the tax rate t enters the cost-minimization problem(6)-(8) is via the reservation level of pay u(t), which is reflected in the way z(·)depends on the tax rate t. The following lemma provides a partial characterizationof z(·), s(·), and C(·):Lemma 1 For all a ∈ A and t ∈ [0, 1),

z(x, a, u(t)) ≡ (1− t)s(x, a, t) = u(t) + z(x, a, 0),

and

C(a, t) =u(t)

1− t+

1

1− t

∫z(x, a, 0)dF (x|a).

Proof. See the Appendix.Lemma 1 shows that the cost-minimizing after-tax payment schedule z depends

on the tax rate t only to the extent that the level of z changes one-for-one with u(t).The slope of z with respect to x is unaffected by the tax rate t. When thinkingof the cost-minimizing incentive contract as consisting of salary and various incen-tive instruments such as restricted stocks and stock options, this result implies thatchanges in the income tax rate affect the after-tax salary, but leave the after-taxpayout of the incentive instruments unaffected for any given stock price realiza-tion. In other words, the principal fully absorbs changes in the tax rate affectingthe incentive instruments so as to leave the after-tax payout of these instrumentsunaffected. Lemma 1 also shows that an increase in the tax rate t affects the costof motivating effort in two ways. First, it affects the fixed cost of motivating anylevel of effort. The sign of this effect is, however, ambiguous in general. Second, itincreases the marginal cost of motivating a higher level of effort.

The partial characterization of the cost function in Lemma 1 leads to the follow-ing result:

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Proposition 1 For any t1, t2 ∈ [0, 1), t1 < t2,

sup

{arg max

a∈A[R(a)− C(a, t2)]

}≤ inf

{arg max

a∈A[R(a)− C(a, t1)]

}.

Proposition 1 shows that increasing the income tax rate results in a weakly lowermanagerial effort level in equilibrium. In other words, the “marginal incidence”of the tax increase penetrates through the contract in a way that reduces effortincentives. The following proof provides an intuition behind this result.

Proof of Proposition 1. Let a∗ be any optimal effort level under the tax ratet1. First note that it must be the case that C(a, t1) > C(a∗, t1) for any a > a∗.This is because since R(a) is strictly increasing in a, it would pay off to increase theeffort level from a∗ to any higher effort level whose cost does not exceed C(a∗, t1),contradicting the optimality of a∗. The optimality of a∗ under t1 implies that R(a∗)−C(a∗, t1) ≥ R(a) − C(a, t1), or, equivalently, C(a, t1) − C(a∗, t1) ≥ R(a) − R(a∗)for any a ∈ A. Now given the partial characterization of the cost function inLemma 1 and the fact that C(a, t1) > C(a∗, t1) for any a > a∗, an increase inthe tax rate from t1 to t2 strictly increases the cost differential C(a, t) − C(a∗, t)for any a > a∗. As a result, C(a, t2) − C(a∗, t2) > R(a) − R(a∗), or, equivalently,R(a∗)−C(a∗, t2) > R(a)−C(a, t2) for any a > a∗. This implies that no a > a∗ canbe optimal under t2. Because a∗ can be any optimal effort level under the tax ratet1, the result follows.

Although the result of Proposition 1 is quite general, given that no distributionalassumptions (apart from the ones assuring existence and uniqueness of the cost-minimizing contract for any effort level a ∈ A) have been made, it is also quitelimited for an empirical application because the manager’s effort is unobservable tothe econometrician. To evaluate its empirical validity, it is therefore necessary toderive implications for the shape of the incentive contract, such as the effect of a taxincrease on the use of stock-option and restricted stock grants that are observablein the data.

A standard way of characterizing the shape of a compensation contract in theprincipal-agent analysis with a continuous set of managerial effort choices A is calleda first-order approach. In this approach, the ICC is replaced by the correspondingfirst-order condition. Mirrlees (1999) and Rogerson (1985), and, alternatively, Jewitt(1988) provide two sets of sufficiency conditions for the validity of this approach.None of the commonly used continuous distributions that satisfy one of these twosets of sufficiency conditions, however, generates a convex compensation contract inthe current setting (Hemmer et al. (2000)). This is in contrast to the widespreadobserved use of stock options in managerial contracts, which introduces convexityinto the compensation schedule.

In order to circumvent this difficulty, the subsequent analysis will restrict atten-tion to the case when there are only two possible effort levels: a low effort levelaL(= a) and a high effort level aH , with aH > aL. In addition, with an eye towardsempirical application, the analysis will also be restricted to three possible realiza-tions of stock price: x1, x2, and x3, with x1 < x2 < x3. This simplification allows fora clear-cut characterization of the impact of taxation on the design of an incentive

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1w

2x 3x1x

2w

3w

PriceStock

PayoutContract

stocks

options

FIGURE 1: Managerial compensation in terms of stocks and options

contract. In particular, suppose that the profit-maximizing pre-tax contract payoutsare wi when the realized stock price is xi, i ∈ {1, 2, 3}. Then assuming monotonic-ity (w1 ≤ w2 ≤ w3) and convexity ((w3 − w2)/(x3 − x2) ≥ (w2 − w1)/(x2 − x1)),the amount of restricted stocks in the compensation contract can be capturedby (w2 − w1)/(x2 − x1), and the amount of stock options can be captured by(w3 − w2)/(x3 − x2) − (w2 − w1)/(x2 − x1), with the options being granted at thestrike price x2.

14 Figure 1 captures this intuition graphically.In this simplified setting, define v ≡ erc(aH), and note that v > 1. Then

U(m, a) =

{ −e−rm if a = aL

−e−rmv if a = aH.

Also denote the probability of realizing a stock price xi by pLi if a = aL, and by pH

i ifa = aH . Table 2 displays the distribution of profit realizations conditional on eacheffort level. Assume that pL

i , pHi > 0 for each i ∈ {1, 2, 3}, and

pL1

pH1

>pL

2

pH2

≥ v >pL

3

pH3

. (10)

This assumption combines the standard monotone likelihood ratio property withthe assumption that pL

2 /pH2 ≥ v. The assumption pL

1 /pH1 > v is necessary to as-

sure that aH is implementable (see the ICC (16)), while the assumption pL2 /pH

2 ≥ v(which, given the monotone likelihood ratio property, implies pL

1 /pH1 > v) is one

14Given three compensation points, there are multiple combinations of restricted stock and stockoption grants that can trace these compensation points. The purpose of the current example is toaid intuition.

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TABLE 2: Distribution of stock price realizations conditional on effort

Stock priceEffort level x1 x2 x3

aL pL1 pL

2 pL3

aH pH1 pH

2 pH3

of the sufficient conditions to assure convexity of the cost-minimizing contract im-plementing aH (see the proof of Lemma 5 in the Appendix). Let xL ≡ ∑3

i=1 pLi xi

and xH ≡ ∑3i=1 pH

i xi be the expected terminal stock price conditional on the effortlevel aL and aH , respectively. Note that the monotone likelihood ratio property(10) implies that xL < xH .15 Also let x0 be the ex-ante stock price and assume thatthe expected capital gains on the stock are positive regardless of the effort level,i.e., x0 < xL. Finally, assume that xH ≤ x2. This assumption is another sufficientcondition to assure convexity of the cost-minimizing contract implementing aH (seethe proof of Lemma 5 in the Appendix).

Also let sLi (t) and zL

i (t), i ∈ {1, 2, 3}, be the pre-tax and after-tax cost-minimizingcontracts implementing aL when the income tax rate is t. Similarly, let sH

i (t) andzH

i (t), i ∈ {1, 2, 3}, be the pre-tax and after-tax cost-minimizing contracts imple-menting aH . Also let CL(t) and CH(t) be the costs of implementing aL and aH ,respectively.

First, consider the least-cost implementation of aL. In this case the cost mini-mization problem (6)-(8) becomes

CL(t) ≡ minzi∈R

1

1− t

3∑i=1

pLi zi (11)

s.t.3∑

i=1

pLi

[−e−rzi] ≥ −e−ru(t) (12)

s.t.3∑

i=1

pLi

[−e−rzi] ≥ v

3∑i=1

pHi

[−e−rzi]. (13)

In this formulation, (12) is the PC, and (13) is the ICC. The following Lemmacharacterizes the solution to this problem:

Lemma 2 The least-cost implementation of the effort level aL involves zLi (t) ≡

(1− t)sLi (t) = u(t), i ∈ {1, 2, 3}, and CL(t) = u(t)/(1− t).

Proof. See the proof of Lemma 8 for the limiting case when t = τ .

15The monotone likelihood ratio property implies that the distribution of stock prices under aH

first-order stochastically dominates the distribution of stock prices under aL (Rogerson (1985),Lemma 1). This in turn implies that xL < xH .

11

The result in Lemma 2 corresponds to the standard result in principal-agenttheory that, since inducing the minimum effort level does not require any incen-tivization, optimal risk sharing between a risk-neutral principal and a risk-averseagent implies a contract consisting of a fixed payment only.

Second, consider the least-cost implementation of aH . This problem is given by

CH(t) ≡ minzi∈R

1

1− t

3∑i=1

pHi zi (14)

s.t. v

3∑i=1

pHi

[−e−rzi] ≥ −e−ru(t) (15)

s.t. v

3∑i=1

pHi

[−e−rzi] ≥

3∑i=1

pLi

[−e−rzi]. (16)

The following Lemma characterizes the solution to this problem:

Lemma 3 The least-cost implementation of the effort level aH involves

zHi (t) ≡ (1− t)sH

i (t) = u(t) + c(aH) +1

rlog

[1 + δ

(1− pL

i

vpHi

)],

and

CH(t) =1

1− t

{u(t) + c(aH) +

1

r

3∑i=1

pHi log

[1 + δ

(1− pL

i

vpHi

)]},

where δ ∈(0,

{(pL

1 /(pH

1 v)− 1

)}−1)

is uniquely implicitly defined by

3∑i=1

pHi

1− pLi

vpHi

1 + δ(1− pL

i

vpHi

) = 0. (17)

Proof. See the proof of Lemma 9 for the limiting case when t = τ .Lemma 3 reflects that to induce a high effort level, the manager needs to be

compensated for the value of the outside option u(t), the cost of effort c(aH), andthen incentivized by the portion of the contract that depends on the likelihood ratiospL

i /pHi and v. It follows from (10) that zH

i (t) and sHi (t) are strictly increasing in i.

Hence a better stock price realization leads to a higher compensation payout, bothpre-tax and after-tax. What is not immediately obvious from Lemma 3 is how CH(t)compares to CL(t), and whether zH

i (t) and sHi (t) are convex in i. The following two

lemmata address these issues.

Lemma 4

CH(t)− CL(t) =1

1− t

{c(aH) +

1

r

3∑i=1

pHi log

[1 + δ

(1− pL

i

vpHi

)]}> 0.

12

Proof. See the Appendix.

Lemma 5 Both zHi (t) and sH

i (t) are strictly convex in i. That is,

zH3 (t)− zH

2 (t)

x3 − x2

>zH2 (t)− zH

1 (t)

x2 − x1

,

and analogously for sHi (t).

Proof. See the Appendix.Lemma 4 gives an intuitive result that inducing the high effort level is more costly

than inducing the low effort level. Lemma 5 shows that the incentive contract isstrictly convex.

Having solved the cost minimization problem, let us now turn attention to theprofit maximization problem. The optimal effort choice is driven by the followingdecision rule: choose a = aH if and only if R(aH) − CH(t) ≥ R(aL) − CL(t), or,equivalently, if and only if R(aH)−R(aL) ≥ CH(t)−CL(t). Using Lemmata 3 and4 leads to the following result:

Proposition 2 Suppose the income tax rate increases from t1 to t2. Then the after-tax slope of the compensation contract strictly decreases if CH(t2)−CL(t2) > R(aH)−R(aL) ≥ CH(t1) − CL(t1), and otherwise remains unchanged. The pre-tax slope ofthe compensation contract may increase (if R(aH)− R(aL) ≥ CH(t2)− CL(t2)), ordecrease (if CH(t2)− CL(t2) > R(aH)−R(aL) ≥ CH(t1)− CL(t1)).

The result for the after-tax slope is a natural reflection of Proposition 1 in that aweak decrease in an equilibrium level of effort is matched by a weakly lower after-taxpay-to-performance sensitivity motivating this effort. However, Proposition 1 alsoshows that the impact on the pre-tax pay-to-performance sensitivity is ambiguousin general.16

3.2 Two-tax System

As discussed in Section 2, a tax system in which capital gains in restricted sharesare subject to the capital gains tax rate may be a better approximation of realityin the U.S. Therefore in this subsection, the model of the previous subsection ismodified in the following way. The (pre-tax) compensation contract now explicitly

16Note that the same set of conclusions, except for the convexity, can be obtained for anydistribution functions FL and FH associated with low and high effort, respectively, as long as aH

is implementable and the likelihood ratio fL(x)/fH(x) is decreasing with x, where fL and fG arethe densities of FL and FH with respect to some common measure. This follows from Lemma 1(changes in u(t) do not affect the slope of the contract), the fact that Lemma 2 (cost-minimizingcontract inducing aL consists of a fixed payment) applies for any admissible pair of FL and FH , andthe “spanning condition” approach of Grossman and Hart (1983) (the cost-minimizing contractinducing aH is increasing).

13

consists of a salary α, a non-negative amount β of (restricted) stocks, and a non-negative amount γ of stock options granted at the strike price xG. In this setting,the pre-tax state-dependent payouts are

si = α + βxi + γ max(0, xi − xG) (18)

for i ∈ {1, 2, 3}. That is, the compensation contract is convex by construction.This is illustrated in Figure 1 for the case when β = (w2 − w1)/(x2 − x1), andγ = (w3 − w2)/(x3 − x2)− (w2 − w1)/(x2 − x1). The salary, the grant value of thestocks, and any option payouts are taxed at the personal income tax rate t ∈ [0, 1).On the other hand, any capital gains or losses in the value of stocks relative totheir grant value are taxed at the capital gains tax rate τ ∈ [0, 1). In this two-taxsystem, the value of the outside option is given by u(t, τ), with u(·) decreasing inboth of its arguments. This reflects an implicit assumption that increases in eitherthe personal income tax rate or the capital gains tax rate have negative incidence onthe supply side of the managerial labor market. Under these tax rules, the after-taxstate-dependent payouts are

zi = (1− t)α + (1− t)βx0 + (1− τ)β(xi − x0) + (1− t)γ max(0, xi − xG), (19)

for i ∈ {1, 2, 3}.As before, it is easier to begin by analyzing the cost minimization problem.

Unlike in the previous subsection, however, conditional on a particular profile ofafter-tax state-dependent payouts zi, different combinations of stocks and optionsgenerate different compensation costs. The following lemma formalizes this obser-vation:

Lemma 6 Suppose that a profile zi, i ∈ {1, 2, 3} of (not necessarily cost-minimizing)after-tax state-dependent payouts induces an effort level aj, j ∈ {L,H}. Then, de-pending on the amount of stocks β used to generate this payout profile, the expectedcompensation cost is given by

1

1− t

3∑i=1

pjizi − β

t− τ

1− t(xj − x0). (20)

Proof. See the Appendix.As a result, the cost minimization problem becomes more complicated than in

the previous subsection. However, if t ≤ τ , this complication does not arise. It isbecause in that case either t = τ , in which case the tax system is a one-tax system, ort < τ , in which case it is optimal to eliminate the use of stocks in the compensationcontract since the latter can be effectively replicated by options with a sufficientlylow strike price and a salary adjustment.17 As a result, the last term in (20) dropsout and the cost minimization problem directly corresponds to programs (11)-(13)

17Conditional on zi, this can be achieved by setting α = z1/(1 − t), β = 0, γ =[(z3 − z2)/(x3 − x2)] /(1− t), and xG = [(z3 − z1)/(z3 − z2)] x2 − [(z2 − z1)/(z3 − z2)] x3.

14

for the low effort level and (14)-(16) for the high effort level with u(t) replaced byu(t, τ), and with the additional constraint of monotonicity and convexity given by

z3 − z2

x3 − x2

≥ z2 − z1

x2 − x1

≥ 0. (21)

However, Lemmata 2, 3, and 5 show that the unrestricted solutions in these twoprograms automatically satisfy the two additional constraints. Therefore these so-lutions are also relevant for the present case. It then follows that when t ≤ τ , anincrease in the capital gains tax rate τ has no impact on the slope of the after-taxcompensation scheme or the cost differential between inducing aH and aL. By exten-sion, it also has no impact on the profit-maximizing choice of effort or the amountof individual compensation instruments used in the profit maximizing contract (as-suming, without loss of generality, that no stocks are used when t = τ). On theother hand, the effect of changes in the personal income tax rate t parallel thosefrom the basic model in the previous subsection (see Propositions 1 and 2).

The rest of this subsection therefore focuses on the more interesting case whent > τ . Denote the cost-minimizing choice of salary, stocks, and options by αL(t, τ),βL(t, τ), and γL(t, τ), respectively, for a = aL, and by αH(t, τ), βH(t, τ), andγH(t, τ), respectively, for a = aH . Also denote the minimum cost of inducing thelow effort level by CL(t, τ), and the minimum cost of inducing the high effort levelby CH(t, τ). The following lemma characterizes the optimal cost-minimizing use ofthe individual compensation instruments when inducing either effort level:

Lemma 7 Suppose that t > τ and the cost minimizing after-tax contract zi inducingan effort level a ∈ {aL, aH} is strictly increasing and convex in i (i.e., it satisfies(21)). Then

β =1

1− τ

z2 − z1

x2 − x1

and γ =1

1− t

x3 − x2

x3 − xG

[z3 − z2

x3 − x2

− z2 − z1

x2 − x1

],

where xG ∈ [x2, x3) if γ > 0.

Proof. See the Appendix.Intuitively, there are multiple ways how the four compensation instrument pa-

rameters (α, β, γ, xG) can be combined to construct any given increasing and convexafter-tax payout profile (z1, z2, z3). Lemma 7 shows that, conditional on zi, stocksare used to a maximum possible extent in any cost minimizing contract when t > τ .

To render the analysis more tractable, in what follows I assume that the taxrate differential t− τ is “small.” First, consider the least-cost implementation of aL.Utilizing Lemma 7, the cost-minimization problem becomes

CL(t, τ) ≡ minzi∈R

1

1− t

{3∑

i=1

pLi zi − t− τ

1− τ

z2 − z1

x2 − x1

(xL − x0)

}(22)

subject to the PC (12) with u(t) replaced by u(t, τ), the ICC (13), and the monotonicity-convexity constraint (21). The following lemma characterizes the solution to thisproblem:

15

Lemma 8 Suppose that t > τ . Then there exists ε > 0 such that, if t− τ < ε, theleast-cost implementation of aL involves βL(t, τ) = k(t, τ)/(1− τ) and γL(t, τ) = 0,where k(t, τ) is implicitly defined by

3∑i=1

pLi

[xi − xL +

t− τ

1− τ(xL − x0)

]e−rk(t,τ)xi = 0, (23)

and it is strictly positive, strictly increasing in t, strictly decreasing in τ when t > τ .In addition, k(t, t) = 0 for any t ∈ [0, 1).

Proof. See the Appendix.Lemma 8 shows that even if the principal wants to induce only the minimum

level of effort, he will use some stocks in the compensation contract, and henceimpose a certain degree of risk on the manager. This marks a stark departure froma fundamental result in principal-agent theory that in case the principal wishesto induce only the minimal level of effort, no incentivization is necessary, and thenefficient risk sharing between a risk-neutral principal and a risk-averse agent involvesa compensation contract that consists of a fixed salary only. This finding is acombination of the result that each risk-averse agent buys a positive amount of arisky asset if this asset has a positive net expected return and the cost-minimizationon the part of the principal. In particular, suppose that the manager’s compensationoriginally consists of salary only. Now suppose that the principal gives the manageran opportunity to replace a part of this salary by stocks of the same pre-tax expectedex-post value. The risk-neutral principal is indifferent between all of the possiblechoices of the manager. For the manager, exchanging a dollar of salary for a dollarworth of stocks in expectation is equivalent to exchanging 1− t dollars of after-taxcompensation that is certain for a lottery with an expected after-tax return of 1−τ .Because 1 − τ > 1 − t, the manager will choose a positive amount of stocks andstrictly improve his expected utility. As a result, the principal can decrease thecompensation cost by offering the manager his preferred positive amount of stocks,and will therefore do so. This result illustrates an inefficiency of the compensationcontract inducing the low effort level originating from the differential taxation ofthe riskless instrument (salary) and a risky instrument (stocks). It is underscoredby the finding that no options are used in the cost-minimizing contract, since theyare risky and they have no tax advantage over salary.

Lemma 8 also reveals that an increase in t leads to more stocks being used in thecost minimizing contract inducing aL, while an increase in τ reduces the effectiveafter-tax amount of stocks in the contract (1 − τ)βL(t, τ), with the effect on theamount of stocks βL(t, τ) being ambiguous.

Now consider the least-cost implementation of aH . Utilizing Lemma 7, the cost-minimization problem becomes

CH(t, τ) ≡ minzi∈R

1

1− t

{3∑

i=1

pHi zi − t− τ

1− τ

z2 − z1

x2 − x1

(xH − x0)

}(24)

subject to the PC (15) with u(t) replaced by u(t, τ), the ICC (16), and the monotonicity-convexity constraint (21).

16

The following lemma characterizes the solution to this problem:

Lemma 9 Suppose that t > τ . Then there exists ε > 0 such that, if t − τ < ε,the least-cost implementation of the effort level aH involves βH(t, τ) and γH(t, τ)such that βH(t, τ) is strictly increasing in t, (1− τ)βH(t, τ) is strictly decreasing inτ , and, holding xG fixed, (1 − t)γH(t, τ) is strictly decreasing in t, and γH(t, τ) isstrictly increasing in τ .

Proof. See the Appendix.Lemma 9 reveals that in the cost minimizing contract inducing aH , and increase

in the personal income tax rate t increases the amount of stocks and decreases theeffective after-tax amount of options (1− t)γH(t, τ), with the effect on the amountof options being ambiguous. On the other hand, an increase in the capital gains taxrate τ increases the amount of options and decreases the effective after-tax amountof stocks (1− τ)βH(t, τ), with the effect on the amount of stocks being ambiguous.

Having solved the two cost minimization problems, let us now turn attentionto the profit maximization problem. As in the previous subsection, the impact ofa tax change on the profit maximizing level of effort depends on the effect of thischange on the cost differential CH(t, τ)−CL(t, τ). First, consider the impact of anincrease in the personal income tax rate t. Using the envelope theorem for the twocost-minimization contracts,18 it follows for j ∈ {L,H} that

∂

∂tCj(t, τ) =

1

1− t

{Cj(t, τ)− βj(t, τ)(xj − x0) + u1(t, τ)

}. (25)

18Note that, apart from the objective functions, the two tax rates enter the cost-minimizationprograms (22) and (24) only via the term u(t, τ) in the PC. Also it follows from the proof that theresult of Lemma 1 does not depend on there being only one tax rate applied to the entire contract,and it applies equally well to the two-tax model. As a result, the Lagrange multiplier κ(t, τ) onthe PC is given by

κ(t, τ) =∂

∂[−e−ru(t,τ)

] u(t)1− t

= − ∂

∂[e−ru(t,τ)

](− log e−ru(t,τ)

r(1− t)

)

=eru(t,τ)

r(1− t).

As a result,

−κ(t, τ)∂

∂te−ru(t,τ) =

u1(t, τ)1− t

and

−κ(t, τ)∂

∂τe−ru(t,τ) =

u2(t, τ)1− t

,

which gives the last terms in curly braces on the right-hand side of (25) and (27), respectively.

17

Because, by Lemma 8, limt↓τ βL(t, τ) = 0,19 it follows that

limt↓τ

∂

∂t

[CH(t, τ)− CL(t, τ)

]

=1

1− t

{[CH(t, τ)− CL(t, τ)

]t=τ

− limt↓τ

βH(t, τ)(xH − x0)

}. (26)

The first term in the curly braces is strictly positive by Lemma 4. By monotonicityof the contract, the second term is nonnegative, and it may be more or less than thefirst term in absolute value.20 As a result, the sign of the left-hand side of (26) isambiguous. By continuity, this result extends to t > τ when t−τ is small. Intuitively,if t − τ is close to zero, the cost advantage of stocks due to their lower taxation issmall, and hence few of them are used in a cost minimizing contract inducing aL.If not too many stocks are used in the cost minimizing contract inducing aH andxH −x0 is small, most of the payouts of either cost minimizing contract are affectedby the personal income tax rate. An impact of an increase in this tax rate is thensimilar to the impact of an increase in a single tax rate affecting all compensationinstruments, which increases the cost differential between inducing high and loweffort (see Lemma 4). However, if a lot of stocks are used in the cost minimizingcontract inducing aH and/or xH−x0 is large, a significant part of the payouts in thiscontract are subject to the capital gains tax rate rather than the personal incometax rate. As a result, while an increase in the personal income tax rate inflates thecost of inducing aL, it has a much more moderate effect on the cost of inducing aH ,hence reducing the cost differential between inducing high and low effort.

This finding leads to the surprising result that, unlike in the previous sectionwhere an increase in the personal income tax rate could only decrease the profitmaximizing effort level, an increase in the personal income tax rate can increasethe profit maximizing effort level. This result also implies, however, that in two-taxsystem, an increase in the personal income tax rate has an ambiguous effect on pre-tax and after-tax pay-to-performance sensitivities generated by stocks and options(see Lemmata 8 and 9).

Second, consider the impact of an increase in the capital gains tax rate τ onthe cost differential CH(t, τ) − CL(t, τ). Using the envelope theorem for the twocost-minimization programs, it follows for j ∈ {L,H} that

∂

∂τCj(t, τ) =

1

1− t

{1

1− τβj(t, τ)(xj − x0) + u2(t, τ)

}. (27)

Because, by Lemma 8, limt↓τ βL(t, τ) = 0, it follows that

∂

∂τ

[CH(t, τ)− CL(t, τ)

]t=τ

=1

(1− t) (1− τ)

[limt↓τ

βH(t, τ)

](xH − x0) > 0, (28)

where the last inequality comes from the strict monotonicity of the contract (andhence βH(t, τ) > 0). By continuity, this result extends to t > τ when t− τ is small.

19This follows from βL(t, τ) = k(t, τ)/(1 − τ) and the fact that k(t, τ) is continuous in t withk(τ, τ) = 0 (see (23)).

20Note that when t = τ , βH(t, τ) is independent of x0 or xH .

18

Intuitively, if t − τ is close to zero, the cost advantage of stocks due to their lowertaxation is small, and hence few of them are used in a cost minimizing contractinducing aL. However, due to strict monotonicity of the contract, a strictly positiveand bounded away from zero amount of stocks is always used in the cost minimizingcontract inducing aH . An increase in the capital gains tax rate therefore inflatesthe cost of inducing aH more than it inflates the cost of inducing aL. As a result,an increase in the capital gains tax rate can only lead to a decrease (or no change)in the profit maximizing level of effort. In addition, if the principal switches frominducing aH to inducing aL, it follows by continuity from the case when t = τ thatthe amount of stocks used in the compensation contract decreases.

These conclusions, combined with Lemmata 8 and 9, lead to the following propo-sition:21

Proposition 3 Suppose that the capital gains tax rate increases from τ1 to τ2 witht > τ2 > τ1. Then there exists ε > 0 such that if t − τ1 < ε, the equilibrium levelof managerial effort decreases weakly and the after-tax sensitivity of compensationgenerated by the stock grant (1− τ)β(t, τ) decreases strictly. The amount of optionsused in the contract may increase (if R(aH) − R(aL) ≥ CH(t, τ2) − CL(t, τ2)) ordecrease (if CH(t, τ2)− CL(t, τ2) > R(aH)−R(aL) ≥ CH(t, τ1)− CL(t, τ1)).

Theoretical results presented in this section show that, when one tax rate appliesto the entire compensation package, an increase in the tax rate weakly decreases theequilibrium level of managerial effort and the after-tax pay-to-performance sensi-tivity of the contract, with the effect on the pre-tax pay-to-performance sensitivitybeing ambiguous. To the extent that in practice the gains on restricted stock grantsmay be taxed at the (lower) capital gains tax rate rather than the (higher) personalincome tax rate, however, the two-tax model may provide a better approximationof the tax treatment of executive compensation in the U.S. This model generatestwo interesting results. First, (risky) stocks are used in the compensation contracteven in the absence of any desire to incentivize the manager. Second, an increase inthe personal income tax rate may increase the equilibrium level of managerial effort.On the other hand, an increase in the capital gains tax rate weakly decreases theequilibrium level of effort and strictly decreases the after-tax pay-to-performancesensitivity generated by the stock grant.

One consequence of these results is that the effect of an increase in the personalincome tax rate on the contract design is ambiguous in the two-tax model. The signof this comparative static result therefore becomes an empirical question. To shedlight on the answer to this question, the second part of the paper investigates theimpact of the 1993 increase in the U.S. federal personal income tax rate and the 1994increase in the Medicare payroll tax rate on the amount of pay-to-stock performancesensitivity generated by annual grants of restricted stocks and stock options, whichcorrespond to the variables β and γ, respectively, in the two-tax model.

21A proof of this proposition is straightforward and it is left as an exercise to the reader.

19

4 Empirical Strategy and Data

In order to empirically identify the impact of a change in the personal income taxrate on the design of executive compensation, this paper uses the variation in thepersonal income tax rate over the period 1992-1996 created by the Omnibus BudgetReconciliation Act (OBRA) of 1993.22 In 1993, OBRA enacted an increase in themarginal personal income tax rate for married couples filing jointly from 31 percentto 36 percent for income between $140,000 and $250,000, and from 31 percent to 39.6percent for income in excess of $250,000. In addition, it also abolished the upperincome limit of $130,200 in 1992 and $135,000 in 1993 for the Medicare payroll taxstarting from 1994. This resulted in an increase in the effective marginal tax rateof 2.9 percent for any income in excess of this limit.23 Another part of this bill,Section 162(m) of the Internal Revenue Code, or the so-called “million dollar rule,”specified that companies could deduct from their corporate income tax base onlyup to 1 million dollars per executive for non-performance-based compensation ofthe CEO and other four most highly compensated executives. During the period1992-1996, the top statutory capital gains tax rate remained constant at 28 percent,and there was only a small increase in the effective top capital gains tax rate from28.93 percent in 1992 to 29.19 percent during 1993-1996. Table 3 summarizes thedetails of the U.S. federal personal income tax system over the period 1992-1996.

Data on executive compensation come from the Execucomp database providedby Standard and Poor’s as a part of its Compustat database. These data are col-lected from proxy statements and 10-K forms of publicly traded companies based onthe U.S. securities regulations. The dataset used in this paper covers companies inthe Standard and Poor’s S&P 500, S&P Mid Cap 400, and S&P Small Cap 600 forthe years 1992 through 1996. The Execucomp dataset provides a rich source of in-formation on executive compensation. It provides information about an executive’ssalary, bonus, restricted stock grants, stock-option grants and exercises, long-termincentive payouts, other compensation, and total holdings of company’s shares andstock options at the end of a fiscal year. This information is provided for the CEOand four other most highly paid executives of each company. The executive-levelinformation is complemented by a variety of financial and demographic data for theindividual companies.

The Execucomp dataset has been used by many researchers of executive com-pensation. In tax studies, it has previously been used by, among others, Gools-bee (2000a, 2000b), Hall and Liebman (2000), Perry and Zenner (2001), and Roseand Wolfram (2002). Because this dataset contains information on both relativelysmaller companies and non-CEO executives, it contains a significant fraction ofexecutives with company-related incomes below the top tax bracket. It thereforeallows identification of the tax effect from (unequal) time variation in the two toptax rates (as opposed to the top rate only) and the use of difference-in-differences

22Ideally, the timespan of the sample would begin several years before the tax change, but theexecutive compensation data coverage only begins in 1992.

23Note that, unlike in the case of the Social Security tax, higher Medicare tax payments do notcorrespond to higher benefits, so the Medicare tax is a true marginal tax.

20

TABLE 3: Summary of the U.S. Personal Income Tax System, 1992-1996

Ordinary Income Tax Cap. Gains TaxYear Top Rate Top Bracket Second Second Top Rate

Rate Bracket1992 31.0% $86,500 28.0 %1993 39.6% $250,000 36.0 % $140,000 28.0 %1994 42.5% $250,000 38.9 % $140,000 28.0 %1995 42.5% $256,500 38.9 % $143,600 28.0 %1996 42.5% $263,750 38.9 % $147,700 28.0 %

Notes:(1) The ordinary income tax rates and brackets are for married taxpayers filing jointly.(2) In 1992, all earned income under $130,200 was subject to the additional Medicare tax of 2.9%.(3) The ordinary income tax rate increase of 1993 is due to the Omnibus Budget ReconciliationAct.(4) The ordinary income tax rate increase of 1994 is due to the removal of the tax base ceiling of$130,200 in 1993 and $135,000 in 1994 for the Medicare tax of 2.9 %.(5) Data sources: The Office of Tax Policy Research (2003) and Urban-Brookings Tax PolicyCenter (2003) (for ordinary income tax rates, payroll tax rates, and brackets), The Department ofTreasury (2002) (for capital gains tax rates).

methodology. Goolsbee (2000a, 2000b) uses an analogous design to identify theeffect of the 1993 and 1994 increases in the top personal tax rate on the elasticityof taxable income and timing of stock-option exercises.

The raw dataset, after eliminating observations with missing identifiers, contains51,360 observations (executive-years). I make several adjustments to the raw data.First, I exclude observations for which there is missing or inconsistent informationon cash and stock-based compensation items, observations with missing informationon the ownership of stocks accumulated in the past, observations where stock andoption ownership are not reported separately, and observation on executives whosimultaneously work for more than one company. This reduces the sample to 27,603observations. I further exclude companies whose fiscal year does not coincide withthe calendar year. This reduces the sample to 19,293 observations. The purpose ofthis reduction is to be able to match the compensation report based on the fiscal yearto the tax treatment based on the calendar year.24 Finally, because the identificationis based on the difference of the compensation design in 1992 and the subsequentperiod, I exclude executives that do not have observations in 1992 and at least oneof the years in 1993-1996. The final sample used in the analysis contains 14,229executive-years. Of these, 3,124 describe CEOs and 11,105 non-CEO executives.

24Although the loss of 8,310 observations appears to be a high price to pay, very few of thelost executives have the relevant data available for 1992, and so they could not be used in theanalysis in any case. This is because the Execucomp coverage is based on enhanced Securities andExchange Commission reporting requirements for fiscal years ending after December 15, 1992. Asa result, very few companies whose fiscal year does not coincide with the tax year have any dataavailable for 1992.

21

The final sample contains information on 3,501 executives in 899 companies.Apart from the Execucomp dataset, I use several other sources of data. To eval-

uate pay-to-performance sensitivity generated by option grants, I use the data onTreasury bill and bond yields from the Department of Treasury (2003). Informationon book value of companies comes from Compustat. To adjust various quantitiesinto constant 1993 dollars, I use the consumer price index from the Bureau of LaborStatistics (2002). Finally, to split the sample by the strength of corporate gover-nance, I use the Gompers at al. (2003) corporate governance index available fromthe Investor Responsibility Research Center. This index is a sum of 24 indicatorvariables, with each indicator being equal to 1 when some shareholder rights arelimited by various company policies and charter provisions.25

The first step in the analysis is to assign executives into the individual tax brack-ets. This requires defining their taxable income. For each executive in each year, Icompute her taxable income as the sum of salary, bonus, long-term incentive pay-ments, stock-option exercises, and dividend income. This measure of income may,however, be itself endogenous to current and anticipated tax rates, as, for example,documented in Goolsbee (2000a) for the timing of stock-option exercises. In orderto avoid this year-to-year endogeneity, I convert these annual taxable incomes into1993 inflation-adjusted dollars, are then average them across all the available ob-servations for each executive so as to arrive at a measure of “permanent income.”Executives with permanent income over $250,000 are then categorized as being inthe top tax bracket, while the other executives are categorized as being in the secondtax bracket (see Table 3).26 Of the total number of 3,501 executives, 3087 (88.17%of the sample) are in the top tax bracket and 414 (11.83%) are in the second taxbracket.

This income measure may to some extent undermeasure the true permanentincome since it does not include spousal and non-company related sources of income,for which no measures are available. However, this unobserved income is also offsetby various deductions and exemptions that an executive may claim on her tax return.In addition, from the perspective of agency theory, it is not just the marginal taxrate on the last dollar of earnings that matters, but rather the marginal tax ratethat applies within the range of income generated by stock and option grants.27

The second step in the analysis is to define dependent variables. Since, as arguedin previous sections, the impact of a tax change on salary is difficult to quantify, Iwill focus on the pay-to-stock-price sensitivity generated by annual grants of stockoptions and restricted stocks as dependent variables. In particular, these sensitiv-

25Therefore a higher value of the index represents weaker corporate governance.26There are also some executives in the sample whose permanent income falls below the threshold

of $140,000. However, they only constitute 1.34% of the total number of executives, and many ofthem do not receive any stock-option or restricted stock grants. For these reasons, they are nottreated as a separate tax bracket group.

27To the extent that executives with taxable income “close” to $250,000 are the ones most likelyto be misclassified with respect to the two tax brackets, I reestimated all the specifications reportedin Section 5 when executives with permanent taxable income between $225,000 and $275,000 aredropped from the sample. The estimates of the net-of-tax-rate elasticities were numerically similarto the reported results and none of the principal conclusions of the analysis were affected.

22

ities are defined by the change in managerial wealth originating from these grantsgenerated a 1 percent increase in stock price; this is the measure used by Coreand Guay (1999, 2002). Other measures that researchers in the area of executivecompensation often use is the increase in annual compensation or total executivewealth generated by a $1,000-increase in market value of the company (for example,Jensen and Murphy (1990), Aggarwal and Samwick (1999a)). However, as pointedout by Hall and Liebman (1998), an executive in a large corporation may have amuch smaller sensitivity of compensation (or wealth) to a $1,000-increase in com-pany value than an executive in a small corporation, and yet a typical variation inthe company stock price generates much higher wealth swings for the first executivecompared with the second one. In this respect, measuring the pay-to-performancesensitivity relative to company size is arguably a more meaningful measure. In ad-dition, pay-to-performance sensitivity measures of the Jensen and Murphy type areoften measured as regression slope coefficients, and hence they are ex-post measuresof incentives. On the contrary, the pay-to-performance sensitivity generated by theannual option and stock grants (or entire portfolios of options and stocks) are exante measures of incentives, hence more explicitly capturing the motivational im-pact on executives. Although the latter measures do not take into account implicitincentives from salary and bonus increases following a good performance, Hall andLiebman (1998), using a panel data on CEOs from 1980 to 1994, find that “changesin CEO wealth due to stock and stock option revaluations are more than 50 timeslarger than wealth increases due to salary and bonus changes.”

When relying on ex ante measures, though, measuring the pay-to-performancesensitivity with respect to the entire current portfolio of stocks and options as op-posed the annual grants only is arguably a superior measure of executive incentives.However, in any given year the company board has discretion only about the an-nual grant. In addition, the sensitivity of the entire portfolio may be affected by theendogenous response of executives who hedge the undiversified position in their com-pany by executing some previously owned options and selling the obtained sharesas well as other previously owned shares (Ofek and Yermack (2000)), or engaging inmore sophisticated hedging transactions such as zero-cost collars and equity swaps(Bettis et al. (2001)). Given that this paper focuses on the behavioral responseof corporations, I am going to focus on the sensitivity generated by annual grantsonly. But to the extent that these grants may depend on the pre-existing amountof incentives originating from previous years’ grants, as documented by Core andGuay (1999), I control for the lagged portfolio pay-to-stock-price sensitivity in theempirical specifications.

Measuring the stock-price sensitivity of stock and option grants requires valuingthese securities. As pointed out by Hall and Murphy (2000, 2002), the marketvalue (the value paid by a diversified investor) of stock and option grants mayoverestimate the value of these securities to an executive since she usually holdsa large undiversified position, and her securities are non-vested and hence non-tradable for some time. However, as an approximation, I use the usual marketprice of these securities to value them. Computing an increase in executive’s wealthoriginating from a restrictive stock grant due to a 1 percent increase in stock price

23

is straightforward. It is simply one hundredth of the market value of the grant.For options, I use the change in the Black-Scholes value of the stock-option grantreflecting a 1 percent increase in the stock price. In particular, for any individualoption grant, this measure is calculated as

∂ option value

∂P

P

100= e−dT N(Z)

P

100,

whereZ =

[ln(P/X) + T (r − d + σ2/2)

]/(σ√

T)

.

In this formulation, P is the stock price, N(·) is the cumulative distribution functionof the standard normal distribution, X is the option strike price, σ is the annual-ized standard deviation of stock returns (excluding the dividends), T is the optionexpiration term in years, r is the natural logarithm of one plus the risk-free interestrate matching the option expiration term (defined as the yield of a Treasury bill orbond of a similar maturity28), and d is the natural logarithm of one plus the stock’sdividend yield (see Core and Guay (2002) for details). Both the stock and the optionsensitivity measure are then adjusted for inflation into constant 1993 dollars.

For continuity with the previous literature, I also examine the dependent variableof Hall and Liebman, defined as the ratio of the Black-Scholes value of the stock-option grant to the sum of salary, bonus, and the Black-Scholes value of the stock-option grant. I will subsequently refer to this measure as “the share of compensationpaid in options.”

Table 4 summarizes the dependent variables over the sample period 1992-1996.Not surprisingly, compared to the executives from the top tax bracket, the executivesfrom the second tax bracket receive smaller option and restricted stock grants, andthe have a lower share of their compensation paid in options. In addition, fewer ofthem receive any option or stock grant at all. The table also reveals that optiongrants are both more frequent and more sizeable in terms of creating pay-to-stock-price sensitivity than restricted stock grants are. When focusing on growth of meansof these measures over time, conditional on the measure being positive, executivesin the top tax bracket experience a higher proportional growth compared to theircolleagues in the second tax bracket. Therefore, using the difference-in-differenceslogic, the first reading of the data would suggest that a higher personal tax rateincreases the growth rate of pay sensitivity generated by stock and option grants,as well as the share of compensation paid in options.

However, there may be a multitude of other factors that are changing over thetimespan of the sample that may account for the observed patterns in the data. Inparticular, the “million-dollar rule” became effective in 1994, and its effect mighthave been to reallocate compensation from salary to stocks and options. In order toallow for this possibility, the multivariate analysis of the next section controls for avariable MILLION defined as a minimum of unity and the ratio of the previous year’ssalary to $1 million in years 1994-1996, and as zero otherwise.29 That is, MILLION

28Details of the calculation are available from the author upon request.29For example, in 1994, for an executive with the 1993 salary of $600,000, MILLION is equal to

0.6, while for an executive with the 1993 salary of more than $1 million it is equal to unity.

24

TABLE 4: Summary Statistics of Compensation VariablesYear

Variable Sample 1992 1993 1994 1995 1996

Observations All 3,501 3,329 2,967 2,430 2,002Observations Top Bracket (TB) 3,087 2,970 2,628 2,176 1,806Observations Second Bracket (SB) 414 359 339 254 196

Option Grant SensitivityMedian All $1,523 $2,081 $2,135 $2,763 $3,740Median Positive Grant $3,800 $4,488 $4,764 $6,644 $7,676Mean Positive Grant $9,866 $10,774 $11,625 $18,638 $22,347Positive Grant All 64.8% 68.6% 68.1% 67.9% 71.1%Median TB $1,974 $2,547 $2,691 $3,527 $4,459Median TB, Positive Grant $4,305 $4,886 $5,351 $7,396 $8,527Mean TB, Positive Grant $10,616 $11,401 $12,562 $19,989 $23,833Positive Grant TB 67.1% 70.4% 70.1% 70.0% 73.0%Median SB $0 $428 $178 $66 $287Median SB, Positive Grant $1,010 $1,213 $1,008 $1,304 $1,293Mean SB, Positive Grant $2,014 $4,020 $1,867 $2,431 $3,691Positive Grant SB 47.8% 54.0% 52.2% 50.0% 53.6%

Stock Grant SensitivityMedian All $0 $0 $0 $0 $0Median Positive Grant $1,423 $1,568 $1,591 $1,636 $2,375Mean Positive Grant $3,666 $3,939 $3,782 $3,825 $6,480Positive Grant All 18.4% 19.7% 19.6% 19.9% 21.8%Median TB $0 $0 $0 $0 $0Median TB, Positive Grant $1,589 $1,749 $1,813 $1,898 $2,758Mean TB, Positive Grant $3,862 $4,139 $4,005 $4,133 $6,898Positive Grant TB 19.6% 20.9% 20.8% 20.4% 22.6%Median SB $0 $0 $0 $0 $0Median SB, Positive Grant $168 $198 $238 $224 $282Mean SB, Positive Grant $448 $388 $396 $402 $398Positive Grant SB 8.9% 9.7% 10.6% 15.7% 14.3%

Share of Compensation Paid in OptionsMedian All 0.160 0.188 0.233 0.205 0.246Median Positive Grant 0.279 0.289 0.340 0.315 0.351Mean Positive Grant 0.323 0.332 0.369 0.350 0.384Positive Share All 65.0% 68.7% 68.1% 68.1% 71.1%Median TB 0.172 0.196 0.250 0.219 0.266Median TB, Positive Grant 0.282 0.290 0.345 0.317 0.359Mean TB, Positive Grant 0.324 0.331 0.374 0.353 0.388Positive Share TB 67.2% 70.5% 70.2% 70.1% 73.0%Median SB 0 0.075 0.068 0.027 0.066Median SB, Positive Grant 0.246 0.288 0.254 0.281 0.232Mean SB, Positive Grant 0.310 0.340 0.319 0.308 0.327Positive Share SB 49.0% 54.0% 52.2% 50.4% 53.6%

Notes: TB = Top Tax Bracket, SB = Second Tax Bracket.

25

proxies for the likelihood of the million dollar rule being a binding consideration inthe design of the compensation contract over the period 1994-1996.30

The set of control variables also includes the (natural) logarithm of the com-pany’s market value, the logarithm of the market-to-book ratio, the logarithm ofthe stock price volatility, an indicator variable for CEOs, the logarithm of one plusthe market rate of return on the stock and the lag of this variable, the logarithmof one plus the rate of return on assets, and individual executive effects. Thesevariables are commonly used as controls in executive compensation regressions.31 Acompany’s market value proxies for the firm size and a stylized fact in the executivecompensation literature is that larger firms pay more in virtually any pay component(Murphy (1999)). Market-to-book ratio proxies for firm growth opportunities, andfirms with higher growth opportunities are expected to provide more incentives totheir managers (Core and Guay (1999)). Stock price volatility proxies for noisinessof firm environment and hence idiosyncratic risk and monitoring costs. As a result,it is not entirely clear how this measure affects provision of incentives.32 The indi-cator variable for CEOs captures the conjecture that CEOs are incentivized morethan other managers. The remaining market and accounting rates of return proxyfor firm performance and they may capture the fact that some stock-based incentivegrants are triggered by a threshold level of performance. All the specifications alsoinclude individual executive effects. Some of the specifications also control for linearindustry-specific time trends33 and year effects.

As documented by Core and Guay (1999) and discussed above, another impor-tant set of variables that may influence the pay sensitivity generated by annual stockand option grants to an executive is the amount of wealth-to-stock-price sensitivitygenerated by his pre-existing holdings of stocks and options originating from previ-ous years’ grants. The measure of stock-holding sensitivity to stock price changesis straightforward and can be calculated analogously to the annual restricted stockgrant. The calculation is more challenging in the case of options, since Execucompdoes not provide a detailed information, such as strike prices and expiration terms,for the option portfolio originating from past grants. This problem is especiallyacute for this empirical application, since it is not possible to go back several yearsto build up the portfolio information. I therefore use the method of Core and Guay(2002) to estimate this sensitivity. A final step in the calculation consists of imputingthese sensitivities for 1992 based on the logarithm of market value, the logarithm ofmarket-to-book ratio, the CEO indicator, the logarithm of stock price volatility, andthe logarithm of one plus the stock return, as there is no information on previous

30The construction of this variable is based on Hall and Liebman (2000) and Perry and Zenner(2001).

31See, for example, Core and Guay (1999), Goolsbee (2000a), Hall and Liebman (1998, 2000),Rose and Wolfram (2002).

32Core and Guay (1999) argue that higher monitoring costs will lead to a higher provision ofincentives. On the other hand, a standard result from agency theory is that a higher level ofidiosyncratic risk is, due to managerial risk aversion, associated with lower incentive provision.Aggarwal and Samwick (1999a) provide evidence of this effect.

33I use seven industries defined by 2-digit SIC codes in the range 10-14, 20-39, 40-49, 50-59,60-67, 70-89, and other.

26

years’ stock and option holdings for 1992.One notable feature of the annual stock and option grant sensitivities apparent

from Table 4 is that they are equal to zero for many executive-years. In order toidentify the impact of the tax change on these sensitivities, one possibility is toconcentrate only on observations with positive amounts of the grant. However, tothe extent that factors influencing the amount of these grants may also influencethe decision to provide a grant, an endogenous selection into the sample of positivesensitivities may be an issue. I therefore use a simplified version of the methodsuggested by Wooldridge (1995) to test and, if necessary, adjust the estimation forthe presence of selection.34 This method is a panel data extension of the Heckman’s(1976) selection model. In the first step, a first-stage probit model using panelmeans of all explanatory variables as additional regressors is estimated and theinverse Mill’s ratio is calculated. In the second step, one can include the inverseMill’s ratio in the fixed effects regression using only the observations with positivesensitivities and test for its significance. A presence of the latter signifies a presenceof selection. If selection is not present, the estimates correspond to a standard fixed-effects model. In the presence of selection, the third step consists of estimating anOLS regression of interest using only observations with positive sensitivities, butincluding the inverse Mill’s ratio and panel means of all the explanatory variables asregressors. In this approach the individual effects are modelled as correlated randomeffects (see Wooldridge (1995)). In either case, the variance matrix of the estimatedcoefficients is adjusted for the sampling variance of the first-stage estimator (inthe presence of selection) and for clustering at executive level.35 In order to avoidentirely relying on the functional form specification built into the first-stage probit,I include the fractions of executives in the particular industry that receive positiveoption and stock grants as an additional explanatory variable in the first stagealongside the variables used in the second stage. These variables proxy for theindustry-specific market pressure to incentivize the company’s executives.

When the share of compensation paid in options is used as the dependent vari-able, all the specifications are estimated using OLS on the entire sample of availabledata with the standard errors adjusted for clustering at executive level. A drawbackof this approach is that it does not take into account the limited dependent natureof the empirical model. However, the main purpose of this exercise is to providecontinuity with the work of Hall and Liebman (2000), and so the method used inthis paper is analogous to their estimating approach.

34I have also estimated the model using Honore’s (1992) panel censored regression approach andKyriazidou’s (1997) panel selection model approach. In the first case, I found the results hard tointerpret, indicating that the selection equation is likely to be different from the main equation ofinterest. In the second case, the results were sensitive to the selection of the bandwidth.

35I have also estimated all the reported empirical specifications using clustering at firm levelrahter than executive level, without any qualitative change in the reported results.

27

5 Results

5.1 Overall Sample

Table 5 displays multivariate estimates for the pay-to-stock-price sensitivity gen-erated by option grants. Both the dependent variable and the net-of-tax-share vari-able are in natural logarithm, so the coefficient of the net-of-tax-share variable isinterpreted as an elasticity.36 Column (1) uses the tax rate variation in both bracketsto identify the tax effect, while column (2) uses only the top tax bracket executivesand identifies out of the time variation in the top tax rate only. The estimates inthese two columns show that the option grant sensitivity increased in response tothe tax increases of 1993 and 1994, but this increase is statistically significant onlyin column (1). These estimates, however, may be biased due to the omitted timetrends in executive compensation that are correlated with tax rate changes. To ac-count for these time trends, columns (3) and (4) repeat the specifications of columns(1) and (2) with the addition of linear industry time trends. With these controls,the net-of-tax-rate elasticity becomes positive (about two thirds), significant, andnumerically similar across the two columns. The comparison to the estimates incolumns (1) and (2) reveals that it is indeed important to control for the under-lying time trends in executive compensation when identifying the tax effect. Theresults in columns (3) and (4) may, however, still be biased due to the restrictiveway (linearity) the time effects are controlled for. To address this problem, columns(5) and (6) repeat the specifications of columns (1) and (2) with the addition ofyear dummies in both columns and linear industry time trends in column (6). Thatis, these two columns identify the tax effect using the difference-in-differences ap-proach. With these controls, the net-of-tax-rate elasticity remains positive, but itsmagnitude approximately doubles and the coefficient estimates become statisticallyinsignificant. One possible explanation for the large standard errors and hence sta-tistical insignificance in columns (5) and (6) is that, as displayed in Table 3, thedifference in the tax rate increase between the two tax brackets is only 3.6 per-centage points, leaving the difference-in-differences approach with little identifyingvariation. A similar caveat applies to all the subsequent results as well. However,reading the results from columns (3)-(6) together suggests a positive net-of-tax-rateelasticity of the option grant sensitivity once the time trends are controlled for. Inthe case of the one-tax model, the elasticity of the after-tax option grant sensitivityis of interest as well (see Proposition 2). This elasticity is obtained by adding one tothe coefficient estimates on the net-of-tax-share variable, with the standard errorsbeing unaffected. In all six columns, this estimate is positive and significant, con-firming the prediction of Proposition 2. The coefficient estimates on other variablesare consistent with predictions outlined in the previous section.

Using the same methodology, Table 6 displays multivariate estimates for thepay-to-stock-price sensitivity generated by restricted stock grants. The net-of-tax-rate elasticity estimates are negative in columns (1) and (2), positive in columns

36Another important reason to use logarithms is to deal with a large variation in the dependentvariable (see Table 4) and the ensuing heteroscedasticity.

28

TABLE 5: Tax Effects on Option Grant SensitivityDependent variable ln(Option Grant Sensitivity)Tax Brackets All Top Only All Top Only All Allln(1-tax rate) −0.419∗ −0.270 0.679∗∗ 0.630∗∗ 1.357 1.461

(0.174) (0.173) (0.221) (0.225) (1.160) (1.166)

ln(market-to-book) 0.161∗∗ 0.162∗∗ 0.189∗∗ 0.198∗∗ 0.197∗∗ 0.193∗∗

(0.045) (0.046) (0.045) (0.047) (0.046) (0.045)

ln(market value) 0.481∗∗ 0.498∗∗ 0.421∗∗ 0.435∗∗ 0.428∗∗ 0.424∗∗

(0.044) (0.047) (0.048) (0.051) (0.046) (0.048)

ln(stock price volatility) −0.167∗∗ −0.183∗∗ −0.042 −0.068 −0.071 −0.049(0.043) (0.045) (0.046) (0.048) (0.046) (0.046)

ln(stock return) 0.324∗∗ 0.327∗∗ 0.303∗∗ 0.296∗∗ 0.310∗∗ 0.313∗∗

(0.049) (0.051) (0.049) (0.052) (0.049) (0.049)

ln(lagged stock return) 0.012 0.020 0.014 0.012 0.006 0.010(0.036) (0.038) (0.036) (0.039) (0.036) (0.037)

ln(return on assets) 0.591∗∗ 0.514 0.552∗∗ 0.546 0.546∗∗ 0.545∗∗

(0.166) (0.351) (0.152) (0.345) (0.151) (0.152)

Corporate governance 0.056∗ 0.051+ 0.030 0.039 0.027 0.029index (0.027) (0.027) (0.027) (0.027) (0.027) (0.027)

CEO 0.279∗∗ 0.202∗∗ 0.200∗∗ 0.199∗∗ 0.204∗∗ 0.205∗∗

(0.050) (0.043) (0.042) (0.042) (0.042) (0.042)

MILLION 0.521∗∗ 0.562∗∗ 0.354∗∗ 0.372∗∗ 0.288∗∗ 0.285∗∗

(0.054) (0.053) (0.060) (0.061) (0.085) (0.085)

ln(1+lagged total −0.027∗ −0.022+ −0.030∗∗ −0.031∗∗ −0.030∗∗ −0.030∗∗

option sensitivity) (0.011) (0.011) (0.011) (0.011) (0.011) (0.011)

ln(1+lagged total 0.014 0.016 −0.000 0.005 0.001 −0.001stock sensitivity) (0.014) (0.014) (0.013) (0.014) (0.014) (0.013)Industry Time Trends No No Yes Yes No YesYear Dummies No No No No Yes YesExecutive Fixed Effets N/A Yes Yes Yes Yes YesObservations 8,612 7,977 8,612 7,977 8,612 8,612Executives 2,679 2,440 2,679 2,440 2,679 2,679R2 0.538 0.252 0.263 0.262 0.26 0.264Selection Yes No No No No NoSelection Test P-value 0.023 0.181 0.14 0.286 0.917 0.084

Notes: (1) Standard errors adjusted for heteroscedasticity and clustering at executive level inparentheses. (2) Significance: ∗∗ at 1% level, ∗ at 5% level, + at 10% level. (3) In columns withoutselection, R2 corresponds to the within regression. In columns with selection, R2 corresponds tothe third-stage OLS regression. In the latter, individual effects are modelled as correlated randomeffects. See Wooldridge (1995) for details.

29

TABLE 6: Tax Effects on Stock Grant SensitivityDependent variable ln(Stock Grant Sensitivity)Tax Brackets All Top Only All Top Only All Allln(1-tax rate) −0.459 −0.377 0.151 0.314 −2.205 −2.982

(0.332) (0.340) (0.438) (0.442) (2.646) (2.653)

ln(market-to-book) −0.193∗ −0.160+ −0.116 −0.083 −0.155+ −0.111(0.086) (0.090) (0.090) (0.095) (0.092) (0.092)

ln(market value) 0.563∗∗ 0.556∗∗ 0.504∗∗ 0.481∗∗ 0.490∗∗ 0.492∗∗

(0.108) (0.118) (0.134) (0.146) (0.117) (0.135)

ln(stock price volatility) −0.088 −0.154+ −0.004 −0.032 −0.043 −0.012(0.079) (0.091) (0.081) (0.095) (0.083) (0.080)

ln(stock return) 0.309∗ 0.265+ 0.238+ 0.191 0.267+ 0.218+

(0.138) (0.140) (0.135) (0.138) (0.136) (0.129)

ln(lagged stock return) 0.221∗ 0.161+ 0.194∗ 0.139 0.165+ 0.134(0.090) (0.089) (0.088) (0.087) (0.091) (0.087)

ln(return on assets) 0.502 0.867 0.403 0.825 0.572 0.483(0.600) (0.705) (0.633) (0.723) (0.633) (0.655)

Corporate governance −0.005 0.001 −0.022 −0.016 −0.010 −0.022index (0.035) (0.035) (0.034) (0.033) (0.036) (0.035)

CEO 0.254∗∗ 0.245∗∗ 0.254∗∗ 0.243∗∗ 0.238∗∗ 0.230∗∗

(0.076) (0.076) (0.078) (0.078) (0.075) (0.078)

MILLION 0.113 0.096 0.019 −0.034 0.225 0.268(0.103) (0.104) (0.109) (0.110) (0.181) (0.182)

ln(1+lagged total 0.000 −0.001 −0.011 −0.013 −0.003 −0.011option sensitivity) (0.020) (0.020) (0.018) (0.018) (0.021) (0.018)

ln(1+lagged total 0.023 0.029 0.015 0.019 0.015 0.018stock sensitivity) (0.031) (0.031) (0.031) (0.031) (0.031) (0.031)Industry Time Trends No No Yes Yes No YesYear Dummies No No No No Yes YesExecutive Fixed Effects Yes Yes Yes Yes Yes YesObservations 2,616 2,464 2,616 2,464 2,616 2,616Executives 1,091 1,022 1,091 1,022 1,091 1,091R2 0.1 0.103 0.125 0.129 0.107 0.131Selection No No No No No NoSelection Test P-value 0.084 0.187 0.28 0.471 0.514 0.559

Notes: (1) Standard errors adjusted for heteroscedasticity and clustering at executive level inparentheses. (2) Significance: ∗∗ at 1% level, ∗ at 5% level, + at 10% level. (3) In columns withoutselection, R2 corresponds to the within regression. In columns with selection, R2 corresponds tothe third-stage OLS regression. In the latter, individual effects are modelled as correlated randomeffects. See Wooldridge (1995) for details.

30

(3) and (4), again negative in columns (5) and (6), and not statistically significantin any of the specifications. These results suggest that changes in the personalincome tax rate do not have a significant impact on the incentives generated byrestricted stock grants. The coefficient estimates on other variables are consistentwith predictions outlined in the previous section except for the coefficient on thelogarithm of market-to-book ratio.

Table 7 displays multivariate estimates for the share of compensation paid inoptions. When the time trends are controlled for by linear industry time trends(columns (3) and (4)), the estimate on the net-of-tax-rate variable is positive andstatistically significant. However, once the time trends are controlled for by yeareffects, the coefficient estimates become negative and statistically insignificant. As aresult, there is little conclusive evidence of the impact of an increase in the personalincome tax rate on the share of compensation paid in options. This corresponds toresults reported by Hall and Liebman (2000), who find that the coefficient on thenet-of-tax-rate variable is numerically small and statistically insignificant.

5.2 The Role of Corporate Governance

The specification used in Tables 5, 6, and 7 assumes that the tax effect is the sameacross all the firms. However, this may not be the case. One important candidatefor heterogeneity of the tax response is the strength of corporate governance. It maybe that executive pay in companies with stronger corporate governance is mostlygoverned by shareholder value-maximization, for which the agency model providesa theoretical underpinning. On the other hand, in companies with weaker corporategovernance, managers may effectively set their own pay, thereby extracting rentsfrom their employers. In the terminology of Bebchuk and Fried (2003), managerialpay in the former group is described by the “optimal contracting approach,” whilein the latter group it is described by the “managerial power approach.”37 Regardingthe response to a personal tax rate increase, in the former group it should be betterapproximated by one of the models in Section 3, which trade off tax savings andincentive contract distortions. On the other hand, the response in the latter group islikely to be more heavily weighted towards tax savings, with little regard to incentivecontract distortions. In particular, one would expect weak corporate governancecompanies to reduce the use of options relative to strong corporate governancecompanies in response to the 1993 (and 1994) tax increase.

The first pair of rows in Table 8 displays the estimates of the net-of-tax-shareelasticity of the option grant sensitivity for the two types of companies. Strongcorporate governance is defined by a panel-average of the Gompers et al. (2003)index of 9 (the median value in the sample) or less, while weak corporate governanceis defined by a panel-average value of more than 9. The estimates of the net-of-tax-share elasticity are positive and statistically significant for the weak corporate

37Bebchuk and Fried also survey empirical evidence in support of this hypothesis when corporategovernance is measured by the effectiveness of company board, the presence of a large outsideshareholder, the concentration of share ownership, and the presence of antitakeover arrangements.See references contained therein for more details.

31

TABLE 7: Tax Effects on the Share of Compensation Paid in OptionsDependent variable Share of Compensation Paid in OptionsTax Brackets All Top Only All Top Only All Allln(1-tax rate) −0.003 −0.003 0.107∗ 0.107∗ −0.104 −0.089

(0.034) (0.035) (0.046) (0.048) (0.206) (0.208)

ln(market-to-book) 0.007 0.006 0.012 0.010 0.013 0.012(0.010) (0.011) (0.011) (0.011) (0.011) (0.011)

ln(market value) 0.034∗∗ 0.031∗ 0.027∗ 0.025+ 0.028∗ 0.028∗

(0.011) (0.012) (0.012) (0.013) (0.012) (0.012)

ln(stock price volatility) 0.012 0.006 0.024∗ 0.020+ 0.022∗ 0.023∗

(0.010) (0.010) (0.010) (0.011) (0.010) (0.010)

ln(stock return) −0.055∗∗ −0.058∗∗ −0.059∗∗ −0.062∗∗ −0.056∗∗ −0.056∗∗

(0.011) (0.011) (0.011) (0.012) (0.011) (0.011)

ln(lagged stock return) −0.005 −0.004 −0.006 −0.005 −0.012 −0.012(0.008) (0.009) (0.008) (0.009) (0.008) (0.008)

ln(return on assets) 0.103∗ 0.128+ 0.110∗∗ 0.138∗ 0.111∗∗ 0.111∗∗

(0.040) (0.071) (0.039) (0.070) (0.040) (0.040)

Corporate governance 0.010∗ 0.010∗ 0.007 0.008 0.008 0.007index (0.005) (0.005) (0.005) (0.005) (0.005) (0.005)

CEO 0.033∗∗ 0.033∗∗ 0.033∗∗ 0.033∗∗ 0.032∗∗ 0.032∗∗

(0.008) (0.008) (0.008) (0.008) (0.008) (0.008)

MILLION 0.053∗∗ 0.052∗∗ 0.029∗ 0.025+ 0.036∗ 0.034+

(0.011) (0.011) (0.013) (0.013) (0.018) (0.018)

ln(1+lagged total −0.009∗∗ −0.009∗∗ −0.009∗∗ −0.009∗∗ −0.009∗∗ −0.009∗∗

option sensitivity) (0.002) (0.002) (0.002) (0.002) (0.002) (0.002)

ln(1+lagged total 0.000 0.001 −0.001 0.000 −0.001 −0.001stock sensitivity) (0.003) (0.003) (0.003) (0.003) (0.003) (0.003)Industry Time Trends No No Yes Yes No YesYear Dummies No No No No Yes YesExecutive Fixed Effects Yes Yes Yes Yes Yes YesObservations 12,524 11,301 12,524 11,301 12,524 12,524Executives 3,161 2,821 3,161 2,821 3,161 3,161R2 0.021 0.022 0.024 0.025 0.025 0.025

Notes: (1) Standard errors adjusted for heteroscedasticity and clustering at executive level inparentheses. (2) Significance: ∗∗ at 1% level, ∗ at 5% level, + at 10% level. (3) R2 corresponds tothe within regression.

32

TABLE 8: Tax Effects: Strong vs. Weak Corporate Governance CompaniesTax Brackets All Top Only All Top Only All AllOption grant sensitivity:Strong Corporate −0.379 −0.378 0.309 0.156c −0.465b −0.499b

Governance Companies (0.249) (0.255) (0.345) (0.354) (1.560) (1.564)

Weak Corporate −0.398+ −0.238 0.953∗∗ 0.971∗∗c 4.679∗∗b 4.819∗∗b

Governance Companies (0.239) (0.244) (0.285) (0.290) (1.456) (1.482)Stock grant sensitivity:Strong Corporate −0.504 −0.304 0.534 0.803 −1.879 −3.308Governance Companies (0.586) (0.609) (0.724) (0.708) (3.339) (3.161)

Weak Corporate −0.592 −0.567 0.023 0.034 −0.213 2.022Governance Companies (0.382) (0.389) (0.530) (0.546) (1.985) (2.215)Share of comp. in options:Strong Corporate −0.097∗a −0.111∗a −0.067a −0.083a −0.433b −0.455+b

Governance Companies (0.049) (0.051) (0.067) (0.071) (0.277) (0.277)

Weak Corporate 0.093+a 0.108∗a 0.305∗∗a 0.318∗∗a 0.402b 0.474+b

Governance Companies (0.048) (0.049) (0.063) (0.065) (0.278) (0.279)Industry Time Trends No No Yes Yes No YesYear Dummies No No No No Yes Yes

Notes: (1) Standard errors adjusted for heteroscedasticity and clustering at executive level inparentheses. (2) Significance: ∗∗ at 1% level, ∗ at 5% level, + at 10% level. (3) Coefficientestimates for the option and stock grant sensitivity are corrected for selection when necessary. (4)Significance for the test of the equality of coefficients: a at 1% level, b at 5% level, c at 10% level.

governance companies once the time trends are controlled for (columns (3)-(6)),while they are insignificant for the strong corporate governance companies. Theelasticity estimates for the former group are also statistically significantly differentfrom the latter group in columns (4)-(6).

The second pair of rows of Table 8 display the estimates of the net-of-tax-shareelasticity of the stock grant sensitivity for the two types of companies. For thismeasure of incentives there are no statistically significant differences in the responseto the tax increase.38

Finally, the third pair of rows of Table 8 displays the estimates of the impact ofthe tax increase on the share of compensation paid in options. In all of the specifica-

38If the pay design in weak corporate governance companies reflects managers paying themselvesand substituting away from options to less heavily taxed instruments of compensation followinga personal income tax increase, a natural question is what these other instruments are. Theresults for the restricted stock grant sensitivity reveal that it is unlikely to be these grants. I havealso tried to investigate whether these managers substitute to non-taxable compensation, definedas inflation-adjusted “all other total compensation.” Although this variable may contain sometaxable items, Goolsbee (2000a) argues that it mostly contains (currently) non-taxable premiumsfor life insurance policies and company matching contributions to 401(k) retirement accounts.The results (not shown here) do not show any statistically significant difference in the reactionof this compensation measure to the tax increase between strong and weak corporate governancecompanies either, however.

33

tions, weak corporate governance companies reduce the share of compensation paidin options in response to the tax increase relative to the strong corporate governancecompanies. In addition, once the time trends are controlled for (columns (3)-(6)),the results show that the weak corporate governance companies reduce the share ofcompensation paid in options in response to the tax increase, while strong corpo-rate governance have either a statistically insignificant reaction, or they increase theshare of compensation paid in options. These results are supportive of the findingsfor the option grant sensitivity, and they also suggest why it may be difficult to finda statistically significant tax effect on the share of compensation paid in optionsusing a pooled sample of companies.

The results of this subsection show that the tax-induced reduction in optiongrant sensitivity documented in Table 5 is concentrated among the weak corporategovernance companies. In addition, the net-of-tax-rate elasticity estimate for thisgroup of companies is in most specifications that control for time trends statisticallysignificantly different from the estimate for the strong corporate governance compa-nies, which display no systematic reaction to the tax increase. These results suggestthat it may indeed be the case that managerial pay is driven by different factors inthe two groups of companies, with the distinction between the optimal contractingapproach and the managerial power approach being a possible explanation for thisheterogeneity.

6 Conclusion

Taxes affect incentives. Public finance economists have demonstrated this facttheoretically, and measured it empirically in a great variety of settings. However,most of this research deals with situations where explicit incentive contracting isnot present. The current paper extends this fundamental line of research into thearea of executive compensation, which serves as a natural laboratory for evaluatingthe impact of personal income taxation on incentives when the disincentive effect oftaxation can be partly “undone” by a designer of the compensation scheme so as topreserve the original incentive structure.

The theoretical model presented in Section 3 shows that, in case of a singletax rate, an increase in the marginal tax rate weakly reduces the after-tax pay-to-performance sensitivity of executive pay after the company has adjusted its compen-sation design and hence acts as a disincentive for executives. However, the theory isambiguous about the impact on the pre-tax pay-to-performance sensitivity. Whenthe model is extended to account for a differential taxation of stocks and options,the theory provides some interesting results, but also ambiguous predictions for theimpact of a change in the personal income tax rate. This is because tax-avoidancesubstitution into less-heavily taxed compensation instruments when inducing a par-ticular effort level is confounded with changes in the optimally induced effort level.However, the theory does predict that an increase in the capital gains tax rate re-duces the after-capital-gains-tax pay sensitivity generated by restricted stock grants,a prediction to be evaluated in future research.

34

The empirical part of the paper uses the U.S. federal personal income tax in-creases of 1993 and 1994 as a natural experiment to evaluate the impact of a personalincome tax increase on the design of executive pay. The results show that these taxincreases reduced the pay-to-stock-price sensitivity generated by option grants, withno statistically significant effects on the analogous sensitivity generated by restrictedstock grants. However, splitting the sample by the strength of corporate governancereveals that there are statistically significant differences in the response of the stock-price sensitivity generated by option grants between strong and weak corporate gov-ernance companies. In particular, the reduction in the option grant sensitivity isconcentrated among the weak corporate governance companies. One plausible in-terpretation for these findings is that the design of executive compensation in thestrong corporate governance companies reflects the considerations present in agencytheory, which predicts an ambiguous impact of a personal income tax rate increase.On the other hand, in the weak corporate governance companies, the compensationdesign mostly reflects a tax-shielding behavior of the managers who, lacking a strongprincipal’s control, redesign their pay away from options to relatively less-heavilytaxed compensation instruments. This interpretation suggests that different modelsof executive compensation may be applicable to companies with different strengthsof corporate governance.

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A Appendix

Proof of Lemma 1. Suppose, by contradiction, that there exist a ∈ A and t ∈ [0, 1) such thatz(x, a, u(t)) 6= u(t) + z(x, a, 0). However, because z′(·) ≡ u(t) + z(·, a, 0) satisfies the constraints(7) and (8) for (a, t), it must be the case that

∫z(x, a, u(t))dF (x|a) < u(t) +

∫z(x, a, 0)dF (x|a). (A-1)

But because z′′(·) ≡ z(x, a, u(t))− u(t) satisfies (7) and (8) for (a, 0), (A-1) implies that z′′(·)implements a under the tax rate 0 at a lower expected cost than z(x, a, 0), contradicting theoptimality of z(·, a, 0) for (a, 0). As a result, it must be the case that z(x, a, u(t)) = u(t)+z(x, a, 0),and hence, using (9),

C(a, t) =u(t)1− t

+1

1− t

∫z(x, a, 0)dF (x|a).

38

Proof of Lemma 4. Assumption (10) and Lemma 3 imply that zH1 (t) < zH

2 (t) < zH3 (t), and

hence the (strict) inequality between geometric and harmonic averages implies that

3∑

i=1

pHi log

[1 + δ

(1− pL

i

vpHi

)]= log

3∏

i=1

[1 + δ

(1− pL

i

vpHi

)]pHi

> − log3∑

i=1

pHi

1

1 + δ(1− pL

i

vpHi

)

= 0,

where the last equality follows from the definition of δ in Lemma 3. The result then follows fromLemmata 2 and 3 and the fact that c(aH) > 0.

Proof of Lemma 5. First note that the assumption xH ≤ x2 implies that

x3 − x2

x2 − x1≤ pH

1

pH3

.

Using this result, it follows that

zH2 (t)− zH

1 (t)x2 − x1

− zH3 (t)− zH

2 (t)x3 − x2

=1

x3 − x2

{x3 − x2

x2 − x1

[zH2 (t)− zH

1 (t)]− [

zH3 (t)− zH

2 (t)]}

≤ 1x3 − x2

{pH1

pH3

[zH2 (t)− zH

1 (t)]− [

zH3 (t)− zH

2 (t)]}

=1

pH3 (x3 − x2)

3∑

i=1

pHi

[zH2 (t)− zH

i (t)]

=1

pH3 (x3 − x2) r

3∑

i=1

pHi [log E2 − log Ei]

=1

pH3 (x3 − x2) r

log

[(E2

E1

)pH1

(E2

E2

)pH2

(E2

E3

)pH3

]

<1

pH3 (x3 − x2) r

log[E2

(pH1

E1+

pH2

E2+

pH3

E3

)]

=1

pH3 (x3 − x2) r

log E2

≤ 0,

where, for i ∈ {1, 2, 3},Ei ≡ 1 + δ

(1− pL

i

vpHi

).

The second inequality follows from the inequality between arithmetic and geometric averages andthe fact that zH

1 (t) < zH2 (t) < zH

3 (t) (which follows from (10)). The third inequality follows fromthe assumption pL

2 /pH2 ≥ v. Since sH

i (t) ≡ zHi (t)/(1− t), the convexity result extends to sH

i (t) aswell.

39

Proof of Lemma 6. From (18), the expected compensation cost is given by

3∑

i=1

pji [α + βxi + γ max(0, xi − xG)] =

3∑

i=1

pji [α + βx0 + β(xi − x0) + γ max(0, xi − xG)]

=3∑

i=1

pji

[α + βx0 +

1− τ

1− tβ(xi − x0) + γ max(0, xi − xG)

]

+3∑

i=1

pji

[β(xi − x0)− 1− τ

1− tβ(xi − x0)

]

=1

1− t

3∑

i=1

pjizi − β

t− τ

1− t(xj − x0),

where the last equality follows from (19).Proof of Lemma 7. It follows from (19) that

z2 − z1 = (1− τ)β(x2 − x1)β + (1− t)γ [max(0, x2 − xG)−max(0, x1 − xG)]≥ (1− τ)β(x2 − x1),

since γ ≥ 0 and x2 > x1. As a result,

β ≤ 11− τ

z2 − z1

x2 − x1. (A-2)

Now suppose that (A-2) holds with a strict inequality. Suppose that xG ≥ x2. Then it followsfrom (19) that z2 − z1 = (1− τ)β(x2 − x1), which is a contradiction to (A-2) holding with a strictinequality. Therefore if (A-2) holds with a strict inequality, it must be the case that xG < x2.Then it follows from (19) that

γ =1

1− t

[z3 − z2

x3 − x2− (1− τ)β

].

Now consider increasing β to

β =1

1− τ

z2 − z1

x2 − x1, (A-3)

decreasing γ to

γ =1

1− t

[z3 − z2

x3 − x2− z2 − z1

x2 − x1

],

and increasing xG to x2, and adjusting α such that (19) is satisfied for i = 1. Then (19) is satisfiedfor all i = 1, 2, 3, and so the after-tax compensation schedule is unaffected. However, by Lemma 6,the new compensation contract is less costly in expectation than the original one, contradicting thecost-minimizing property of the original contract. Therefore any cost-minimizing contract mustsatisfy (A-3). Then if (z3 − z2)/(x3 − x2) = (z2 − z1)/(x2 − x1), (19) implies that γ = 0. If(z3 − z2)/(x3 − x2) > (z2 − z1)/(x2 − x1), (19) implies that

γ =1

1− t

x3 − x2

x3 − xG

[z3 − z2

x3 − x2− z2 − z1

x2 − x1

]

and xG ∈ [x2, x3).Proof of Lemma 8. First, the PC must hold with equality. Suppose not. Then each zi

can be reduced by a small positive amount ω, which does not affect the validity of any of theconstraints and strictly reduces the expected compensation cost, which is a contradiction.

Second, the convexity constraint must hold with equality. Suppose, in contradiction, that thecost-minimizing after-tax contract is strictly convex, i.e., it satisfies

z3 − z2

x3 − x2>

z2 − z1

x2 − x1.

40

Then construct an alternative linear after-tax contract z′ such that

z′2 − z′1x2 − x1

=z′3 − z′2x3 − x2

=z2 − z1

x2 − x1(A-4)

and3∑

i=1

pLi z′i =

3∑

i=1

pLi zi. (A-5)

It then follows that z′1 > z1, z′2 > z2, and z′3 < z3. As a result, z is a strict mean-preserving spreadof z′. Therefore

3∑

i=1

pLi

[−e−rz′i

]>

3∑

i=1

pLi

[−e−rzi]

= −e−ru(t),

i.e., z′ satisfies the PC with a slack. By construction, it also satisfies the convexity and themonotonicity constraint. Finally,

3∑

i=1

pLi

[−e−rz′i

]= e−r(z′1−z1)

3∑

i=1

pLi

[−e−rzi]+ pL

3

[(−e−rz′3

)−

(−e−r(z3+z′1−z1)

)]

≥ e−r(z′1−z1)v

3∑

i=1

pHi

[−e−rzi]+ pH

3 v[(−e−rz′3

)−

(−e−r(z3+z′1−z1)

)]

= v

3∑

i=1

pHi

[−e−rz′i

],

where the inequality follows from the ICC for z, z′1 > z1, z′3 < z3, v > 1, and pH3 > pL

3 (which followsfrom (10)). As a result, z′ also satisfies the ICC. Therefore z′ is feasible for the cost-minimizationproblem (22), and, by construction, it generates the same expected pre-tax compensation cost asz. However, because z′ satisfies the PC with a slack, each z′i can be reduced by a small positiveamount ω, which does not affect the validity of any of the constraints and strictly reduces theexpected compensation cost. As a result, aL can be implemented at a strictly lower expectedcompensation cost than under z, which is a contradiction to z being a cost-minimizing contract.

Because the convexity constraint holds with equality, the after-tax cost-minimizing contractimplementing aL must be linear, i.e., it must satisfy zi = kxi + l. The PC them implies that

l = u(t, τ) +1r

log3∑

i=1

pLi e−rkxi ,

and hence the cost-minimization problem becomes equivalent to

mink≥0

k

[1− t

1− τ(xL − x0) + x0

]+

1r

log3∑

i=1

pLi e−rkxi

s.t.3∑

i=1

pLi

[−e−rkxi] ≥ v

3∑

i=1

pHi

[−e−rkxi]. (A-6)

Note that the ICC is satisfied with a slack for k = 0 (−1 > −v), and hence, by continuity, it issatisfied for any k ∈ [0, k∗) for some k∗ > 0. Also note that the derivative of the objective withrespect to k is given by

1− t

1− τ(xL − x0) + x0 −

∑3i=1 pL

i xie−rkxi

∑3i=1 pL

i e−rkxi

,

and the second derivative is given by

r

[∑3i=1 pL

i x2i e−rkxi

] [∑3i=1 pL

i e−rkxi

]−

[∑3i=1 pL

i xie−rkxi

]2

[∑3i=1 pL

i e−rkxi

]2

41

and it is strictly positive by the Cauchy-Schwarz inequality since x1 < x2 < x3. Therefore theobjective function is strictly convex in k. In addition, the first derivative evaluated at k = 0 is− [(t− τ) /(1− τ)] (xL−x0) < 0 when t > τ . Finally, because x1 < x2 < x3, when t > τ and t− τis “small”,

limk→∞

{1− t

1− τ(xL − x0) + x0 −

∑3i=1 pL

i xie−rkxi

∑3i=1 pL

i e−rkxi

}=

1− t

1− τ(xL − x0) + x0 − x1

> 0

As a result, the optimal choice of k is characterized by k(t, τ) implicitly defined by the first-ordercondition

1− t

1− τ(xL − x0) + x0 −

∑3i=1 pL

i xie−rk(t,τ)xi

∑3i=1 pL

i e−rk(t,τ)xi

= 0. (A-7)

This is because when t > τ and t − τ is “small,” the unconstrained solution of (A-6) will, bycontinuity, be “close” to zero, and hence the ICC will not bind. The equation (A-7) can be moreconveniently restated as

3∑

i=1

pLi

[xi − xL +

t− τ

1− τ(xL − x0)

]e−rk(t,τ)xi = 0. (A-8)

Because xL > x0 and (t− τ) / (1− τ) is strictly increasing in t and strictly decreasing in τ , itfollows that k(t, τ) is strictly increasing in t and is it strictly decreasing in τ . Finally, (A-8) impliesthat k(t, t) = 0 for any t ∈ [0, 1).

Proof of Lemma 9. Begin by ignoring the convexity and the monotonicity constraints. Itwill be shown that limit of the solution when t ↓ τ satisfies both of these constraints with a slack,and so, by continuity, so will all the solutions when t− τ is small.

Using the approach of Grossman and Hart (1983), the initial optimization problem may beconverted into a convex programming problem by changing the choice variables to ui ≡ −e−rzi ,implying that zi = − log(−ui)/r. The cost-minimization problem is then equivalent to

minui∈R−

−3∑

i=1

qi(t, τ) log(−ui)

s.t.3∑

i=1

pHi uiv ≥ −e−ru(t,τ)

s.t.3∑

i=1

pHi uiv ≥

3∑

i=1

pLi ui,

where the functions qi(t, τ), i ∈ {1, 2, 3}, are defined by

q1(t, τ) ≡ pH1 +

t− τ

1− τ

xH − x0

x2 − x1,

q2(t, τ) ≡ pH2 − t− τ

1− τ

xH − x0

x2 − x1,

q3(t, τ) ≡ pH3 .

Because the objective is strictly convex in the choice variables and the constraints are linear,standard Kuhn-Tucker approach can be used to characterize the solution. With λ and µ being the

42

multipliers on the two constraints, the necessary and sufficient conditions are

qi(t, τ)ui

+ λpHi v + µ

(pH

i v − pLi

)= 0, i ∈ {1, 2, 3} (A-9)

λ

[3∑

i=1

pHi uiv + e−ru(t,τ)

]= 0, λ ≥ 0 (A-10)

µ

[3∑

i=1

(pH

i v − pLi

)ui

]= 0, µ ≥ 0. (A-11)

The equation (A-9) gives that

ui = −qi(t, τ)pH

i

1λv + µ

[v − pL

i /pHi

] . (A-12)

Substituting this result into (A-10) and (A-11) gives

3∑

i=1

qi(t, τ)λv

λv + µ[v − pL

i /pHi

] = λe−ru(t,τ) (A-13)

and3∑

i=1

qi(t, τ)µ

[v − pL

i /pHi

]

λv + µ[v − pL

i /pHi

] = 0, (A-14)

respectively. Adding (A-13) and (A-14) together and rearranging gives

λ = eru(t,τ). (A-15)

One implication of this finding is that the PC binds at the optimum, with the intuition being thesame as in the proof of Lemma 8. Substituting (A-15) into (A-12) gives

ui = −qi(t, τ)pH

i

1eru(t,τ)v + µ

[v − pL

i /pHi

] . (A-16)

Now suppose that µ = 0. Then ui = −(qi(t, τ)/pHi )/(veru(t,τ)), and the ICC becomes, since

q1(t, τ) > pH1 , q2(t, τ) < pH

2 and q3(t, τ) = pH3 ,

v ≤3∑

i=1

pLi

qi(t, τ)pH

i

. (A-17)

However, if t = τ , qi(t, τ) = pHi for all i ∈ {1, 2, 3}, and hence (A-17) implies v ≤ 1, a contradiction.

Since v > 1, the same result also extends, by continuity, to the case when t > τ and t−τ is “small.”As a result, in optimum it must be the case that µ > 0, and hence the ICC binds.

Using this result and substituting from (A-15) into (A-14) gives that µ is uniquely implicitlydefined by

3∑

i=1

qi(t, τ)1− pL

i /(pH

i v)

1 + µe−ru(t,τ)[1− pL

i /(pH

i v)] = 0. (A-18)

on the interval(0, eru(t,τ)

(pL1 /

(pH1 v

)− 1)−1

). To see this, first note that the left-hand side (LHS)

of (A-18) is continuous in µ on[0, eru(t,τ)

(pL1 /

(pH1 v

)− 1)−1

). Second, note that at µ = 0, the

left-hand side (LHS) of (A-18) is equal to 1 − 1/v > 0 for t = τ and hence, by continuity, itis strictly positive for t > τ and t − τ “small.” Third, the LHS of (A-18) is strictly decreasingin µ on the interval

[0, eru(t,τ)

(pL1 /

(pH1 v

)− 1)−1

). Fourth, the LHS of (A-18) tends to −∞ as

43

µ → eru(t,τ)(pL1 /

(pH1 v

)− 1)−1. These four arguments show that there is a unique solution of

(A-18) for µ on the interval(0, eru(t,τ)

(pL1 /

(pH1 v

)− 1)−1

). Finally, it cannot be the case that

µ ≥ eru(t,τ)(pL1 /

(pH1 v

)− 1)−1, since then eru(t,τ)v + µ

[v − pL

i /pHi

] ≤ 0, which, using (A-16),contradicts ui ∈ R−.

To summarize, ignoring the convexity and monotonicity constraints, the solution to the cost-minimization problem inducing aH for t ≥ τ , t− τ small, is given by

zHi (t, τ) = u(t, τ) + c(aH) +

1r

log{

pHi

qi(t, τ)

[1 + δ(t, τ)

(1− pL

i

vpHi

)]}, (A-19)

with the expected cost function being

CH(t, τ) ≡ 11− t

{u(t, τ) + c(aH) +

1r

3∑

i=1

qi(t, τ) log(

pHi

qi(t, τ)

[1 + δ(t, τ)

(1− pL

i

vpHi

)])},

where δ(t, τ) is implicitly defined by

3∑

i=1

qi(t, τ)1− pL

i /(pH

i v)

1 + δ(t, τ)[1− pL

i /(pH

i v)] = 0. (A-20)

Now note that for t = τ , the cost-minimizing contract corresponds to the contract describedin Lemma 3 (because qi(t, t) = pH

i for any t ∈ [0, 1)), for which strict monotonicity and strictconvexity have been established. Then, by continuity, same is also true for the case t > τ whent− τ is small.

Now consider an increase in t. Such increase increases q1 and decreases q2 by the same absoluteamount, without affecting q3. Note that from the assumption (10) it follows that

1− pL1 /

(pH1 v

)< 1− pL

2 /(pH2 v

) ≤ 0 < 1− pL3 /

(pH3 v

), (A-21)

and hence1− pL

1 /(pH1 v

)

1 + δ(t, τ)[1− pL

1 /(pH1 v

)] <1− pL

2 /(pH2 v

)

1 + δ(t, τ)[1− pL

2 /(pH2 v

)] .

As a result, an increase in t, holding δ constant, strictly decreases the LHS of (A-20). Because theLHS of (A-20) is also strictly decreasing in δ over the range

[0,

{(pL1 /

(pH1 v

)− 1)}−1

), it follows

that δ(t, τ) is strictly decreasing in t.Now consider an increase in τ so that t ≥ τ is preserved. Such increase decreases q1 and

increases q2 by the same absolute amount, without affecting q3. An analogous argument thenshows that δ(t, τ) is strictly increasing in τ .

Using these comparative statics and (A-21), (A-19) implies that zH2 (t, τ) − u(t, τ) is strictly

increasing in t and zH3 (t, τ)− u(t, τ) is strictly decreasing in t. To see the impact of an increase in

t on zH1 (t, τ)− u(t, τ), rewrite (A-20) as

3∑

i=1

pHi

1− pLi /

(pH

i v)

pHi

qi(t,τ)

{1 + δ(t, τ)

[1− pL

i /(pH

i v)]} = 0.

The inequality (A-21) and the monotonicity properties of zH2 (t, τ) − u(t, τ) and zH

3 (t, τ) − u(t, τ)then imply that both the second and the third term in this sum increase with an increase in t.As a result, the first term must decrease, and hence zH

1 (t, τ) − u(t, τ) is strictly decreasing in t.An analogous argument shows that the comparative statics properties of zH

i (t, τ) − u(t, τ) withrespect to an increase in τ are exactly the opposite.

Then, using Lemma 7, it follows that, when t > τ and holding xG fixed, βH(t, τ) is strictlyincreasing in t, and (1− t)γH(t, τ) is strictly decreasing in t, (1− τ)βH(t, τ) is strictly decreasingin τ , and γH(t, τ) is strictly increasing in τ .

44