Discussion Papers In Economics
And Business
Taxing mobile and overconfident top earners
Andreas HAUFLER
Yukihiro NISHIMURA

Discussion Paper 21-26

February 2022

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANTaxing mobile and overconﬁdent top earners∗
Andreas Hauﬂer†

Yukihiro Nishimura‡

University of Munich and CESifo

Osaka University and CESifo

Revised version, February 2022

Abstract
We set up a simple model of tax competition for mobile, highly-skilled andoverconﬁdent managers. Firms endogenously choose the compensation scheme formanagers, which consists of a ﬁxed wage and a bonus payment in the high state.
Managers are overconﬁdent about the probability of the high state and hence ofreceiving the bonus, whereas ﬁrms and governments are not. When governments
maximize tax revenues, we show that overconﬁdence unambiguously reduces thebonus tax rate that governments set in the non-cooperative tax equilibrium, while
increasing tax revenues. When the government objective incorporates the welfareof resident managers, however, bonus taxes also serve a corrective role and may
rise in equilibrium when overconﬁdence is increased.
Keywords: Overconﬁdence, bonus taxes, tax competition, migration
JEL classiﬁcation: H20, H87, G28∗

We are grateful to the editor, Ron Davies, and three referees for their constructive and detailed

comments. This paper was presented at the Belgian-Japanese Public Finance Workshop in Louvainla-Neuve and at the 2020 online conference of the IIPF. We thank David Agrawal, Traviss Cassidy,
Daniel Gietl, Bernhard Kassner, Mohammed Mardan and Pierre Pestieau for helpful comment and Susanna Hassinen for excellent research assistance. For ﬁnancial support we thank the German Research
Foundation through CRC TRR 190 (Hauﬂer) and MEXT/JSPS KAKENHI JP18H00866 (Nishimura).
The paper was started when Yukihiro Nishimura visited the University of Munich. He wishes to thankthe members of CES for their hospitality.
†

Seminar for Economic Policy, Akademiestr. 1/II, D-80799 Munich, Germany. Phone +49-89-2180-

3858, fax: +49-89-2180-6296, e-mail: Andreas.Hauﬂer@econ.lmu.de‡

Graduate School of Economics, Osaka University, e-mail: ynishimu@econ.osaka-u.ac.jp

1

Introduction

During the last decades, top income earners have been able to increase their share in
total national income in most countries.1 An important contributor to this increased
income concentration is the prevalence of bonus payments and other forms of incentive
pay. For example, Bell and Van Reenen (2014) show that the top percentile of income
earners in the United Kingdom received 35% of their total pay as bonus income in2008, and the bonus share was even 44% in the ﬁnancial sector. Similarly, Lemieux etal. (2009) ﬁnd for the United States that performance pay accounts for most of theincrease in wage inequality above the 80th percentile during the period 1976-1998.2
At the same time, top income tax rates have been reduced in many countries. Egger etal. (2019) have shown for the OECD countries that income tax systems have becomeless progressive since the mid-1990s. They explain this development with the increasing
international mobility of high income earners. Moreover, several countries speciﬁcallytry to attract international top earners by means of tax cuts that are only available
to foreign residents (Kleven et al., 2020). There is a substantial literature indicatingthat the international mobility of top managers has grown substantially over the past
two decades (e.g. Staples, 2007; Greve et al., 2015).3 Theoretical work has shown thatthe international mobility of top income earners reduces the optimal progressivity ofincome tax schedules in the countries competing for the high-skilled, mobile population(Simula and Trannoy, 2010; Lehmann et al., 2014). These eﬀects are conﬁrmed inempirical studies demonstrating that foreign residents respond to tax incentives with
an elasticity that is far larger than the response of domestic residents (Kleven et al.,2013; Kleven et al., 2014; Akcigit et al., 2016).
In this paper we introduce overconﬁdence as a behavioral trait of mobile top earners andask how this aﬀects the tax competition for them. Speciﬁcally we focus on overestima1

See Atkinson et al. (2011) for an international comparison and Piketty et al. (2018) for a detailed

study of income distribution in the United States, based on national accounts data.
2

More generally, Bryson et al. (2012, Figures 1 and 2) show that the share of private sector em-

ployees with an incentive pay contract has risen substantially over time in most OECD countries, andis highest in the Scandinavian countries (around 30%) and in the U.S. (over 40%).
3

Staples (2007) investigates 70 of the world’s largest transnational corporations and ﬁnds thatthe percentage of ﬁrms with at least one non-national board member rose from 36% in 1993 to 75%
in 2005. Greve et al. (2015) show that the internationalization of management boards is positively
associated with the globalization strategies of ﬁrms.

1

tion as the most common form of overconﬁdence.4 The psychology literature has shownthat many individuals overestimate their own abilities and talents, as well as the probabilities of advantageous events (Taylor and Brown, 1988). This behavioral pattern isparticularly pronounced among high-income individuals, who have experienced successin their previous career and attribute this success largely, or even exclusively, to theirown abilities (Gervais and Odean, 2001). Empirical research has convincingly shownthat company CEOs and top managers exhibit systematic and persistent patterns ofoverconﬁdent behavior.5 Thus Malmendier and Tate (2005) show that overconﬁdent
CEOs make excessive investments when their liquidity is high, and Malmendier andTate (2008) show that overconﬁdent CEOs engage in value-destroying mergers. Ho etal. (2016) demonstrate that banks with overconﬁdent CEOs weakened lending standards prior to the 2007-2009 ﬁnancial crisis, and performed worse in the crisis. Finally,
Humphery-Jenner et al. (2016) show empirically that ﬁrms ‘exploit’ this overconﬁdenceby adjusting their compensation structure and increasing the share of incentive pay.
There are some well-known examples of top income earners that are both internationally mobile and overconﬁdent. Perhaps the most obvious case is Elon Musk, a South
African native who migrated to the United States. As the current CEO of Tesla, a large
share of his wealth is invested in Tesla stock, a clear signal of overconﬁdent beliefs in
his ﬁrm (cf. footnote 5). Another example is Josef Ackermann, a Swiss national who
chaired the Deutsche Bank from 2002-2012 and in 2003 famously declared a 25% rateof return to equity as his target. A third example is Masayoshi Son, the Korean-born
CEO of the Japanese SoftBank, who bought the US real estate company We-Work for
USD 47 billion in 2019, and revised its value to USD 2.9 billion just one year later.6
More generally, the psychology literature has shown that migration decisions are empirically linked to personality traits such as openness and extraversion (Jokela, 2009;

4

Moore and Healy (2008) distinguish three notions of overconﬁdence: overestimation, overplacement(relating to comparisons with others) and overprecision. While overprecision relates to the margin oferror in stochastic decisions, overestimation relates to the probability with which (typically positive)
outcomes occur.
5

In these studies, the dominant approach to empirically identify managerial overconﬁdence is thefailure of top managers to diversify their personal investment portfolio. Speciﬁcally, managers areclassiﬁed as ‘overconﬁdent’ when they do not sell stock options in their own ﬁrm that are “in themoney” (Malmendier and Tate, 2005).
6

See https://www.cnbc.com/2020/05/18/softbank-ceo-calls-wework-investment-foolish-valuation-

falls-to-2point9-billion.html

2Canache et al., 2013).7 Just these personality traits have in turn been shown to bepositively associated with overconﬁdence (Schaefer et al., 2004; Dessi and Zhao, 2018).
Against this background, the present paper studies tax competition for managers thatare both mobile and overconﬁdent. We cast our analysis in a framework where managers
hold an incentive contract and receive a compensation that consists of both a ﬁxed
wage and a bonus in case of success. Governments levy a bonus tax, but this tax canalso be seen more generally as a higher tax rate (or surtax) on top incomes.8 In ourbenchmark model we assume that governments maximize tax revenues and use thereceipts from bonus taxation for redistributive purposes. In this setting we analyze
how overconﬁdence of top managers aﬀects both the compensation structure of theﬁrms employing them, and the bonus tax rate that governments competing for themobile high-skilled levy in the non-cooperative tax equilibrium.
Our ﬁrst main ﬁnding is that overconﬁdence increases the share of the bonus component
in total manager compensation, relative to the ﬁxed wage rate. This higher prevalenceof bonuses makes the ex-post distribution of incomes more unequal and it increases thetax base for the bonus tax. Nevertheless, we ﬁnd that higher levels of overconﬁdence
reduce revenue-maximizing bonus tax rates in the symmetric Nash equilibrium.
The intuition for this result is that overconﬁdent managers overestimate the likelihood that they will receive the bonus, and hence pay the bonus tax. This increases
their migration elasticity, and rational governments have to factor in this behavioral
response. Since the true likelihood that the bonus is paid and the bonus tax is indeed
collected is lower than anticipated by managers, overconﬁdence makes the bonus tax aless attractive instrument from the perspective of governments. Hence, in addition to
the international mobility of top earners, their overconﬁdence may also contribute to
explaining both the increased reliance of ﬁrms on bonuses and other forms of incentive
pay, and the fall in top income tax rates faced by highly skilled migrants.
The falling tax rate on bonus incomes does not imply, however, that overconﬁdence
is detrimental for the government’s tax revenue collections. In fact, our second main
7

The important role of personality traits is also mirrored in the economics literature. Borjas et al.

(2018) ﬁnd, using Danish data, that more than half of the diﬀerence between the expected earningsof migrants and non-migrants is due to diﬀerences in unobserved characteristics (‘personality’).
8

Many countries incorporate bonuses in the tax base for the general income tax. The United States

is an exception as it taxes bonuses as ‘supplementary income’ under a separate tax schedule. This
schedule has fewer tax brackets than the general income tax schedule, but the top tax rates are aligned.

3

result shows that managerial overconﬁdence raises bonus tax revenues in equilibrium,
despite the falling bonus tax rate. This is because overconﬁdence shifts the manager’s
compensation towards the taxed bonus component of his pay, and this increase in thebonus tax base dominates the eﬀects of a lower tax rate. In fact, governments can“exploit” the overconﬁdence of managers in equilibrium, in the sense that the rewards
to the overly high eﬀort caused by the manager’s overconﬁdence partly accrue to thegovernment as tax revenue.
We then consider two modiﬁcations of our benchmark model. First, we assume thatgovernments incorporate the utility of resident managers in their objective function, in
addition to tax revenues. This gives the bonus tax a corrective component, as governments use bonus taxes to limit the managers’ excessive work eﬀort that follows from themisperceived success probability. Therefore, equilibrium tax rates may now be rising in
the managers’ overconﬁdence level, if this corrective component is suﬃciently strong.
In a second extension, we show that our results for revenue-maximizing governments
are robust to introducing a two-tier income tax where the ﬁxed wage income is taxed
at a positive rate and bonus income is subject to an additional tax – or, alternatively,
to a higher marginal tax rate. In this setting it remains true that, whenever the extra
tax on bonus income is positive in equilibrium, it is reduced by overconﬁdence, andtax progressivity accordingly falls.
Our paper contributes to the literature on the optimal taxation of mobile high-income
individuals (Hamilton and Pestieau, 2005; Simula and Trannoy, 2010; Bierbrauer et al.,2013; Lehmann et al., 2014; Blumkin et al., 2015; Lipatov and Weichenrieder, 2015;
Wilson, 2015). Most of this literature considers non-linear income taxation and focuseson marginal migration costs as a determinant of optimal marginal tax rates.9 This
literature shows that marginal tax rates generally fall as a consequence of international
mobility, and that marginal tax rates on top earners may even become negative. We
simplify the tax schedule by focusing on (piecewise) linear income taxation and showthat adding overly optimistic beliefs will increase the elasticity with which high-income
migrants respond to taxation, and hence further reduce optimal tax rates.
We also contribute to the recent literature on optimal income taxation with behavioral
agents (Bernheim and Taubinsky, 2018; Farhi and Gabaix, 2020; Moore and Slemrod,
9

The empirical literature on tax-induced mobility conﬁrms that high-income earners respond tohigher taxes by relocating to other regions or countries (Schmidheiny and Slotwinski, 2018; Agrawal
and Foremny, 2019).

42021). Much of this literature focuses on issues of tax salience and individual misperceptions of tax schedules. One exception is Gerritsen (2016), who derives optimalcorrective tax policies when heterogeneous individuals work too much or too little.
By incorporating overconﬁdence, we introduce individual misperceptions of their own
abilities, which is a major theme in behavioral ﬁnance (Malmendier and Tate, 2015).
Moreover we take the behavioral public economics literature to an international setting
where countries compete in attracting mobile and overconﬁdent managers.10
Our analysis of bonus payments and their taxation draws on the framework of Besley
and Ghatak (2013). Gietl and Hauﬂer (2018) have extended this setting to analyze
international competition in bonus taxes when governments bail out failing banks.
Gietl and Kassner (2020) introduce overconﬁdence, but analyze bonus taxation in aclosed economy. In this paper, we use a simpler framework without default risks andgovernments bailouts, which is therefore not speciﬁc to the banking sector. We combine,
however, the international mobility of managers with their overconﬁdence.
In the following, Section 2 describes our model, analyzing in turn the eﬀort and migration decisions of high-skilled managers and the ﬁrm’s choice of the managerial compensation structure. Section 3 addresses international competition in bonus taxes when
governments maximize tax revenues. Section 4 analyzes the case where governments
maximize national welfare, including managers’ utility. Section 5 extends the analysisof revenue maximizing governments by considering a two-tier income tax in which all
incomes are taxed at a positive, exogenous rate and an endogenous surcharge is leviedon bonus income. Section 6 concludes.

2

The model

2.1

The basic setup

We structure our analysis as a sequential four-stage game. In the ﬁrst stage, governments non-cooperatively choose their bonus taxes, anticipating the responses of bothﬁrms and their managers to these taxes. In the second stage, ﬁrms choose their proﬁtmaximizing remuneration scheme, consisting of a ﬁxed wage and a bonus payment, and10

See Kotakorpi (2009) for one of the few existing analyses of tax competition in a framework with

non-rational agents. Her focus is very diﬀerent from ours, however, and analyzes Pigouvian taxation
in a setting where consumers have self-control problems.

5taking as given the bonus taxes that governments have set in the ﬁrst stage. In thethird stage, managers decide in which country to work, on the basis of the remuneration
schemes oﬀered to them by ﬁrms, as well as the taxes levied by their potential countryof residence. Finally, in the fourth stage, managers choose their level of eﬀort provisionin the country (and hence ﬁrm) of their choice. We thus model a strict hierarchy ofdecisions where optimizing governments behave as ﬁrst movers towards ﬁrms, whereasﬁrms behave as ﬁrst movers vis-à-vis managers.
The framework of our analysis is a region of two symmetric open economies i ∈ {1, 2},which are small in the world market. In each of the two countries, there is a representative ﬁrm of variable size, where ﬁrm size corresponds to the number of identicaldivisions within the ﬁrm. Running a division requires the speciﬁc knowledge of a ﬁrm
manager, which is the limiting resource in our model. Each ﬁrm employs exactly one
manager per division and the number of managers a ﬁrm hires equals the number ofits divisions. Hence each ﬁrm tries to attract internationally mobile managers in order
to increase the number of its divisions, and hence proﬁts.
Each division of a ﬁrm in country i has a total amount of ﬁxed assets equal to one,which is lent in the world market. Lending operations are risky. We assume that there
are two possible returns for each of the identical divisions of a ﬁrm, which can be high(h) or low (l). The division realizes a high return Y h with probability ph > 0 and a low
return Y l < Y h with probability pl = 1 − ph . Even the low return Y l is suﬃcient for theﬁrm to pay all its obligations, and to avoid default. Returns are ﬁxed from the ﬁrm’s
perspective, as outputs are sold in a large world market. Hence, the representative ﬁrms
in both countries produce with constant returns to scale.
We employ a standard principal-agent problem between the ﬁrms’ shareholders andtheir managers. Managers have private eﬀort costs and thus choose lower eﬀort thanwould be optimal from the perspective of shareholders. Bonuses thus serve as a secondbest instrument for the ﬁrm to solve this principal-agent problem.11
In each country i, managers choose an eﬀort level ei and all managers behave in identicalways with respect to their eﬀort choice. We assume that the probability that the ﬁrm

11

There is an active discussion of whether existing incentive contracts are compatible with modern

shareholder value theories (Edmans and Gabaix, 2016), or reﬂect “rent extraction” by CEOs (Piketty
et al., 2014). Our discussion restricts attention to the productivity enhancing eﬀects of incentive
contracts.

6

receives the high return Y h is a linear function of each manager’s eﬀort level:12
phi = βei ,pli = 1 − βei ,β, phi , pli > 0.

(1)

In our model, ﬁrm managers overestimate their skills and thus overestimate the returnto their eﬀort. We denote parameters as perceived by an overconﬁdent manager witha hat (ˆ). The probabilities of the high and the low state, as perceived by managers,
are then given byp̂hi = (1 + θ)βei ,p̂li = 1 − (1 + θ)βei .

(2)The parameter θ > 0 in eq. (2) measures the degree of managers’ overconﬁdence.
Overconﬁdence thus leads each manager to overestimate the likelihood of the highstate, p̂h > ph , and hence to overestimate the expected reward to his eﬀort. We conﬁne
the level of overconﬁdence to the range 0 ≤ θ < 1.13 In the following, changes in θ will
be at the heart of our analysis of how overconﬁdence aﬀects the ﬁrms’ compensation
schemes for their managers, as well as the governments’ optimal bonus taxes.

2.2Managers’ eﬀort choice

We proceed by backward induction to obtain the subgame perfect Nash equilibrium.
In Stage 4, risk-neutral managers choose their eﬀort levels. For analytical tractability,
we assume that the cost of eﬀort provision is quadratic and given by c(e) = ηe2 /2. Dueto these private costs, managers will not exert enough eﬀort from the point of view ofﬁrm owners. Eﬀort decisions are not observable. However, ﬁrm owners can mitigate theprincipal agent problem by a bonus payment zi in the high return state, which occurs
with a higher probability ph (e) when the manager’s eﬀort level e is increased.
Managers located in country i maximize their perceived location-speciﬁc utility ûi ,which is the excess of expected bonus payments and the ﬁxed wage wi over the private
costs of eﬀort.14 Using (2) givesûi = p̂i h zi + wi − c(ei ) = (1 + θ)βei zi + wi −

ηe2i
.
2

(3)

12

Since managers are identical, manager-speciﬁc indices are omitted from all variables.

13

The upper bound placed on θ is needed to ensure that the ﬁrms’ optimal bonus choice is well-deﬁned; see Section 2.4 below. The constraint that p̂h < 1 must hold in (2) will then be met by placing

an upper bound on the parameter β.
14The perceived utility is also labelled decision utility in the behavioral economics literature. In

contrast, the utility evaluated at the true success probability phi is known as experience utility (e.g.
Farhi and Gabaix, 2020).

7

Maximizing (3) with respect to the managers’ choice variable ei yields
ei =(1 + θ)βzi
.
η

(4)

Hence each manager’s eﬀort level ei increases in the bonus payment zi and in his levelof overconﬁdence θ. The latter occurs because an overconﬁdent manager overestimates
the likelihood that the high state will occur, and hence the expected return to his eﬀort.
In contrast, the ﬁxed wage wi does not aﬀect managers’ optimal eﬀort.
Substituting the managers’ eﬀort decision (4) into (1), we can derive the true equilibrium probabilities of the states h and l:ph∗i = βei = (1 + θ)
β2zi ≡ (1 + θ)γzi ,ηpl∗i = 1 − βei = 1 − (1 + θ)γzi .

(5a)
(5b)

In eq. (5a), we have introduced the parameter γ > 0 to summarize the marginal eﬀectof the bonus payment on the probability of a high return. This eﬀect is multiplied by
the overconﬁdence factor (1 + θ). Therefore, the true success probability is rising in theoverconﬁdence of managers, because of the higher eﬀort level that is induced by thebonus when managers overestimate their return to eﬀort.
The managers’ perceived success probability is then p̂hi = (1 + θ)ph∗= (1 + θ)2 γzi .
i
Substituting this along with (4) and (5a) into (3) gives the location-speciﬁc perceived
utility of a manager working in country i. This perceived utility is increased by both ahigher bonus and a higher ﬁxed wage:
2
∗
2 γzi
+ wi .
ûi = (1 + θ)

2

(6)

Again, an overconﬁdent manager (θ > 0) overvalues the inﬂuence of the bonus payment
zi on his utility in country i, as he overestimates his success probability.

2.3

Managers’ migration decision

In Stage 3 managers take the bonuses zi and ﬁxed wages wi as given and choose
whether to work in country 1 or in country 2. Managers maximize their gross utility,
which consists of the location-speciﬁc utility in (6), and the non-monetary attachment
to a particular country. We normalize the total number of managers in the region to 2.

8

All managers are employed in one of the two countries. Hence N1 + N2 = 2, where Ni isthe (continuously divisible) number of managers working in country i in equilibrium.
Managers diﬀer in their country preferences. More precisely, managers are of type m,
where m is the relative attachment to country 1 and we assume that m is distributed
uniformly along [−1, +1]. Other things equal, all managers with m > 0 prefer to workin country 1, whereas managers with m < 0 prefer to work in country 2.
The utility weight of the location preference parameter m is given by the constanta > 0. The gross utility Ui of a manager of type m in country i is thenU1 (z1 , w1 , m) = û∗1 (z1 , w1 ) + am,

U2 (z2 , w2 ) = û∗2 (z2 , w2 ).

(7)

All managers choose to work in the country that gives them the higher gross utility.
We characterize the manager that is just indiﬀerent between working in country 1 orin country 2 by the critical location preference mc . Equating U1 and U2 in (7) andusing (6), we derive mc as a function of diﬀerences in bonus payments and ﬁxed wages
between the two countries:
mc =


1 γ
(1 + θ)2 (z22 − z12 ) + (w2 − w1 ) .
a 2

(8)

Managers with m ∈ [mc , 1] work in country 1 and managers with m ∈ [−1, mc ] workin country 2. Using (8) then determines the number of managers in country i as afunction of the diﬀerences in bonus payments and wages:

1 γ
2 2
2
Ni = 1 +∀ i, j ∈ {1, 2}, i = j.
(1 + θ) (zi − zj ) + (wi − wj )
a 2

(9)

The larger is the bonus of country i, relative to that of country j, the more managers
will work in country i in equilibrium. The same holds for the ﬁxed wage. As managers
value the bonus payment zi in each country by their perceived probability of the highstate p̂h , the eﬀect of bonuses (z1 , z2 ) on the location decision of managers is increasedby managers’ overconﬁdence, whereas the eﬀect of the ﬁxed wage is not.

2.4

Firms’ compensation choices

In Stage 2, we turn to the remuneration decisions made by the owners of the representative, scalable ﬁrm in each country. The representative ﬁrm in country i sets thebonus zi ≥ 0 and the ﬁxed wage wi ≥ 0 to maximize its expected after-tax proﬁts. Bothbonuses and ﬁxed wage payments are constrained to be non-negative. Since all divisions
9

are equal, total proﬁts Πi are obtained by multiplying the proﬁts of a representative
division, πiD , with the number of divisions, which equals the number of managers Ni .
Firms assess their expected division proﬁts using the true probabilities of the high andl∗
the low state (ph∗i , pi ). The bonus is paid only in state h, and its gross costs to theﬁrm are increased by the bonus tax ti . Therefore, total after-tax expected proﬁts are


h
l∗ l
[Y
−
z
(1
+
t
)]
+
p
Y
−
w
Πi = Ni πiD = Ni ph∗
.
i
i
i
i
i

(10)

The ﬁrm maximizes its proﬁts in (10) with respect to the bonus zi , taking account of the
managers’ migration decision (9) and their equilibrium eﬀort levels, which determine
probabilities by (5a)–(5b). This gives

∂Πi
γzi D
πi + Ni (1 + θ)γ (Y h − Y l ) − 2zi (1 + ti ) = 0.
= (1 + θ)2
∂zi
a

(11)

The ﬁrst eﬀect in eq. (11) gives the eﬀect of the bonus on the number of ﬁrm divisions.
This eﬀect is unambiguously positive. As we discussed in (9) above, it incorporates
the managers’ migration decision that is based on their perceived success probabilityp̂h∗ = (1 + θ)2 γzi . The second eﬀect in eq. (11), which describes the marginal eﬀect of
the bonus on the proﬁts of a representative division, must therefore be negative in the
ﬁrm’s optimum. For this second eﬀect, the true success probability ph∗ = (1 + θ)γzi isrelevant, as this determines the probability with which the bonus is actually paid.
Maximizing ﬁrm proﬁts in (10) with respect to the ﬁxed wage gives∂ΠiπD= i − Ni ≤ 0 ∀ i.
∂wi
a

(12)

The ﬁrst-order condition (12) holds with equality, if and only if the ﬁxed wage ispositive in the ﬁrm’s optimum; otherwise the ﬁxed wage is zero. In line with existing
manager compensation schemes, we assume that wi > 0 holds in the ﬁrm’s optimum.
This implies πiD /a = Ni . Substituting this into (11) we can simplify the ﬁrst-ordercondition for the optimal bonus payment. The optimal bonus can then be expressed
as:15
zi∗ =
15

Ω
1 − θ + 2ti∀ i,Ω ≡ Y h − Y l.

(13)

To ensure positive bonuses, the denominator in eq. (13) must be positive. This will be guaranteed

if θ < 1 and ti ≥ 0. This is the reason for placing an upper bound on θ (cf. footnote 13). The secondorder condition for the ﬁrm’s optimization problem with respect to zi and wi will also hold if θ < 1.
The proof is available from the authors upon request.

10

Hence the equilibrium bonus is rising in the output gap between the high and thelow state, Ω, and it is falling in the bonus tax rate ti . Moreover, for a given tax
rate ti , the bonus payment is unambiguously increasing in the managers’ degree ofoverconﬁdence θ. Intuitively, since managers overestimate the probability of the goodstate h, they also overestimate the probability of receiving the bonus. This makes thebonus an attractive form of compensation from the perspective of the ﬁrm, which holds
rational expectations about the likelihood of the high state.16
In Appendix A.1, we use equations (12) and (13) for both countries to derive theoptimal ﬁxed wage, as a function of both countries’ tax rates. This is given by1γ(1 + θ)Ω2
4ti − 5θ − 1∗lwi = Y − a +.
+6(1 + 2tj − θ) (1 + 2ti − θ)2(14)

The second term in (14) shows that an increase in manager mobility (a decrease in a)
increases the ﬁxed wage. In contrast, the parameter a does not appear in the optimal
bonus equation (13). This shows that the ﬁxed wage is the ﬁrms’ marginal instrument to
attract mobile workers in our analysis, whereas the bonus payment is used to incentivize
eﬀort. The overall eﬀects of the overconﬁdence parameter θ on the ﬁxed wage will beanalyzed once we have solved for the equilibrium bonus tax rate ti .

3

Bonus tax competition with revenue maximizing
governments

3.1

Cooperative tax setting

We now turn to the eﬀects that the mobility of overconﬁdent managers has on tax
competition between the two symmetric countries. In this section we assume that governments aim to maximize tax revenues Ri . This assumption corresponds to maximizing
the transfer to the local resident population, and hence local welfare, while assumingthat governments do not include footloose managers in their welfare objective. Revenue maximization is also an attractive government objective for reasons of analyticaltractability, and it allows us to derive closed-form solutions for optimal tax rates underboth coordinated and non-coordinated tax setting.
16

Humphery-Jenner et al. (2016) provide empirical evidence for this eﬀect by showing that ﬁrms

‘exploit’ overconﬁdent CEOs and other executives by tailoring their compensation structure towards
incentive pay.

11

Like ﬁrms, governments form an unbiased expectation about the probabilities of thediﬀerent states. Hence they maximize:
Ri = ti ph∗
i zi N i∀ i ∈ {1, 2}.

(15)

We ﬁrst study the eﬃcient benchmark case when countries coordinate their tax policies.
In our symmetric model this corresponds to joint-revenue maximization Ri + Rj =h∗ti ph∗
i zi Ni + tj pj zj Nj . The optimal coordinated tax rate is then implicitly deﬁned by∂(Ri + Rj )
∂(ph∗
∂Ni
∂Nj
i zi )= ph∗Ni + ti ph∗+ tj ph∗= 0.
i zi Ni + ti
i zi
j zj
∂ti
∂ti
∂ti
∂ti

(16)

Since the total number of managers in our model is ﬁxed, Ni + Nj = 2, symmetry
implies that the last two terms on the left-hand side of (16) sum to zero. Using (5a)and (13) and noting that ph∗j and zj do not depend on ti the ﬁrst-order condition forthe cooperative, revenue-maximizing tax rate (superscripts C, R) simpliﬁes to
4ti∂(Ri + Rj )= zi2 γ(1 + θ)Ni 1 −∂ti
1 − θ + 2ti

=0

=⇒

1
tC,R
= (1 − θ).
i
2

(17)

Eq. (17) yields a very simple expression for the coordinated tax rate, which is fallingin the degree of overconﬁdence. Intuitively, overconﬁdent managers overestimate thelikelihood of receiving a bonus, and hence they also overestimate the negative eﬀect of
the bonus tax on their utility. This reduces the optimal coordinated tax, relative to asetting where managers behave fully rationally.
Substituting (5a), (13), and the cooperative tax rate from (17) into the revenue expression (15) yields
RiC,R =

(1 + θ)γΩ2,8(1 − θ)(18)

which is unambiguously rising in θ. Higher overconﬁdence of managers increases boththeir eﬀort and the bonus payment as an incentive device. Together, these increases
in the bonus tax base more than compensate for the lower bonus tax rate so thatcoordinated tax revenue increases in the overconﬁdence level θ.

3.2

Non-cooperative tax setting

We now turn to the non-coordinated tax equilibrium where each country chooses its
optimal bonus taxes ti in isolation. In the ﬁrst step, we determine the response of the12equilibrium number of managers in country i when this country raises its bonus tax.
Diﬀerentiating (9) with respect to ti and taking account of (13) and (14) leads to−γ(1 + θ)Ω2
1 γ(1 + θ)2 ∂(zi2 ) ∂wi ∂wj
∂Ni
=
=
+
−
< 0.
(19)
∂ti
a
2
∂ti
∂ti
∂ti
3a(1 − θ + 2ti )2
From (19) we see that the outﬂow of managers in response to a higher bonus tax unambiguously rises in the managers’ overconﬁdence level θ. This is because the managers’
migration decision is based on their overly high expectation of receiving the bonus,
combined with their correct anticipation that the tax will reduce the bonus payment.
In the second step we maximize tax revenue in (15) with respect to ti , using the
migration response (19), along with the eﬀects of the tax on the success probability
ph∗i in (5a) and on the bonus payment zi in (13). This gives
4titi γ(1 + θ)Ω2∂Ri
zi2 γ(1 + θ)aNi 1 −−.
=∂tia
1 − θ + 2ti
3(1 − θ + 2ti )2(20)

In eq. (20), the ﬁrst term in the squared bracket gives the change in tax revenues for agiven number of managers. This term is in turn composed of the mechanical eﬀect ofa tax increase at an unchanged tax base, and the fall in the expected tax base inducedby the higher bonus tax (resulting from both the lower bonus payment and the reduced
success probability of the project). The negative second term in the squared bracket
gives the loss in the tax base that results from the outmigration of managers in responseto the tax.
Evaluated at ti = 0, the derivative ∂Ri /∂ti is positive in eq. (20). Also, continuity ofRi (ti , tj ) is guaranteed in our setting because all the relevant functions are continuous inti and tj . Finally, Appendix A.2 derives the second-order condition for the governments’optimal choice of bonus taxes, and shows that this is fulﬁlled. Hence, a Nash equilibriumwith positive bonus tax rates in both countries must exist in our model.
In the following, we focus on a symmetric Nash equilibrium. Setting (20) equal to zeroat ti = tj , which in turn implies Ni = 1, yields a quadratic equation in t. Solving gives
an explicit expression for the non-cooperative, revenue maximizing tax rate in bothcountries (superscripts N, R):γ Ω2 (1 + θ)
=
−
+
tN,R
i
24a

γΩ2 (1 + θ)
24a

2
+

1
(1 − θ)2 > 0.
4

(21)Comparing the non-cooperative tax rate in (21) to the cooperative tax rate in (17)yields tN,R
< tC,R . Hence, as in models with mobile, but fully rational individualsi
13

(Lehmann et al., 2014; Lipatov and Weichenrieder, 2015), the competition for mobile
managers reduces optimal bonus tax rates below their cooperative levels. In turn, thelower non-cooperative tax than under cooperation leads to higher bonus payments andlower ﬁxed wages.
It is also straightforward to establish that a higher degree of international mobility (alower attachment-to-home parameter a) reduces the non-cooperative bonus tax rates
in equilibrium:
∂tN,R
i
∂a

2

=

⎡

2

⎛

γ Ω (1 + θ) ⎝γ Ω (1 + θ) ⎣
1−
2
24a
24a

γΩ2 (1 + θ)

⎞−1 ⎤

2

24a

+

1
(1 − θ)2 ⎠ ⎦ > 0.
4(22)This result is in line with the literature on the optimal taxation of mobile individuals
that face relocation costs (e.g. Simula and Trannoy, 2010).
The simple structure of (21) further allows us to unambiguously sign the derivative of
the Nash equilibrium tax rate with respect to the overconﬁdence parameter θ:


2 4
2
Ω
(1
+
θ)
−
144(1
−
θ)a
∂tN,R
γ
1
i
< 0.
=
−Ω2 γ + ∂θ24a
γ 2 Ω4 (1 + θ)2 + 144(1 − θ)2 a2(23)

Equation (23) shows that overconﬁdence of mobile managers reduces the optimal tax
rate that both governments choose in the Nash equilibrium. Note that this result occurs
even though the bonus tax base is increasing in the overconﬁdence level θ for any given
tax rate ti [eq. (13)]. However, this eﬀect is overcompensated by a twofold increasein the elasticity with which the bonus tax base responds to taxation. First, even in
the absence of manager migration, ﬁrm’s bonus payments respond more elasticallyto the bonus tax when θ increases. This is seen from the positive ﬁrst eﬀect in thesquared bracket of the ﬁrst-order condition (20), whose negative component is risingin θ. Moreover, a higher level of overconﬁdence leads to a larger migration responseof managers to the bonus tax rate, as is seen from the negative second eﬀect in thesquared bracket of (20).
Intuitively, the negative eﬀect of overconﬁdence on the governments’ optimal bonus tax
rate arises because revenue-maximizing governments have to incorporate the higher
migration elasticity with which overconﬁdent managers respond to the bonus tax. At
the same time, a rational government will calculate the expected bonus tax revenue
only on the basis of the true success probability. Therefore, a higher overconﬁdence level
raises the elasticity of the bonus tax base more quickly than it raises the bonus tax
14base itself. This makes the bonus tax a less attractive instrument from the government’s
perspective.
Given that the optimal bonus tax rate is falling in the overconﬁdence level θ, we
can now determine how the ﬁrms’ compensation schemes for managers change as aresult of overconﬁdence. Inserting this result in the bonus payment (13) shows thatthe positive direct eﬀect that an increase in θ has on the equilibrium bonus payment
zi increases further due to the indirect eﬀect that results from the reduced bonus tax
rate. Therefore, equilibrium bonus payments unambiguously increase when managers
become more overconﬁdent.
Next we derive the equilibrium change in the ﬁxed wage payment of ﬁrms that resultsfrom overconﬁdence. Evaluating the ﬁxed wage expression (14) at the common Nash
= tN,Rin (21) and diﬀerentiating gives
equilibrium tax rate tN,R
i
j


N,R
(2t
−
3θ
−
1)
∂wi∗
∂t
i
1 + ti − (1 + θ) i
= γΩ2
< 0.
3
∂θ
(1 + 2ti − θ)
∂θ

(24)

−3θ−1 < 2tN,R−2tC,R< 0 holds in the Nash equilibrium.
This is negative because 2tN,Riii

Also, the squared bracket is unambiguously positive since ∂tN,R/∂θ < 0 from (23).
i
Therefore the ﬁxed wage declines when managers become more overconﬁdent, and thecompensation structure unambiguously shifts towards more incentive pay.
We can also determine how managerial overconﬁdence aﬀects each government’s equilibrium level of tax revenues. Overconﬁdence has counteracting eﬀects on total tax
collections, because it increases the bonus tax base, but reduces the equilibrium tax
rate under tax competition, as given in (23). The total eﬀect of θ on tax revenues isderived in Appendix A.3 and given by
8t2i γΩ2
∂Ri∗
=
> 0.
∂θ
(1 − θ + 2ti )2 [(1 − θ)2 + 4t2i ](25)

Hence, in the Nash equilibrium, the enlarged tax base dominates the eﬀects of moreaggressive tax rate competition between countries. To understand this result, recall that
there are oﬀsetting eﬀects on the equilibrium bonus tax rate that follow from the higher
tax base and the higher elasticity of the tax base. Combined with the unambiguous
increase in the tax base, this leads to higher tax revenue collections in each of the twocompeting countries as a result of higher managerial overconﬁdence.
With non-cooperative tax setting, overconﬁdence thus lowers the bonus tax rate, butraises equilibrium tax revenue. The same qualitative results have also been shown under15

cooperative tax setting; cf. eqs. (17) and (18). It is then natural to ask how the level ofoverconﬁdence θ aﬀects the revenue losses that arise from tax competition. From (18)and (A.8) an the appendix, we get:1 ∂RiC,R
1 ∂Ri∗2(2ti + θ − 1)2−=> 0,Ri∗ ∂θ
(1 + θ)[(1 − θ)2 + 4t2i ]RiC,R ∂θ

(26)

which shows that higher managerial overconﬁdence aggravates the revenue losses arisingfrom tax competition for mobile managers. This is a direct consequence of the resultthat higher levels of overconﬁdence increase the migration response of managers, andtherefore make tax competition more aggressive.
Our results in this section are summarized in:
Proposition 1 When governments maximize tax revenues, increased overconﬁdenceof mobile managers (a rise in θ) leads to:(i) ﬁrms choosing a compensation structure with higher bonus payments and lower ﬁxed
wages;(ii) governments choosing lower bonus taxes in the non-cooperative equilibrium, despite
the increased bonus tax base;(iii) equilibrium tax revenues rising for all initial levels of θ.
(iv) a larger shortfall of tax revenues, in comparison to the cooperative allocation.
Note that Proposition 1(iii) implies a redistributive eﬀect from the manager to thegovernment in equilibrium. This redistributive eﬀect arises because the manager’s overconﬁdence leads to a higher reliance on the taxed bonus component of his pay, whereas
the untaxed ﬁxed wage component is reduced. In this sense the manager’s overconﬁdence is “exploited” by the government in our model. This diﬀers from the existing
literature in closed economies, where a higher bonus share allows the ﬁrm to meet theparticipation constraint of an overconﬁdent manager at a lower expected total salary(Gervais et al., 2011; De la Rosa, 2011). In our model ﬁrms’ proﬁts are ﬁxed in equilibrium, however, by the international competition for scarce and mobile managers.17
As a result, the gains from the managers’ misconception are passed on to governmentsin the form of higher tax revenue.
17

This is seen from (12) when the ﬁxed wage is part of the compensation package and (12) thus holdswith equality. The ﬁrms’ competition for mobile managers will, in equilibrium, ﬁx division proﬁts πiD
at a level that depends only on the managers’ international mobility, as measured by the parameter a.
Total proﬁts in equilibrium are then ﬁxed by Π∗i = πiD = a.

16

We conclude this section by emphasizing that employing an overconﬁdent manager
does not have any negative eﬀects on the level of ﬁrm proﬁts in our benchmark model.
In Appendix A.4 we analyze an extended setting that incorporates such a negativeeﬀect of overconﬁdence. Speciﬁcally, we assume that overconﬁdence leads managers
to take fewer precautions, leading to higher costs for the ﬁrm in the case of adverse
events. We model this as a loss to the ﬁrm arising in the low state that is proportional
to the manager’s degree of overconﬁdence. Hence, in the low state, the return to theﬁrm is given by Y l − κθ, where κ ≥ 0. We show that this additional eﬀect increases
the ﬁrm’s optimal bonus payment in equilibrium, and also that it strengthens thedownward eﬀect on the equilibrium tax rate. On net, the increase in the bonus tax
base dominates and tax revenues rise further under this model extension. In otherwords, all our results in Proposition 1 are strengthened when we incorporate a loss
from overconﬁdent managerial behavior that arises in the low state.
The intuition for these results is straightforward. The loss that overconﬁdence createsin the low state makes the ﬁrm even more willing to incentivize the manager’s eﬀort,
in order to increase the likelihood that the high state with Y h occurs. Hence managerial pay will be directed even more towards higher bonuses and lower ﬁxed wages, ascompared to our benchmark model. These changes simultaneously raise the bonus tax
base and increase the elasticity with which managers respond to the bonus tax rate,
hence amplifying all eﬀects that overconﬁdence has in our benchmark model.

4

Welfare maximizing governments

In this section we ask how our above results are changed when governments, in addition to caring about transfers to the local population, also include the actual utilityof mobile managers into their objective function. Hence we adopt a paternalistic perspective where governments see through the behavioral bias of managers and aim atcorrecting this bias via bonus taxation. We assume, however, that governments do notincorporate ﬁrms’ proﬁts in their objective function, for example because these proﬁts
accrue entirely to non-residents.18
18

While ﬁrm proﬁts are ﬁxed in equilibrium in our model (see footnote 17), proﬁts are still perceived

as endogenous by non-cooperatively behaving governments. This is why we need an explicit assumptionthat competing governments place a weight of zero on ﬁrm proﬁts. We are grateful to an anonymousreferee for pointing out this complication to us.

17Speciﬁcally, we assume that governments maximize a weighted sum of tax revenues
and the welfare of a representative manager in their jurisdiction, where the welfare
weight of the representative manager is exogenously given by λ > 0. This objectivefunction is in line with other models where the government’s objective includes thewelfare of mobile individuals (Mansoorian and Myers, 1993; Hindriks and Myles, 2006,
Chapter 18). Evaluating the representative manager’s net utility in (3) at the actual
probability of receiving the bonus, phi , the welfare function is given by

ηe2i
∗
h
h
Wi = Ri (ti , tj ) + λui = ti pi zi Ni + λ pi zi + wi −
∀i.
2

4.1

(27)

Cooperative tax policy

We start again with the cooperative case. The cooperative tax rate of welfaremaximizing governments is derived from a global welfare function W = Ri + Rj +λ(ui + uj ), i = j. The eﬀect of a change in ti on joint tax revenue Ri + Rj is givenin (16) above. For the utility of the representative manager in country i, we substitute (4), (5a) and (13) in (27). This yields:u∗i = γ zi2(1 + θ)2
1+θ−+ wi ,
2(28)

where wi (ti , tj ) is given in (14). To derive the tax rate that maximizes the joint welfareof the representative managers in both countries, we make use of symmetry, which
implies ∂uj /∂ti = ∂ui /∂tj . Diﬀerentiating (28) and (14) with respect to ti and tj , thecooperative tax rate that maximizes the manager’s utility, tC,u , is given by5θ − 1tC,u =.
(29)
2
Equation (29) shows that the cooperative tax rate tC,u is increasing in the manager’s
behavioral bias θ. Intuitively, overconﬁdence distorts the manager’s eﬀort decision in
the direction of ‘excessive’ eﬀort, when evaluated at the actual success probability. A
higher bonus tax changes the managers’ compensation package, increasing the ﬁxed
wage and reducing the bonus, thus reducing the incentive to provide excessive eﬀort.
This corrective eﬀect of the tax will beneﬁt managers, and it can even dominate thenegative income eﬀect of the tax when θ is suﬃciently high.
The optimal cooperative tax rate of welfare-maximizing governments is derived by
diﬀerentiating W with respect to ti . This yields the ﬁrst-order condition∂Wγ(1 + θ)Ω2=[(1 − θ − 2ti ) + λ(5θ − 1 − 2ti )] = 0,∂ti
(1 − θ + 2ti )3
18

(30)

which can in turn be written as a weighted average of the revenue-maximizing tax rate
derived in the last section [eq. (17)] and the tax rate that maximizes managerial utilityin (29). Denoting this cooperative tax rate by tC,W gives:
tC,W =

1
λtC,R +
tC,u .
(1 + λ)(1 + λ)(31)

Equation (31) shows that the overconﬁdence parameter θ has counteracting eﬀects on
the cooperative, welfare-maximizing tax rate, since tC,R is falling in θ from (17), buttC,u is rising in θ from (29). If the welfare weight λ of a representative manager is large
enough, then the cooperative, welfare-maximizing tax rate may indeed be rising in θ,
in contrast to our result for tax revenue maximizing governments.
The result that optimal taxes should be corrected upwards when individuals provide
excessive work eﬀort has been studied by Gerritsen (2016) in a closed-economy setting,and for a non-linear income tax. Gerritsen also empirically estimates, using British life
satisfaction data, that high-income individuals do indeed tend to work “too much”.
His empirical ﬁndings are therefore in line with the setup of our analysis.

4.2

Non-cooperative tax policies

When bonus taxes are set non-cooperatively, the eﬀect of a tax increase in country i on
its tax revenues Ri is given in (20). The eﬀect of an isolated tax increase in country ion the welfare of a representative manager in this country diﬀers from the cooperative
case analyzed above. Diﬀerentiating (28) and wi (ti , tj ) in (14) with respect to ti , while
holding tj constant, and denoting the non-cooperative tax rate that maximizes the
gives
manager’s utility by tN,ui7θ − 1.
(32)
2Comparing (32) to (29) shows that, when looking at the isolated maximization of mantN,u=iagerial utility, the non-cooperative tax rate rises more steeply in θ than the cooperative
tax rate. Intuitively, an isolated increase in the bonus tax puts ﬁrms in country i undercompetitive pressure and induces them to change the managers’ compensation package
more strongly in the direction of higher ﬁxed wages and lower bonuses than is true
under a coordinated tax increase. This is because the ﬁxed wage is the marginal instrument for each country to attract mobile managers; cf. our discussion of eq. (14).
In sum, therefore, an insolated tax increase in country i is a more potent instrument19

to correct the distortion arising from managerial overconﬁdence than is a coordinatedincrease in ti and tj .
To derive the non-cooperative tax rate of welfare maximizing governments, we diﬀerentiate (27) with respect to ti . This yields
γ(1 + θ)Ω2
ti γ(1 + θ)Ω2∂Wi
2λ=(1 − θ − 2ti ) −+ (7θ − 1 − 2ti ) = 0. (33)∂ti
(1 − θ + 2ti )33a(1 − θ + 2ti )3Comparing (33) to the ﬁrst-order condition in the cooperative case [eq. (30)] showsthat the ﬁrst terms in the squared bracket are identical. In (33) there is, however, anegative second eﬀect that derives from tax competition and is not present in (30). On
the other hand, the isolated last term in the squared bracket of (33) implies a higher
tax rate than the last term in (30), corresponding to the higher level of tN,uin (32), asi

compared to tC,uin (29). Therefore, non-cooperative tax rates can even be higher thanicooperative tax rates for welfare-maximizing governments, if the manager’s welfare
weight λ is suﬃciently high.19
The eﬀects of higher overconﬁdence levels θ on the optimal non-cooperative tax rates
will generally be ambiguous when governments maximize national welfare. We know
from our analysis of Section 3 that the eﬀect of θ on non-cooperative, revenuemaximizing tax rates tN,R
is negative [see eq. (23)], whereas the eﬀect of a higher θ onithe tax rate that maximizes the managers’ utility is positive [eq. (32)]. Appendix B.2
shows that a suﬃcient condition for the net eﬀect of θ on tN,W
to be negative is thatithe welfare weight of the manager is λ < 3/14. Conversely, tN,W
will be rising in theioverconﬁdence level θ when the managerial welfare weight λ is suﬃciently high.

Taking these tax changes into account, we can then ask how overconﬁdence aﬀects theequilibrium utility level of a representative manager. Diﬀerentiating (28) with respect
and tN,W
symmetrically gives
to θ and incorporating the endogeneity of tN,WijγΩ2 (2t2i + ti − 5ti θ − 4θ)
∂tN,W
∂u∗i
=
+Δ i ;∂θ(1 − θ + 2ti )3∂θΔ≡γΩ2 (1 + θ)(5θ − 1 − 2ti ). (34)(1 − θ + 2ti )3

The ﬁrst term in (34) shows that managers’ utility is generally hump-shaped in theiroverconﬁdence level θ. For θ = 0, managers’ utility is rising in θ if ti > 0.20 Intuitively, for low levels of overconﬁdence, the extra eﬀort supplied by the manager has
19

The argument comparing implicit optimal tax rates in (30) and (33) relies in the second-ordercondition of the optimal tax problem Wi (ti ) to hold. Appendix B.1 shows that the second-ordercondition is satisﬁed for any λ > 0, if θ is suﬃciently low (θ ≤ 0.25).
20

In Appendix B.2, we show that tN,R
> 0 when λ ≤ 1.5.
i20

a ﬁrst-order eﬀect on relaxing the moral hazard constraint and increasing the aggregate surplus from employing the manager. Therefore, higher total earnings more thancompensate the manager for his increased eﬀort. At the same time, the distortion of
the manager’s eﬀort level through his behavioral bias is only of second order when theinitial level of overconﬁdence is very low. For high levels of θ, however, the distortion of
the manager’s eﬀort decision arising from overconﬁdence will cause a ﬁrst-order welfare
loss when measured by his actual utility. Therefore, high levels of overconﬁdence reducethe manager’s true utility.
These eﬀects are modiﬁed by the changes in the bonus tax rate that are induced byincreased overconﬁdence. For low levels of overconﬁdence we have Δ < 0. In this casethe second term in (34) is positive when the bonus tax rate falls in response to higher
overconﬁdence, as is the case when revenue considerations dominate in governments’optimal tax choices. For high levels of overconﬁdence and suﬃciently low level of λ, theterm Δ is instead positive. In this case, the second term is positive when the corrective
motive dominates in the setting of bonus tax rates and ∂tN,W/∂θ > 0.
i
Finally, we derive the aggregate welfare changes resulting from overconﬁdence. Appendix B.2 derives suﬃcient conditions for tax revenue to rise in response to changes
in overconﬁdence, ∂Ri∗ /∂θ > 0, as in the case of revenue maximizing governments.
When these conditions are met, a suﬃcient condition for Wi∗ to be rising in θ is that∂u∗i /∂θ in (34) is also positive. As we have discussed above, this will unambiguously be
the case when the overconﬁdence level is low in the initial equilibrium (so that the ﬁrst
eﬀect in (34) is positive and Δ < 0), and when the bonus tax rate is decreased by a risein θ. Interestingly, the latter will occur when governments value the managers’ utility
only moderately (λ < 3/14), and hence reduce the tax rate in response to an increasein managerial overconﬁdence. Intuitively, in this case the governments’ dominant concern about tax revenues will lead them to reduce the bonus tax rate, and the resulting
increase in their net income beneﬁts managers when their initial level of overconﬁdence
is suﬃciently low. Conversely, aggregate welfare may be falling in θ when the initiallevel of managerial overconﬁdence is high, and when the welfare weight of managers in
the governments’ objective function is so high that this negative eﬀect dominates the
positive eﬀect that higher levels of θ have on equilibrium tax revenue.
We summarize the conditions under which our results from the previous section extend
to the case of welfare-maximizing governments as follows:
Proposition 2 When governments maximize national welfare, but the managerial wel21

fare weight is suﬃciently low (λ < 3/14), higher overconﬁdence θ of mobile managers
leads to: (i) falling optimal tax rates; and (ii) rising equilibrium tax revenues.
Proof: See Appendix B.2.
The results summarized in Proposition 2 are illustrated in Figure 1. The panels on theleft-hand side of Figure 1 [case (a)] assume a low welfare weight of managers (λ = 0.1),
satisfying the condition λ < 3/14.21 All panels compare cooperative tax setting (blackline with diamonds) and non-cooperative tax setting (blue line with ﬁlled circles). The
upper panel of case (a) shows that both cooperative and non-cooperative tax rates are
falling in the level of overconﬁdence θ. The middle panel of case (a) shows that tax
revenues are rising in θ under both cooperative and non-cooperative tax setting, but the
gap between cooperative and non-cooperative tax revenues widens as θ is increased.
In the lower panel of case (a), aggregate welfare is monotonously rising in θ undercooperative tax policies, but it is hump-shaped in θ when taxes are set cooperatively.
This is because true managerial welfare drops steeply when θ becomes large, and tax
rates under tax competition are too low to correct this distortion substantially.
**************************************************** Figure 1 about here ****************************************************The right-hand panels of Figure 1 [case (b)] assume instead that the government values
managerial welfare higher (λ = 0.5), thus violating the condition λ < 3/14 in Proposition 2. In the upper panel of case (b), cooperative tax rates are now rising in θ, whereas
non-cooperatively tax rates are largely ﬂat. As θ increases, the downward pressure on
tax rates due to the more intense competition for mobile managers is roughly compensated by the upward eﬀect on tax rates that arises from the motive to correct the
manager’s distorted beliefs. Tax revenues continue to rise in θ for both cooperative andnon-cooperative tax setting, and Proposition 2(ii) continues to hold for this higher levelof overconﬁdence. Finally, as in case (a), aggregate welfare levels under cooperative andnon-cooperative tax setting diverge as θ increases, implying that the welfare costs oftax competition become more severe, the higher is managerial overconﬁdence. Hence,
our ﬁnding for revenue-maximizing governments [Proposition 1(iv)] carries over to themore general welfare function in this section.
21

Exogenous parameters in the numerical examples are set at Y l = 1, Ω = 8, γ = 0.02; a = 0.1.

22

5

Two-tier income taxation

Our benchmark model in Section 3 has considered only a tax on bonus payments, but
has left untaxed the ﬁxed salary earned by managers in each country. This has direct
implications for tax revenue when ﬁrms change the mix of bonus versus ﬁxed wage
compensation. In this section we show that qualitatively the same results are obtained
when we allow both the ﬁxed wage and the bonus to be taxed, but the bonus is taxed
at a higher rate under a general and progressive income tax. To keep the complexityof the resulting framework manageable, we revert to the simpler case of tax revenue
maximizing governments.
We thus analyze a two-tier tax system where all income is taxed at a ﬂat rate τi ,but an additional surtax ti is levied on bonuses. Since all managers receive the same
compensation package in our model, this is equivalent to a directly progressive twotier income tax system where the ﬁxed wage falls in the lower (‘general’) income tax
bracket with a tax rate of τi , and the bonus payment in case of success falls in thehigher bracket with tax rate (τi + ti ). Moreover, while managers are identical ex ante,their incomes diﬀer ex post due to the stochastic environment of our model. In such asetting we ask how an optimal progressive tax is determined from the perspective oftax revenue maximization.
It is well-known from the optimal income tax literature that replicating the directly
progressive tax schedules that exist in most countries through a two-tier income tax
with two endogenously chosen tax rates is extremely diﬃcult, even if redistributive
motives between individuals are explicitly accounted for.22 In the following, we therefore
follow a two-step procedure. In the ﬁrst step we provide analytical results for the casewhere the general income tax rate τi is exogenously ﬁxed, and only the bonus tax ti is
endogenized. We derive the optimal bonus tax in this extended setting, and ask how
it is aﬀected by overconﬁdence for any given level of the lower-bracket tax rate τi . Ina second step we treat both tax rates τi and ti as endogenous and solve the model
numerically for this case.
With two-tier taxation of the manager’s remuneration, the after-tax proﬁts of the ﬁrm
22

Slemrod et al. (1994) have shown that no directly progressive income tax schedule will result

in settings where standard assumptions are made with respect to social welfare functions and the
distribution of abilities. Conversely, Apps et al. (2014) study the conditions under which two-tierincome tax systems are directly progressive, but these schemes are not easily tractable analytically.

23[eq. (10)] change to


h
l∗ l
Π̃i = Ñi p̃h∗i [Y − z̃i (1 + t̃i + τi )] + p̃i Y − w̃i (1 + τi ) ,(35)

where the tilde symbol (˜) is used for endogenous variables in this extension.
Maximizing (35) with respect to the ﬁrm’s compensation variables wi and zi gives
π̃ D
∂ Π̃i
= i − Ñi (1 + τi ) = 0,∂ w̃iaz̃i∗ =

Ω(1 − θ)(1 + τi ) + 2t̃i

∀ i,

(36)

where the second expression in (36) assumes again that the ﬁxed wage is positive inequilibrium. In comparison to the benchmark case [eqs. (12) and (13)], the equilibrium
level of division proﬁts π̃iD is now larger, because the ﬁxed wage is taxed in equilibrium.
Also, the larger denominator in the expression for z̃i∗ reﬂects the additional taxation of
the bonus by the general income tax τi . The optimal ﬁxed wage in this setting is given
in Appendix C.1 [eq. (C.3)].
Tax revenue for government i is now
W̃i = Ñi [p̃h∗i (t̃i + τi )z̃i + τi w̃i ]

∀ i.

(37)

Appendix C.2 derives the optimal bonus tax t̃N,Rwhen the income tax rate τi is exi
ogenous. This gives:
=
t̃N,R
i

B+

√

B 2 − AC
,
A

(38)




A = 8 3 + τi 7 + 4τi + Yal > 0,
B = −4τi (1 − θ)(1 + τi ) 4(1 + τi ) + Yal − (1 + θ)(1 + 2τi )γΩ2a < 0,


C = 2(1 + τi )(1 − θ) (1 − θ)(1 + τi )(Yal τi + 4τi2 + τi − 3) + τi (1 + θ)γΩ2a ,where Yal ≡ Y l /a and Ω2a ≡ Ω2 /a.
We can infer from the structure of (38), together with A > 0 and B < 0 for all θ ∈ (0, 1),that the bonus tax will be positive if and only if the term C is negative. In this case it
will thus be optimal to tax bonus income at a higher rate than income from the ﬁxed
wage. Inspection of the term C shows that this will unambiguously be the case whenτi = 0, in which case we return to our benchmark setting.23 As τi is continuously raised,
this tends to reduce the optimal bonus tax t̃N,R. Therefore, t̃N,Rwill be lower than theii23This can easily be checked by setting τi = 0 in expressions A–C. In this case the optimal tax

in (38) collapses to eq. (21) in Section 3.

24

corresponding bonus tax rate in the benchmark case, tN,R
, for all τi > 0 and θ ∈ [0, 1].
i
At some point the general income tax rate will reach a critical level, denoted by τicrit ,at which the term C turns positive and hence t̃N,R< 0. Our following analysis focusesion the parameter range τi ∈ [0, τicrit ] for which the bonus surtax t̃N,R
is positive, andigenerates additional tax revenue.
Our main interest is in the question of how an increase in the overconﬁdence level ofmanagers aﬀects the optimal bonus tax under this extension. This is summarized in:
Proposition 3 When all incomes are taxed at an exogenous rate τi ∈ [0, τicrit ], and the

optimal bonus tax rate t̃N,R
is positive in the symmetric, non-cooperative equilibrium,ithen increased overconﬁdence of mobile managers (a rise in θ) leads to:(i) higher bonus payments and lower ﬁxed wages by ﬁrms;(ii) managers responding more elastically to bonus taxation;(iii) lower optimal bonus taxes t̃N,Rin both countries.
i
Proof: See Appendix C.3.

Proposition 3 states that our results for revenue-maximizing governments (Proposition 1) carry over to an extended setting with a positive general rate of income taxation, whenever the endogenous bonus surtax is positive in the optimum. In this casethe intuition from our benchmark model carries over as well. The higher bonus payments by ﬁrms will increase the bonus tax base. However, the higher overconﬁdenceof managers simultaneously increases their migration response, making the bonus tax
base respond more elastically to a higher bonus tax rate. In equilibrium, the eﬀect of
the higher tax base elasticity dominates, and the optimal bonus tax must fall. To put it
diﬀerently, a directly progressive income tax system will become less progressive in thegovernment’s optimum, when the overconﬁdence level of mobile managers increases.
In the next step, we endogenize the general income tax rate τ . In our setting, theanalysis of tax competition with two endogenous variables is too complex, however, toyield tractable analytical results. Therefore we solve the model numerically. Figure 2
summarizes our results for diﬀerent values of the parameter a, which inversely measures
the international mobility of managers.24*******************************************
24

The numerical analysis in Figure 2 is based on the following parameter values: Y l = 0.75, Ω = 0.5,

γ = 0.5 (which diﬀer from the values used in Figure 1).

25

********* Figure 2 about here ****************************************************
is higher, for any given level of θ, in the low-mobility
In Figure 2, the bonus tax rate t̃N,Ricase shown in panel (a), as compared to the intermediate mobility case in panel (b). This
corresponds to our results for the benchmark model [eq. (22)]. However, the qualitative
response of both taxes to changes in the overconﬁdence parameter θ is the same inboth panels. The bonus tax rate (upper-tier tax rate) is monotonously falling in θ, asin Propositions 1 and 3. The new insight from the numerical analysis is that the generalincome tax rate (lower-tier tax rate) is instead rising in θ. The reason for this is seenfrom the ﬁrm’s optimal compensation structure in (36): a higher overconﬁdence level θ
reduces the eﬀect that a higher income tax rate τi has on equilibrium bonus paymentsz̃i∗ , and hence on the bonus tax base. This in turn results from the fact that a higher
income tax must increase division proﬁts in equilibrium, from the ﬁrm’s ﬁrst-ordercondition for the ﬁxed wage [the ﬁrst expression in (36)].
In both panels of Figure 2 it is also true that the total tax rate on managers’ bonus
income, (τi + ti ), is falling in the overconﬁdence level θ. Hence the downward eﬀect
that overconﬁdence has on the isolated bonus tax rate ti dominates the upward eﬀect
on the general wage tax τi . This implies that the negative eﬀect of overconﬁdence that
we derived in our benchmark analysis for an isolated tax on bonus income carries over
to a two-tier income tax system in which the ﬁxed wage is also taxed optimally.25
Finally, there is a third case, not shown in Figure 2, in which the mobility of managers
is high (a = 0.4). In this case the bonus tax rate ti is negative for higher levels of θ andthe overall tax schedule thus turns regressive. This case is familiar from the analysisof two-tier taxes (Slemrod et al., 1994) and from the non-linear income taxation formobile top earners (Simula and Trannoy, 2010; Lehmann et al., 2014). In our setting,the negative eﬀect of managers’ international mobility on the bonus tax rate [eq. (22)]becomes so strong that governments choose a bonus subsidy and thus a regressive tax
structure. In this case, higher levels of overconﬁdence will reduce the equilibrium bonus
25

Note that the total taxation of bonus income (ti + τi ) in panel (b) exceeds that in panel (a) for

all given levels of θ. This result is surprising at ﬁrst glance, as the international mobility of managers
is higher in case (b). The reason is that higher manager mobility (a fall on a) induces the ﬁrms toincrease total manager compensation by increasing both the bonus and the ﬁxed wage payment. Thisis seen from the fact that the ﬁrm’s division proﬁts fall in equilibrium when the parameter a is reduced[the ﬁrst expression in (36)]. For the government this implies that the total tax base is increased, andthis eﬀect dominates the tighter tax competition between governments.

26

subsidy, and thus increase ti . Intuitively, overconﬁdent managers now overestimate theexpected bonus subsidy that they receive in a given location, and this misperception
makes them less mobile internationally.
To summarize, our analysis in this section has shown that the negative eﬀect of overconﬁdence on the bonus tax rate ti carries over to a two-tier tax system, whenever theupper-tier bonus tax rate is positive, and the tax system is therefore progressive. This
result can be established theoretically for the case where the general income tax rateis held ﬁxed (Proposition 3), and it carries over to our numerical analyses, where theentire tax structure is optimized.

6

Conclusion

Our analysis has shown that overconﬁdence as a behavioral trait of high-skilled managers can contribute to explaining several important developments that have been
observed over the past decades. On the one hand, it oﬀers an additional rationalization for the increased use of incentive pay contracts in most OECD countries. Higher
shares of incentive pay are in turn an obvious driver of wage income inequality. On
the other hand, overconﬁdence in connection with incentive pay provides a mechanismthat increases the migration elasticity of the mobile highly skilled. This adds to the
explanation for the rather high migration elasticities that have been found empiricallyfor this segment of the labor market (Kleven et al., 2020). When governments focus
on tax revenue maximization, the higher migration elasticities in turn oﬀer a rationale
for the fall in top income tax rates that has been observed since the mid-1990s (Egger
et al., 2019). A counteracting eﬀect arises, however, when governments incorporate thetrue utility of overconﬁdent managers in their objective function and use bonus taxes
to correct the managers’ distorted incentives.
Our theoretical results have been derived in a simpliﬁed setting where general wage
income remained either untaxed, or it was taxed at an exogenously given tax rate. A
ﬁrst extension of our analysis would therefore be to study the eﬀects of (potentiallyheterogeneous) overconﬁdence under a more general system of optimal non-linear income taxation. A further extension would be to study optimal contracts by ﬁrms when
only some part of their managers are overconﬁdent, and this is private information. We
leave these and other extensions to future research.

27

Appendix
A. Appendix to Section 3
A.1 Derivation of equation (14)Firm i chooses the bonus zi and the ﬁxed wage wi , which depends on zj and wj . Hencethe system of ﬁrst-order conditions in (12) is interdependent, and given by∂Πi
1  h∗ hl=pi [Y − zi (1 + ti )] + pl∗
i Y − wi
∂wi
a

1 γ
(1 + θ)2 (zi2 − zj2 ) + wi − wj = 0,
−1−
a 2

∂Πj
1  h∗ hl=
pj [Y − zj (1 + tj )] + pl∗
i Y − wj
∂wj
a

1 γ
(1 + θ)2 (zj2 − zi2 ) + wj − wi = 0.
−1−
a 2

(A.1)

(A.2)

Substituting in the equilibrium bonuses zi and zj from (13) and the equilibrium probabilities from (5a)–(5b) yields
⎞
⎛
γ
2 2
2
l

(Ω
−
z
(1
+
t
))
+
Y
−
a
−
(z
−
z
)
(1
+
θ)
ph∗
i
i
i
j
2 1 ⎜ i
⎟
2
wi =
⎠
⎝
γ
2 2
2
l
3 3
(1
+
θ)
(Ω
−
z
(1
+
t
))
+
Y
−
a
−
(z
−
z
)
ph∗
j
j
j
i
j
 2

2
2
1 γ(1 + θ)Ω
1 + tj2 γ(1 + θ)Ω
1 + til= Y −a+1−+1−
3 (1 + 2tj − θ)
1 + 2tj − θ
3 (1 + 2ti − θ)
1 + 2ti − θ1γ(1 + θ)2 Ω2
1
−
+
.
(A.3)
2
6
(1 + 2tj − θ)
(1 + 2ti − θ)2
Simplifying (A.3) leads to eq. (14) in the main text.
A.2 Second-order condition for optimal bonus taxes
Diﬀerentiating the ﬁrst-order condition for bonus taxes (20) with respect to ti gives
4ti∂ 2 Riti γ(1 + θ)Ω2
∂zi2 γ(1 + θ)−aNi 1 −
=
∂t2i
∂ti
a
1 − θ + 2ti
3(1 − θ + 2ti )2γ(1 + θ)Ω2
4
4(1 − θ)z 2 γ(1 + θ)−a+ i
1 − ti− aNi
2a3a(1 − θ + 2ti )1 − θ + 2ti(1 − θ + 2ti )2γ(1 + θ)Ω2
4ti γ(1 + θ)Ω2 
.
(A.4)
−
+
3(1 − θ + 2ti )2 3(1 − θ + 2ti )328

At ti = tN,R
[eq. (21)], the ﬁrst term on the right-hand side is zero. The remainingiterms are rearranged to yield:
4(1 − θ)zi2 γ(1 + θ) 
Ω2 (−1 + 2ti + θ)2γ(1 + θ) 
∂ 2 Ri
−
aN
.
=
+
i
∂t2i
a
(1 − θ + 2ti )23(1 − θ + 2ti )3(A.5)

A suﬃcient condition for (A.5) to be negative is −1 + θ + 2ti < 0, which is true since(1 − θ)/2 = tC,R
> tN,R. Hence the second-order condition for an optimum is fulﬁlled.
ii.
A.3 Derivation of eq. (25)
All derivatives include ∂ti /∂θ. Evaluating this at the equilibrium tax rate ti = tN,Riyields−ti [2(1 + θ)(1 − θ) + (1 − θ)2 − 4t2i ]∂ti=.
∂θ ti =tN,R(1 + θ)[(1 − θ)2 + 4t2i ]

(A.6)

i

2
From (5a) and (13), tax revenue is Ri∗ = ph∗i ti zi Ni = ti (1 + θ)γzi =ti (1 + θ) γΩ2
Ni .
(1 − θ + 2 ti )2

Diﬀerentiating and dividing by Ri∗ gives
3 + θ + 2ti
1 − θ − 2 ti ∂tN,R
1 ∂Ri∗
i=+.
Ri∗ ∂θ
(1 + θ)(1 − θ + 2ti ) ti (1 − θ + 2ti ) ∂θ

(A.7)

Substituting (A.6) in (A.7) and simplifying terms gives
1 ∂Ri∗
8ti
=
> 0.
∗
Ri ∂θ
(1 + θ)[(1 − θ)2 + 4t2i ](A.8)

Multiplying (A.8) by Ri∗ gives (25) in the main text.
A.4 Loss in the low state caused by overconﬁdence
We extend our benchmark model by assuming that, in the low state, the return to theﬁrm is given by Y l − κθ, where κ ≥ 0. The ﬁrm’s after-tax expected proﬁts are then


h
l∗
l
Πi = Ni πiD = Ni ph∗
i [Y − zi (1 + ti )] + pi (Y − κθ) − wi .

(A.9)

Diﬀerentiating with respect to zi and assuming that (12) holds with equality yields
zi∗ =

Ω̂
1 − θ + 2tiΩ̂ ≡ Y h − (Y l − κθ),

∀ i,

29

(A.10)

which is higher than the optimal bonus in the benchmark model [eq. (13)]. Hence part(i) of Proposition 1 is strengthened by this extension.
The structure of the optimal tax rate in (21) is unchanged by this extension, but Ω isreplaced by Ω̂ in (A.10). Therefore, the derivative with respect to θ changes to
⎧
⎫
N,R
⎨
2 3
2⎬
γ Ω̂ (1 + θ)[Ω̂ + 2(1 + θ)κ] − 144(1 − θ)a∂ti
1$−Ω̂γ[Ω̂ + 2(1 + θ)κ] +
< 0.
=
⎭
∂θ
24a ⎩
2
4
2
2
2
γ Ω̂ (1 + θ) + 144(1 − θ) a
(A.11)The sum of the terms added by the parameter κ in (A.11) is negative. Hence, a given
increase in overconﬁdence lowers the equilibrium tax rate by more when overconﬁdence
causes a loss to the ﬁrm in the low state. Hence Proposition 1(ii) is also strengthened
when overconﬁdence causes a loss in the low state.
Finally, we turn to tax revenue collections. With losses in the low state, we get∂ R̂i∗∂Ri∗ ∂Ri∗
κ
≡
+
∂θ
∂θ
∂ Ω̂

(A.12)

where the ﬁrst term on the right-hand side is positive from (A.8). Evaluating the second
term on the RHS of (A.12), accounting for the endogeneity of tN,R
with respect to Ω̂i
and introducing δ ≡ γ/(a) gives:




1
∂Ri∗
∂
ti Ω̂2∂ti Ω̂2∂ti=|ti =t̄i +2
2(1 + θ)γ ∂ Ω̂∂ti (1 − θ + 2 ti )∂ Ω̂ (1 − θ + 2 ti )∂ Ω̂2κti (κ θ + Ω̂) 2κ(κ θ + Ω̂)3 (1 − θ − 2 ti ) δ 2 (κ θ + Ω̂)2 (1 + θ)2=+− δ(1 + θ) ,(1 − θ + 2 ti )2χ
24 (1 − θ + 2 ti )3(
where χ ≡)*4
δ 2 κ θ + Ω̂ (1 + θ)2 + 144 (θ − 1)2 .

Dividing this expression by 2κ(κθ + Ω̂)/(24(1 − θ + 2ti )2 ) gives:∂Ri∗ 24(1 − θ + 2ti )2

1(1 + θ)γ ∂ Ω̂2κ(κθ + Ω̂)

= −

)
*2
δ kθ + Ω̂ (1 + θ) (2 − 2 θ)
⎡

+

1 ⎢δ
⎣
χ

2

)

1 − θ + 2 ti
*4κ θ + Ω̂ (1 + θ)2 (2 − 2 θ)
1 − θ + 2 ti⎤⎥
+ 144 (θ − 1)2 ⎦.

Multiplying the last formula by χ and noting that −δ(κθ + Ω̂)2 (1 + θ) + χ = 24ti , we30

ﬁnally obtain:∂Ri∗ 24(1 − θ + 2ti )2

1(1 + θ)γ ∂ Ω̂2κ(κθ + Ω̂)

χ = −

)
*2
δ κ θ + Ω̂ (1 + θ) (2 − 2 θ) 24ti

1 − θ + 2 ti

2
)
*2
+ 24 ti + δ κ θ + Ω̂ (1 + θ) − δ 2 (κθ + Ω̂)4 (1 + θ)2= (24ti )2 +96t2i δ(κθ + Ω̂)2 (1 + θ)
> 0.
1 − θ + 2 ti

(A.13)

Hence Proposition 1(iii) also carries over to the case where overconﬁdence leads to an
extra loss κ in the low state.

B. Appendix to Section 4
B.1 Second-order condition for optimal bonus taxes
Diﬀerentiating the ﬁrst-order condition for bonus taxes (33) with respect to ti gives
1 − θ − 2ti∂ 2 Wi1 ∂zi2ti γ(1 + θ)Ω2
2λ (7θ − 1 − 2ti )=γ(1 + θ)−+∂t2i
1 + λ ∂ti
1 − θ + 2ti 3a(1 − θ + 2ti )2
3 (1 − θ + 2ti )
8λθ
1 2
−
zi γ(1 + θ)
1+λ(1 − θ + 2ti )24(1 − θ)1 zi2 γ(1 + θ) 
Ω2 (1 − θ − 2ti )2γ(1 + θ) 
− aNi
+
−
1+λ
a
(1 − θ + 2ti )23(1 − θ + 2ti )3At ti = tN,W
, the ﬁrst term on the right-hand side is zero. The second term is negative.
iNote that 1−θ+2tN,W> 0, since tN,R
> 0 and tN,u +(1−θ)/2 = (7θ−1)/2+(1−θ)/2 >ii
0. Finally, the third term on the right-hand side is negative when (1 − θ)/2 ≥ tN,W
, ori

, tN,u }. From Section 3, we know that (1− θ)/2 =from (33), when (1−θ)/2 ≥ max{tN,RitC,R> tN,R. Also, tN,u − (1 − θ)/2 = (7θ − 1)/2 − (1 − θ)/2 = 4θ − 1. Therefore, aiisuﬃcient (but not a necessary) condition for the second-order condition to be fulﬁlled
is that θ ≤ 0.25.
B.2 Proof of Proposition 2
We diﬀerentiate (33) with respect to θ to get


14λ
∂tN,W
∂ 2 Wiγ(1 + θ)Ω2
2ti (1 + ti )γΩ2isign
. (B.1)= sign∝−1−∂θ∂ti ∂θ
(1 + λ)(1 − θ + 2ti )2 33a(1 − θ + 2ti )231

Since the third term in the squared bracket is unambiguously negative, a suﬃcient (butnot a necessary) condition for the sum of all terms to be negative is that λ < 3/14.
Part (i) of Proposition 2 follows immediately from (B.1). For part (ii), diﬀerentiating
Ri∗ with respect to θ gives
3 + θ + 2ti(1 − θ − 2 ti ) ∂tN,W
1 ∂Ri∗
i
=
+
.
∗
Ri ∂θ
(1 + θ)(1 − θ + 2ti ) ti (1 − θ + 2ti ) ∂θ

(B.2)

where the only diﬀerence to (A.7) lies in the diﬀerent equilibrium tax rate. Evaluating (33) at ti = 0 gives ∂Wi /∂ti |ti =0 > 0 for all λ > 0 and θ ≥ 1/7. For θ < 1/7,
∂Wi /∂ti |ti =0 > 0 follows if λ ≤ 1.5. Therefore, tN,W
> 0 must hold when λ ≤ 1.5.
iWhen λ < 3/14 we have 0 > ∂tN,W/∂θ > ∂tN,R/∂θ. Therefore, if (1 − θ − 2ti ) < 0 inii

the second term of (A.7), the second term is positive and ∂Ri∗ /∂θ > 0 is unambiguous.

If (1 − θ − 2ti ) > 0 a suﬃcient condition for ∂Ri∗ /∂θ to be larger under tN,W
than underitN,Ris that tN,W
≈ tN,R. But this is met when λ is suﬃciently low. Since ∂Ri∗ /∂θ > 0iii
holds for tN,R
from (A.8), it must then also hold for tN,W.ii

C. Appendix to Section 5
C.1 Derivation of the ﬁxed wageDiﬀerentiating (35) with respect to ﬁxed wages gives1  h∗ h∂ Π̃il=p̃i [Y − z̃i (1 + t̃i + τi )] + p̃l∗i Y − w̃i (1 + τi )
∂ w̃i
a

1 γ
2 2
2
− (1 + τi ) −(1 + θ) (z̃i − z̃j ) + w̃i − w̃j (1 + τi ) = 0,
a 2

∂ Π̃j
1  h∗ hl=p̃j [Y − z̃j (1 + t̃j + τj )] + p̃l∗i Y − w̃j (1 + τj )
∂ w̃j
a

1 γ
− (1 + τj ) −(1 + θ)2 (z̃j2 − z̃i2 ) + w̃j − w̃i (1 + τj ) = 0,
a 2

(C.1)

(C.2)

where the only diﬀerence to (A.1)–(A.2) lies in the tax factors (1 + τi ) and (1 + τj ).
Substituting in equilibrium bonuses from (36) and equilibrium probabilities from (5a)–(5b) and simplifying leads to


2
1
∗
w̃i =
(C.3)
+
Yl−a
3(1 + τi ) 3(1 + τj )γ(1 + θ)Ω2
1
4t̃i − 5θ(1 + τi ) − (1 + τi )+.
+6
(1 + τj )[(1 − θ)(1 + τj ) + 2t̃j ] (1 + τi )((1 − θ)(1 + τi ) + 2t̃i )2
32

C.2 Derivation of optimal bonus tax
Diﬀerentiating (9) with respect to t̃i , taking account of (36) and (C.3), leads to−γ(1 + θ)Ω2
1 γ(1 + θ)2 ∂(z̃i2 ) ∂ w̃i ∂ w̃j
∂ Ñi
=
=
+
−
< 0.
a
2
∂ t̃i
∂ t̃i
∂ t̃i
∂ t̃i
3a(1 + τi )[(1 − θ)(1 + τi ) + 2t̃i ]2(C.4)
Diﬀerentiating (37) with respect to t̃i , using (C.4) and (C.3), gives∂ W̃i (t̃i , t̃j )4(1 + θ)γ z̃i2 (t̃i + τi )4τi γ(1 + θ)Ω2 (2τi (1 + θ) + 1 + τi − t̃i )2= Ñi z̃i γ(1 + θ) −+∂ t̃i
(1 − θ)(1 + τi ) + 2t̃i
3(1 + τi )[(1 − θ)(1 + τi ) + 2t̃i ]3γ(1 + θ)Ω2 [(1 + θ)γ z̃i2 (t̃i + τi ) + w̃i τi ]−= 0 ∀ i = j.
(C.5)
3a(1 + τi )[(1 − θ)(1 + τi ) + 2t̃i ]2
We evaluate the above formula at the symmetric equilibrium where t̃N,R= t̃N,Randijτi = τj . The ﬁxed wage (C.3) is thenw̃i∗ =Ylγ(1 + θ)Ω2 [t̃i − (1 + τi )θ]−a+.
1 + τi(1 + τi )[(1 − θ)(1 + τi ) + 2t̃i ]2(C.6)

Using this in (C.5) and dividing by γ(1 + θ)Ω2 gives
1∂ W̃i (t̃i , t̃j )(1 − θ)(1 + τi ) − 2t̃i − 4τi4τi [2θ(1 + τi ) + 1 + τi − t̃i ]=+23γ(1 + θ)Ω∂ t̃i[(1 − θ)(1 + τi ) + 2t̃i ]3(1 + τi )[(1 − θ)(1 + τi ) + 2t̃i ]3



Yl
(1 + τi )(t̃i + τi ) + τi [t̃i − (1 + τi )θ] τi [ 1+τi − a]γ(1 + θ)Ω2
= 0.
−+
γ(1 + θ)Ω2
(1 + τi )[(1 − θ)(1 + τi ) + 2t̃i ]23a(1 + τi )[(1 − θ)(1 + τi ) + 2t̃i ]2(C.7)
Multiplying eq. (C.7) by 3[(1 − θ)(1 + τi ) + 2t̃i ]4 (1 + τi )2 and rearranging terms yields
a quadratic equation for t̃N,Rgiven byiA
C
− t̃2i + B t̃i − = 0,
2
2

(C.8)

where the terms A − C are in eq. (38) in the main text. The solution to (C.8) is thengiven in (38).
C.3 Proof of Proposition 3
Denote B = B(θ) and C = C(θ) as functions of θ. The optimal bonus tax t̃N,Rin (38)iis decreasing in θ iﬀ:
2B(θ)B  (θ) − AC  (θ)
B  (θ) + < 0.
2 (B(θ))2 − AC(θ)
33

(C.9)

For C(θ), the following can be readily conﬁrmed:
C  (θ)1−θ
= −C(θ) + 2τi (1 + τi )(1 − θ)γΩ2a > 0.
2

(C.10)

Therefore, if B  (θ) < 0 (which holds, for example, if τi = 0), then condition (C.9) is
fulﬁlled immediately.


If B  (θ) > 0, condition (C.9) holds iﬀ B  (θ)[ (B(θ))2 − AC(θ) + B(θ)] < AC  (θ)/2.
Using (C.10) and rearranging, this is equivalent to:A−(B (θ)) 2τi (1 + τi )(1 − θ)γΩ2a < (C  (θ))2 − C  (θ)B  (θ)


2

4



1−θ 
B(θ) +
B (θ) (C.11)
2

Since B(θ) is a linear function of θ, we have B(θ) = (1 − θ)B(0) + θB(1) and B  (θ) =B(1) − B(0). Hence
B(θ) +

1−θ
1−θ 
1+θ
B (θ) =
B(0) +
B(1) < 0 for all θ ∈ (0, 1),2
2
2

since B(0) < 0 and B(1) < 0. Therefore, the LHS of (C.11) is negative and the RHS/∂θ < 0of (C.11) is positive. Hence, condition (C.9) also holds for B  (θ) > 0 and ∂ t̃N,Ri
is true for all levels of B(θ).
The equilibrium bonus payment is immediately seen to rise in θ from eq. (36). Also,
evaluating the ﬁxed wage expression (C.3) at the common Nash equilibrium tax ratein (38) givest̃N,Ri∂ w̃i∗
∂θ

= γΩ2



∂ t̃N,Ri(2t̃i − 3θ − 1 − 3τi θ − τi )1 + t̃i + τi − (1 + θ)∂θ((1 − θ)(1 + τi ) + 2t̃i )3,(C.12)

which is negative since t̃N,R(τi , θ) ≤ t̃N,R(0, θ) = tN,R
(θ). Finally, (C.4) shows that theiiioutﬂow of managers in response to a higher bonus tax rises in θ. 34

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39

0.5

0.5
tC,W
(θ)
i
tN,W
(θ)
i

0.4

0.3

tN,W
, tC,WiitN,W, tC,W
i
i

0.4

0.2

0.1

0

tC,W
(θ)
i
tN,W
(θ)
i

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0

0.5

0

0.1

0.2

θ

0.3

0.4

0.5

0.3

0.4

0.5

0.3

0.4

0.5

θ

0.5

0.5
Ri (tC,W)i
Ri (tN,W)i

0.4

Ri (tC,W)i
Ri (tN,W
)
i

0.4

Ri

0.3

Ri

0.3

0.2

0.2

0.1

0.1

0

0

0.1

0.2

0.3

0.4

0

0.5

0

0.1

0.2

θ

θ

0.7

1.1
Wi (tC,W)i
Wi (tN,W)i

0.6

Wi (tC,W)i
Wi (tN,W
)
i

1

Wi∗

0.9

Wi∗

0.5

0.4

0.8

0.3

0.7

0.2

0

0.1

0.2

0.3

0.4

0.6

0.5

θ

0

0.1

0.2
θ

(a) Low λ case (λ = 0.1)(b) High λ case (λ = 0.5)

Figure 1: Changes in θ under cooperative and non-cooperative tax policies

40

t∗i (θ)τi∗ (θ)(t∗i + τi∗ )(θ)

0.5

0.5

0.4
t∗i , τi∗t∗i , τi∗

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

t∗i (θ)τi∗ (θ)(t∗i + τi∗ )(θ)

0

0.1

0.2

0.3

0.4

0

0.5

θ

0

0.1

0.2

0.3

0.4

θ

(a) Low mobility (a = 0.6)(b) Intermediate mobility (a = 0.5)

Figure 2: Tax eﬀects of changes in θ under two-tier income taxation

41

0.5