A Theory of Dynastic Cycle
Yinan(Leo) Liy
October 31, 2009
Abstract
This paper proposes a dynamic politico-economic theory on dynastic
cycle. I characterize the Markov Perfect Equilibrium of the dynamic game
and derive the analytical solution to the equilibrium. The main conclusion is that the demise of any dictatorial regime is inevitable if there are
discontinuity of power caused by dictator’s physical death and the delegation of the dictator’s unbalanced power, which are two common properties
shared by all dictatorial regimes. Consistent with historical evidence, the
model shows the overall pattern of the evolution of dictatorial regime is
increasing real burden on the citizen caused by increasing bureaucrats’
tax surcharge due to weakering dictator, and the decreasing …scal revenue
of the dictator due to the decreasing of tax base, as will cause the demise
of dictatorship in the long run.
1
Introduction
At least till now, human beings are ruled by dictators in most time of history.
However, compared to the advanced economic literature on political economy of
democracies (reviewed in Persson and Tabellini 2000), not much attention is paid
to dictatorship. Among the small and growing economic literature in this …eld,
most papers (reviewed in Acemoglu and Robinson 2006) treat dictatorship as a
form of governance by unitary ruling elites, whose only constraint of predation,
which often binds due to rent maximization, comes from the citizens’revolution.
I thank Professor John Hassler for his excenllent guidance and encouragement in this
project. I also thank Professor Hans Wijkander and Professor Magnus Henrekson for their
help and support at di¤erent stages of this project. Financial suport from Finanspolitiska
Forskningsinstitutet is greatly acknowledged. All errors are mine.
y Depertment of Economics, Stockholm University, Stockholm, Se10691, Sweden. Email:
[email protected].
1
Models with this simpli…cation of the internal structure of dictatorship have
particularly deep insights in democratization (e.g. Acemoglu and Robinson
2000), as the main con‡ict of the society is between the ruling elites and the
citizens during the time around the institutional changes. However, historical
evidence shows dictators have much more threats from inside the ruling elites
than from public uprisings1 . Therefore, there are two potential questions: (i)
What are the constraints that a dictator faces from inside the ruling elites? (ii)
How the economy evolves before public uprising?
To me, the above two questions can not be answered without exploring the
internal organization of dictatorship, because …rstly, policy distortions in such
regimes are often re‡ections and outcomes of the interest con‡ict among the
ruling elites, and secondly, these distortions can a¤ect the evolution of such
regime that leads to public upsings …nally. In this paper, I develop a tractable
and positive theory of the evolution of dictatorship— –“the dynastic cycle characterized by peace and prosperity in the upswing when a new line of emperors
is established, and by civil war, misery, and population decline in the downswing when the dynasty becomes old and feeble”2 — –by relaxing the citizens’
non-revolution constraint and focusing instead on the interest con‡ict inside the
ruling elites.
The model economy is populated by four kinds of two-period-lived overlapping generations of sel…sh and rational agents: the citizens, the dictator, the
dictator’s successor candidates and the bureaucrats. All the citizens undertake
a costly investment at birth and yield the returns in each living period. The
incumbent dictator, who is among the successor candidates at birth, is designated as the successor in his …rst period of life by the previous dictator and
is supposed to be the ruler in his second period of life. Once taking power,
the dictator sets an age-independent tax rate before young citizens make the
investment decision to maximize total tax revenue from the young and the old
1 Svolik
(2008) shows an overwhelming majority of authoritarian leaders lose power as a
result of a successful coup rather than a popular uprising.
2 This de…nition of dynastic cycle is borrowed from Usher(1989).
2
citizens. The intergenerational con‡ict between the incumbent dictator and his
successor plays out as follows. If the dictator designates a successor when alive,
this reduces the dictator’s safety, since the successor always has an incentive to
take the place of the incumbent to get the dictator’s rent earlier. Moreover, as
the result of power struggle between the incumbent and the successor is probabilistic, depending on their relative strengths, the stronger the successor, the
less safe the incumbent will be.
However, the incumbent can not simply choose the weakest successor, as the
functioning of dictatorial regime depends a lot on the quality of the dictator3 .
No matter how strong the dictator is, he has to rely on some agents to carry out
his policies. This is modelled as some bureaucrats collecting the tax from the
citizens for the dictator. The asymmetric information between the dictator and
bureaucrats create the possibility for corruption. Moreover, in dictatorship, the
dictator delegates his power to the bureaucrats. There can not be any source
of independent check and balance of the bureaucrats’power4 since this means
the erosion of the dictator’s power. The unbalanced power plus the asymmetric
information between the dictator and the bureaucrats makes corruption hardly
be eradicated5 . In the model economy, bureaucratic corruption is modelled
as the surcharge of tax by the bureaucrats. That is, a bureaucrat can say a
citizen, who actually has paid the tax, has not paid; or a bureaucrat can say
a citizen, who actually has not paid the tax, has paid. In equilibrium, the
bureaucrats can charge more than the tax rate announced by the dictator6 .
Since the bureaucrats’surcharge distorts the citizens’investment decision and
decreases the tax base of the dictator, it is not in the interest of the dictator. The
size of the bureaucrats’surcharge depends on the dictator’s ability in regulating
3 Jones
and Olken (2005) shows the e¤ects of individual leaders on growth are strongest in
autocracy. See also Fisman (2001) for an interesting study about stock market reaction to
rumors about dictator’s health.
4 See Persson, Roland and Tabellini(1997) for the importance of separation, check and
balance of power in democracy.
5 See Shleifer and Vishny(1993), Svensson(2005) and Yi(2007) for a detailed discussion.
6 See Acemoglu and Verdier (2000) for a microeconomic study about equibibrium surcharge.
3
the bureaucrats, which is positively correlated to the ability to …ght in the
power struggle with the successor (dictator), since these two abilities are both
the re‡ections of the leader’s political skills.
Given the model’s setup, the incumbent dictator has a trade-o¤ between his
safety and the tax base. If the successor is too strong, then although the tax
base will be larger, as forward-looking young citizens will make more investment
because bureaucrat’s surcharge will be lower in the next period, the incumbent
will be in danger because he is more likely to be replaced by the successor in his
ruling period. I call this safety e¤ ect; if the successor is too weak, then although
the incumbent will be safe, the tax base will be smaller, because forward-looking
young citizens will make less investment as bureaucrat’s surcharge will be higher
in the next period. I call this second e¤ect as tax base e¤ ect. With two opposing
e¤ects, the incumbent will not tend to choose the strongest successor. Under
the reasonable assumption that all the dictators put primary concern on their
own safety rather than the tax base, the strengths of the dictator will become
lower and lower within one dictatorial dynasty and thus, bureaucrats’surcharge
will become higher and higher.
There are two sources of dynamic ine¢ ciencies in the model that have implications on both the short run and the long run evolution of dictatorial regime,
respectively:
In the short run, as the incumbent sets the age-independent tax rate when
the old citizens’investment is sunk, the tax rate set by the dictator will increase
with the sunk investment. This will not only discourage the young citizens’
investment, but also generate an oscillatory pattern on the equilibrium law of
motion of tax rate between generations. The intuition is as following. If the last
period tax rate is relatively high, then seen at the current period, the investment made by the old citizens will be relatively low. Facing such a situation,
the incumbent will set a relatively low tax rate to encourage the young citizens’
investment in order to increase the tax base. While if the last period tax rate is
relatively low, then seen at the current period, the investment made by the old
citizens will be relatively high. Facing such a situation, the incumbent will set
4
a relatively high tax rate to maximize the tax revenue, although this relatively
high tax rate will reduce the young citizens’ investment. This oscillatory pattern has three important implications: (i) Growth-enhancing economic reforms
in dictatorial regime will probably to be reversed with the change of the ruler,
if there is no institutional reform that balances the power of the ruler, because
without institutional reform, the power to set the policies stays on the dictator,
and as the tax base becomes larger due to the growth-enhancing economic reforms, the new dictator has an incentive to tax heavily on the sunk investment.
This will reverse the growth-enhancing economic reform; (ii) Bureaucratic corruption and economic growth can be positively correlated in dictatorial regime.
The intuition is as following. When the tax base is low due to less sunk investment, the dictator has an incentive to lower the tax rate, which is growthenhancing to increase the tax base. However, the lower tax rate itself can not
put any constraint on bureaucratic corruption. On the contrary, this increases
the rent base of the bureaucrats to get corrupt income. Thus, bureaucratic and
growth can be positively correlated. This explains the high corruption and high
growth puzzle in east Asia after Second World War after which not much capital
is left. (iii) As the oscillatory tax rates between generations can be seen as the
variations of economic policies that are growth-enhancing or growth-retarding
and can be controlled by dictators, it is wrong to use variables that re‡ect economic institutions as an indicator of political institutions in empirical analysis7 .
In the long run, my model predicts dictatorial government’s revenue will
become lower and lower, as a result of endogenous deterioration of state capacity
caused by weakering dictators8 . The intuition is as following. Other things
given, the weaker the dictator, the worse is he in controlling his bureaucrats
and the higher the bureaucrats’ surcharge will be. This will increase real tax
rate that the citizens face and shift the La¤er curve to the left, which means the
tax rate set by the dictator will be lower. As the dictator becomes weaker and
7 See
8 See
Glaeser(2004) for a detailed discussion .
Tilly(1990) for a discussion about the importance of state capacity and Persson and
Besley(2008) about orgin of state capacity.
5
weaker within one dictatorial dynasty, the real tax rate faced by the citizens
tends to increase and the tax rate charged by the dictator will be lower and
lower. This means dictatorial government’s revenue will be lower and lower
because on the one hand, the increasing real tax burden will reduce the citizens’
investment, which decreases the dictator’s tax base and on the other hand, the
dictator’s share of the pie becomes lower and lower.
Combining the two sources of dynamic ine¢ ciencies, my model shows the
real tax rate faced by the citizens has an oscillatory pattern with an upward
trend, and the tax rate charged by the dictator has an oscillatory pattern with
a downward trend. This leads to the …nal conclusion of the paper:
1. If there is a possibility of discontinuity of power due to the physical death
of the dictator; and
2. If the dictator concerns primarily on his own safety, rather than his tax
base; and
3. If the dictator has to rely on some agents, whose power can not be e¤ectively balanced and checked; and
4. If the functioning of dictatorial regime depends a lot on the quality of the
dictator; then
5. Dictatorial regime is doomed to demise once being set up as a result
of decreasing …scal income because of diminishing tax base caused by
increasing bureaucratic corruption due to weakering dictator. .
To the best of my knowledge, no previous work explained how bureaucratic
corruption caused by a deteriorating state capacity leads to the demise of dictatorship. Two most related studies may be Usher(1989) and, Gennaioli and
Caselli (2005). Usher(1989) argues that it is population growth that leads to
a gradual fall in income per capita, until eventually the surplus over bare subsistence is insu¢ cient to provide for the ruling class and it is more pro…table
to be a bandit than a farmer. In an agricultural society, this argument can be
6
translated as the continuous population pressure on cultivated and cultivable
land leads to rural uprisings that cause the demise of dictatorial regime. However, this is not supported by the historical evidence. For example, according
to Wang(1973), although various rural uprisings took place around 1850, the
population increased only 5 per cent while cultivated land went up by over 25
percent between 1750 and 1850 in China; Perkins(1969) shows that only by the
early twentieth century had China reached the point where there was no more
new cultivable land and even later the point at which traditional methods could
no longer increase per unit yields on land already under cultivation, although
the rural uprisings had happened long before this point. In a study at …rm level,
Gennaioli and Caselli (2005) show that due to the imperfections of contractual
enforcement in developing countries, the ownership and the control of private
…rms often pass across generations within the same family. However, as it is
impossible that there is always a member in the family with managerial talent
and the ownership and control are always transferred to the right person, family
…rms in developing countries will end up in the wrong hands sooner or later.
Although the long run outcomes of family …rms in Gennaioli and Caselli (2005)
and dictatorial regime in my paper is similar, the mechanism is di¤erent because
in my paper, the incumbent dictator intentionally chooses a su¢ ciently weak
successor as future dictator from candidates with all possible strengths due to
safety concern.
Methodologically, my paper is closest to Hassler et.al (2003)9 , who provide
an analytical characterization of Markov Perfect Equilibria in a model with repeated voting. Like that paper, I focuse on Markov Perfect Equilibria where
the strategies of all the agents are conditioned only on their pay-o¤-relevant
state variables and characterize the analytical solution to the equilibria. Unlike
9 Hassler
et al. (2005,2007) use similar structures to analyze democratic public good provi-
sion and the dynamics of democratic goverment. Azzimonti Renzo (2007), Song, Storesletten
and Zilibotti (2007) also characterize the analytical solution to a MPE, but with a di¤erent
microfoundation. Some other papers (Marco Bassetto, 1999, Krusell, Vincenzo Quadrini, and
Rios-Rull, 1996; Krusell and Rios-Rull, 1996,1999; and Gilles Saint Paul, 2001) imbed interest
con‡ict into repeated voting and yield numerical solutions.
7
that paper, the political game is di¤erent because the politics is di¤erent in
dictatorship. As Acemoglu et.al (2004) points out, “The qualitative nature of
politics appears to di¤ er markedly between strongly and weakly-institutionalized
polities: when institutions are strong, citizens punish politicians by voting them
out of power; when institutions are weak, politicians punish citizens who fail
to support them. When institutions are strong, politicians vie for the support
and endorsement of interest groups; when institutions are weak, politicians create and control interest groups. When institutions are strong, citizens demand
rights; when institutions are weak, citizens beg for favors.” In my model, the
policy is made by a sel…sh dictator, rather than re‡ecting the preference of
the decisive voters; the leadership turnover depends on the relative strengths
between the incumbent dictator and his successor, rather than via democratic
process; the economic policies are implemented costly by sel…sh bureaucrats
whose power is not balanced and checked, rather than by an e¢ cient and costless bureaucracy. I believe these changes in the political game capture the main
di¤erence between the politics between democracy and dictatorship.
In addition to providing a theory on the evolution of dictatorship, my paper also contributes to the economic literature on the internal organization of
dictatorship. This small and growing literature can be divided in two strands.
From a micro perspective, Egorov and Sonin (2006) formalize the loyalty and
competence trade-o¤ that the dictator faces when choosing agents and explore
the incentive for a dictator to keep incompetent agents; Acemoglu, Egorov and
Sonin (2008) show the size of ruling coalitions is determined by a trade-o¤ between the “power” and “self-enforcement”. Ruling coalitions must not only be
powerful enough to be able to impose their wishes on the rest of the society,
but also self-enforcing so that none of their subcoalitions be powerful enough
and wish to split from or eliminate the rest of this coalition. Egorov and Sonin
(2005) explore the trade-o¤ that a winner of the throne faces after the power
struggle. If the winner kills the loser, the threat of power is reduced. But the
winner builds up a tough reputation and will be probably killed by his contender
when losing the power struggle in the future. While if the winner only spares the
8
loser, the loss is that the loser may compete for power again and the gain is the
slighter punishment when losing in future struggle. From a macro perspective,
Acemoglu, Robinson and Verdier (2004) argue that the survival of a dictator
depends a lot on his ability to implement the “Divide and Rule”strategy among
his subordinates. Debs (2007,2008), shows that growth is positively related to
dictator’s strength as more able dictator can control more able agents, who are
more productive. Padro-i-Miquel (2007) shows successful dictator can expropriate not only the citizens outside the ruling group but also the his supporters
inside the ruling group while still keeping his supporters’support because once
the leader is replaced due to the loss of support from his supporters, there is a
chance that the citizens outside the ruling group can get the power and the core
supporters of the current dictator will get expropriated. Besley and Kudamatsu
(2007) show that autocratic government works well when the power of the selectorate does not depend on incumbent leader. My paper extend the existing
literature in two important ways. Firstly, I explore not only the interest con‡ict
between the incumbent dictator and his agents, but also the intergenerational
con‡ict between the current dictator and the future dictator. Secondly, I derive the macroeconomic implication of this intergenerational interest con‡ict on
bureaucratic corruption and the evolution of dictatorship.
The organization of the paper is as follows. Section 2 provides some historical evidence. Section 3 describes the model environment. Section 4 de…nes
and solves analytically the Markov Perfect Political Equilibrium. Section 5
concludes. All the proofs are in the technical appendix.
2
Historical evidence
In this section, I present some historical evidence related to the evolution of
the dictatorship. I note that in the historical literature, some works about
palace politics focus on the interest con‡ict between current dictator and future
dictator, but not on the implications of this con‡ict on bureaucratic corruption;
some other works focus on the interaction between the bureaucrats and the
9
citizens and point out it is the decay of the dictator’s power that leads to
increase of bureaucratic corruption and thus the burden on the citizens, which
leads to the …nal demise of dictatorial government, although these works do not
explain the reason of the decaying of the dictator’s power. In the following two
subsections, I provide historical evidence from Qing Dynasty, the last imperial
dynasty of ancient China, from the above two perspectives.
2.1
The interest con‡ict between current dictator and future dictator10
Emperor Kangxi was regarded as the founder of Qing Dynasty, because he united
China after conquering Mongolia, Taiwan, and Tibet, getting rid of the warlords’
threats from three provinces in south China and defeating Tzars Russia, though
there were three other emperors before him in this dynasty. In the year of 1676,
Kangxi’s designated his second surviving son Yinreng, who was at age two, as
the Crown Prince of the Great Qing Empire.
Even though Kangxi favoured Yinreng and had always wanted the best for
him, Yinreng did not prove cooperative. Yinreng’s supporters, led by Suoertu,
had gradually formed a "Crown Prince Clique", which tried the best to make
Yinreng be the emperor as soon as possible, with any possible method. Emperor
Kangxi was perfectly aware of Yinreng’s misbehavior. The relationship between
the father and the son became gradually worse and worse. In the 46th years of
Kangxi’s reign (1707), Kangxi decided that "after twenty years, he could take
no more of Yinreng’s actions”, which he partly described in the Imperial Edict
as "too embarrassing to be spoken of", and decided to demote Yinreng from his
position as Crown Prince. Yinzhi, Kangxi’s eldest surviving son, who had many
times attempted to sabotage Yinreng, even employing witchcraft, was placed
to watch Yinreng during home arrest. With such an important task, Yinzhi
thought he had got trust from Kangxi and would be made the new Crown
1 0 This
subsection is adapted from the introduction of Kangxi, Yongzheng and Qianlong in
Wikipedia and Feng(1985).
10
Prince. To ensure his position as Crown Prince, Yinzhi even asked Kangxi for
permission to execute Yinreng. This enraged Kangxi and Yinzhi was arrested
immediately and kept home arrest till his death.
With a vacant position of Crown Prince, debate began among o¢ cials and
members in the royal family. Everyday, rather than working, everyone in the
central government and the palace just speculated who might be the new Crown
Prince and spread various rumors, although Emperor Kangxi advised the o¢ cials and the nobles to stop such debate. The 8th Prince, Yinsi, who was widely
known as “wise prince”, turned out to get the most support from the o¢ cials.
However, Kangxi did not favor Yinsi because the emperor was aware of Yinsi’s
strength and was afraid of abnormal death caused by Yinsi once choosing him.
Facing such a situation, Kangxi re-established Yinreng as Crown Prince as a
temporary solution to avoid malfunction of the government and more importantly, to prevent Yinsi from being chosen the Crown Prince. The o¢ cial reason
of the reestablishment was that Yinreng’s former fault was the result of mental
illness caused by Yinzhi’s (the …rst Prince) witchcraft and Yingreng need some
time to recover.
However, Yinreng did not “recover” at all. In 1712, during Kangxi’s visit
of South China, Yinreng ruled as regent in charge the routine a¤airs of the
central government in Beijing. With more power than before, Yinreng decided
to mount a coup against Emperor Kangxi. This coup was unsuccessful because
Emperor Kangxi had received the information in advance from several sources.
When Kangxi returned to Beijing, he removed Yinreng from the Crown Prince
for the second time. Since then, Yingreng had been kept home arrested till his
death.
Emperor Kangxi’s health was badly hurt by the Crown Prince problem. To
prevent further debate on this issue, Kangxi o¢ cially declared that he would not
designate Crown Prince until his death and he would instead put his political
testament about Crown Prince inside a box, which could only be opened after
his death, in one palace of the Forbidden City.
However, Kangxi’s choice of Crown Prince through the secret arrangement
11
was not unpredictable. After Yingreng’s abolition, Kangxi carried out a political
purge. Yinxiang (the 13th Prince), the supporter of the Yinzhen (the 4th Prince)
was placed under home arrest for “cooperating with Yinreng”. Yinsi (the 8th
Prince) was declared not be eligible for Crown Prince due to his guile and his
mother’s humble origin. The 14th Imperial Prince Yinti, whom many considered
to have the best chance in succession, was sent to quell rebels in Western China
far away from Beijing. It turned out that Yinzhen, the 4st Prince, was the only
adult prince who had some chance of being chosen as Crown prince survived in
the purge and the purpose of Emperor Kangxi’s purge was to pave the way for
Yinzhen to get the crown.
On December 20, 1722, Emperor Kangxi died after ruling China for 61 years
and Yinzhen became the new emperor. Historians previously believed that
Yinzhen forged Kangxi’s testament and killed the old emperor. According to
some new evidence11 , the current consensus among historians is that Kangxi
designated Yinzhen as the successor, but Kangxi’s death still remains a myth.
Yinzhen’s strategy to get the crown was noteworthy. Fully aware of the fact
that the Crown Prince must face the threats from all the other princes and
suspicion of the old emperor, Yinzhen worked hard for Emperor Kangxi, showing
intentionally that he had no interest in striving for the power though the fact
was the opposite, and tried to keep a good relationship with all the princes.
With the strategy of neutralism, Yinzhen became the sole bene…ciary of the
con‡ict among the other princes and Emperor Kangxi.
The power struggle for the throne did not stop with Emperor Kangxi’s death.
Upon getting the throne, Yinzhen released his long-time ally, the 13th prince
Yinxiang, who had been kept home arrested because his old father was afraid
that Yinxiang’s striving power for Yinzhen would cause trouble that could obstruct the plan to transfer the power to Yinzhen. With the help of Yinxiang,
the new emperor, Yinzhen, continued to keep Yinzhi (the 1st Prince) and Yinreng (the former Crown Prince) home arrested. Yinti (the 14th Prince) was
1 1 Emperor
Kangxi’s testament, which was witten in three di¤erent languages, was publicly
shown in the Forbidden City recently .
12
placed under home arrest at the Imperial Tombs after coming back to Beijing
from the west for Kangxi’s funeral, under the pretext of watching over Kangxi’s
tomb. The biggest challenge of the new emperor was to destroy Yinsi’s (the
8th Prince) clique, which mainly consisted of Yinsi himself, the 9th Prince, the
10th Prince, and their many subordinates in the government. Yinzhen did this
step by step. Firstly, Yinsi was nominated as Prime Minister. By doing this,
Yinzhen could keep a close watch over Yinsi himself. Secondly, the 9th Prince
was sent to West China under the control of Yinzhen’s trusted general, with
the pretext of supervising the army. Thirdly, the 10th Prince, was rid of all
the titles and sent outside Beijing. Both princes died soon after leaving Beijing.
Finally, Yinsi was rid of all the titles and died lonely.
With the end of the old struggle for Crown Prince among Yinzhen and his
brothers, the new struggle for Crown Prince started between two of the three
Yinzhen’s sons, although Yinzhen used the same secret method to designate
his successor as his father. The con‡ict was between the fourth Prince, Hongli,
who was favored by Emperor Kangxi and Yinzhen, and was also believed by the
o¢ cials as the successor, and the third Prince, Hongshi, who was supported by
his eighth Uncle, Yinsi. Hongshi lost in the power struggle against Hongli and
even Yinzhen, and was forced by his father to commit suicide in 1727 at the age
of 24.
In 1735, Yinzhen died suddenly at the age of 57 and Hongli got the power at
the age of 24. The reason of Yinzhen’s death was believed by historians to be
the result of either too much hard working or irregular use of medicine produced
by Taoist. There was no documented con‡ict between Yinzhen and Hongli.
In 1796, after ruling China for 61 years, Hongli transferred power to his son,
Emperor Jiaqing, in order not to rule longer than his grandfather, Emperor
Kangxi. However, Hongli changed his mind soon after the power transfer. He
named himself Supreme Emperor and kept a tough control of everything till his
death in 1799.
From the above examples about power transfer in dictatorship, it is clear that
(i) The incumbent dictator concerns primely about his own safety and thus (ii)
13
The successor may not necessarily be the strongest among all the candidates.
2.2
The decay of ruling elites, the rise of land tax and the
fall of dynasties12
The record of the Qing dynasty, beginning with the redistribution of land and
the lightening of taxes and ending with the degeneration of the ruling class, the
swollen accumulation of estates in the hands of private, privileged, tax-evading
landholders, extortionate taxation of the poor peasantry, and helplessness in the
face of foreign invasion, is an epitome of Chinese economic and social history.
In the late years of Ming Dynasty (1368-1644), excessive taxation and corruption in the levying of the taxes provoked peasant uprisings all over China.
The Manchus conquered China and set up Qing Dynasty by taking advantage of
the collapse of the Central government of Ming caused by the rebelling peasant
army, who actually took the capital of the country and caused the suicide of the
last emperor of Ming Dynasty. Upon ruling, the new Manchu rulers redistributed land to the peasants and reduced the land tax rate. The reward of these
e¤orts was the social stability in earlier period of Manchus’ rule. Hoping to
restore such stability forever, the emperor of Kangxi set the “permanent settlement” decree in 1713, committing that the tax burden will be never increased.
Good intentions of Kangxi did not lead to good outcomes. Like in any
dictatorship, the Manchus, or the ruling elites, became a privileged class over
the society and no imperial decrees could stop their exploitation on the rest
of the society. Members of the ruling elites gradually robbed the wealth and
power of the central government. They could not possibly be restrained because
although their job is to protect the interests of the nation, they are also private
individuals who are the sole bene…ciaries of corruption. While some of them, as
o¢ cials, understood what was wrong, the most that they could accomplish as a
class was to try to protect both the government interest and their class interest
by trying to make up for the taxes which they themselves evaded by increased
1 2 This
subsection is adapted from Wang(1936).
14
taxation of the poor and unprivileged class. The whole process may be brie‡y
summarized in the following paragraph:
As the basic source of wealth was from the land, the interest of the central
government was to obtain the greatest possible volume of land tax. But since
the interest of the privileged class (including the landlords who had connections
with the privileged class) was to extract rent and to evade taxation on their own
lands, the volume of land tax revenue could only be kept up by an increased
rate of levy on the peasants. The burden of the peasants became even heavier
as local governors can surcharge the land tax and pocket these surcharged income due to the general slackness in administration caused by decay of central
government’s political power. By and by, peasants started to sell their land to
the privileged class and became their tenants. This further increased the burden
on the remaining peasants. The disproportionate concentration of land in the
privileged class increased their power. The more powerful they became, the less
they paid, and the less they paid, the more insistent became the pressure on
the decreasing number of small peasant proprietors. By the end of the dynasty,
the original strong centralized power of the Manchus had broken down into a
system of arbitrary and suicidal exploitation by the whole of the ruling class,
for the individual and competitive bene…t of the separate members of the class.
As a result, Qing dynasty fell down with peasant insurrections and the invading
of western colonists.
The following three examples document the extent of the corruption at different levels in late Qing Dynasty and a comparison the extent of corruption at
di¤erent time in the dynasty.
1. Corruption at low levels. When the date for the collection of land tax
had been proclaimed, the petty o¢ cials and their hangers-on went to each
village, forced their way into the cottages of the peasants, and compelled
them to make immediate payment of the tax. If there was any delay,
the peasants would be lashed till the blood spurted, unless they paid, as
bribe, what was known as pao-erh-ch’ien or “pocket money,”in earnest of
15
full payment later. Payments of this kind might have to be made more
than once, and might even, in the end, amount to more than the total tax
due. But as they were not discounted against the tax, the full amount
remained still to be paid. Peasants who had enough grain to pay their
tribute promptly, brought it to the Yamen(local government), the whole
family of each peasant attending, including the women. They had to appear actually before the due date, so that there should be no delay on the
day of payment. If it rained while they were waiting, they had to protect
their rice as best they could, for fear that the dampness would make the
color changed. Even if the collectors received it on time, various demands
for “wastage charge”, “light weight charge,” “cargo charge,” “transport
charge”and so forth might still have to be met, so that it was regarded as
not abnormal for a peasant to pay his tax at the rate of 250 per cent of the
assessed amount. When the collectors measured the grain, they usually
managed to get a considerable surplus (later to be deducted privately for
their own bene…t), by “trampling the measure,” to pack it tight, and by
heaping a cone on the top of it so that, in the biblical phrase, it should
be “pressed down and running over.” When this had been done, even the
spare grain that the peasant had brought to meet the surcharges was likely
not to be enough. If the grain was measured with a discount of 30 percent (a frequent practice), the storage would be all the greater. Disputes
between taxpayer and tax collectors were therefore common, which gave
the collectors a further opportunity to extort hush-money, on the ground
that the peasant had refused to pay.
2. Corruption at high levels. In the transport of grain tribute to Peking,
the Provincial Grain Intendant demands his ts’ao-kuei (grain fee, grain
perquisite); the Grain Commissioner (equal in rank to a viceroy, and
charged with the transport and disposal of tribute grain from the eight
provinces adjacent to the Yangze, to be shipped to Peking by the Grand
Canal) demands it; even the Deputy Prefects and Magistrates— – all de-
16
mand it. The o¢ ce of the Prefect demands a lodging fee; the o¢ ce of the
Provincial Treasurer demands a lodging fee; the petty o¢ cers of the Grain
Commissioner— –they all demand it.
3. The change in the extent of corruption. In the past, when the collection of land tax began, the local o¢ cials used to send several strong men
to guard the o¢ cial grain measure. Now, however, they openly declare a
discount of 20 per cent (in measuring the grain); and on top of this another
20 percent is demanded. Besides heaping up the surface of the measure,
trampling it down, and “seizing the pig13 ,”they demand food-money and
a transport fee a tax-roll fee, a fee for stamping the seal, a fee for sifting
rice, a granary door fee, and a granary fee, amounting in all to two tou
(20 per cent on the shih, the unit of measurement). The taxpayer has to
pay more than 2.5 shih for each shih.
These examples are enough to show that toward the end of the Manchu dynasty, the total of taxation centering around the land tax had swollen to the
almost incredible proportion of 20 to 30 times of the “permanent and unalterable”tax determined at the beginning of the dynasty, and the conditions which
had caused the fall of the Ming dynasty had been reproduced. Therefore, the
people, growing full of hatred, rise and rebel. Sporadic insurrections began in
the reign of Tao Kuang(1821-1850), the most serious of them being in Hunan
1844, and at the same time there were scattered risings in Chekiang, where the
slogan of the peasants was refusal to pay the land tax, as it had been at the end
of the Ming dynasty two centuries before. The great Taiping Rebellion began
in 1851, in Kuangsi, and before its defeat in 1865 had occupied two thirds of
the country. In 1853 began the rebellion of the Nien Min, starting in Shantung
and spreading widely through the north, where it dragged on for years; and
in 1871 there was another general rising in Shantung against the collection of
the land tax. The Boxer Rising of 1900 stemmed, therefore, from what was by
1 3 ”Seizing
the pig,”refers to the ”squealing”of the peasant when seized by the tax collectors
to force him to pay up.
17
then an established tradition of peasant revolts, and there is no doubt that the
Boxers were recruited mainly from poor peasants who had rebelled, originally,
against payment of the land tax. In the end, Qing Dynasty fell down for the
same reason as that of Ming Dynasty and the whole country was divided by
several warlords.
3
The model
The model economy has a two-period OLG structure and in every period, there
are four types of risk neutral agents: the citizens, the dictator, the dictator’s
successor candidates and the bureaucrats. The mass of each generation of citizens is unitary. Each of the citizens undertakes an investment when young,
which costs
i2
2,
and yields a return i in both periods of lives. The dictator is
the ruler of the economy. He sets an age-independent tax rate to maximize
the tax revenue from the investment returns of the young and the old citizens.
No matter how strong a dictator is, he must face the following two problems
about power: (i) The discontinuity of power caused by the physical death of
the dictator; (ii) The delegation of power.
The dictator has a dilemma when solving the …rst problem. If the dictator
does not designate anyone to be his successor when alive, there will be some
chaos, in which
of the citizens’ investment will be destroyed, caused by the
power struggle for the crown after the dictator’s death. Such a bad state ex post
will decrease the citizens’investment ex ante and thus decreases the dictator’s
tax base. Alternatively, the dictator can designate his successor when alive.
Although this can preclude the possibility of chaos after the dictator’s death
and thus increases the dictator’s tax base, such a method reduces the dictator’s
safety when alive, since the successor always has an incentive to take the place
of the incumbent earlier to enjoy the dictator’s rent. I assume
= 1; such that
designating the successor when alive always dominates leaving no successor after
death14 .
1 4 Herz(1952)
provides a detailed discussion about this problem and shows designating a
18
Assume some successor candidates with mass m (m < 1) are born in every
period. These candidates are the only people in the economy that have the privilege to be the future dictator. Every incumbent dictator designates his successor
from one of the successor candidates in the beginning of the incumbent’s second period of life and transfers the power to the successor before death. Given
the above assumptions, the timing of the power transfer, unless there is a coup
against the incumbent dictator, is as follows: at the beginning of any period t;
the incumbent dictator, who is in his second period of life and is designated as
successor by the previous dictator, becomes the ruler and designates the successor from the successor candidates born at period t; at the end of period t, the
incumbent dictator transfers the power to the successor.
The strength of the successor candidate has a uniform distribution in [0; m] ;
such that a candidate j can be marked by his strength
ability of the incumbent dictator
i,
j
2 [0; m] : The prob-
who is among the successor candidates in
the previous period and thus can also be marked by his strength
power struggle with his successor candidate
P(
i
wins) =
8
>
< 1
>
:
if
if
if
1
2;
0;
j
i;
to win the
is
i
j
i
d
<d
i
i
>d
i
d
j
i
j
The intuition of con‡ict technology is that if the incumbent is su¢ ciently
stronger than the successor, the incumbent will win for sure; if the di¤erence
between the dictator’s strength and the successor’s strength is not big enough,
the probability that each side wins is one half; if the dictator is su¢ ciently
weaker than the successor, then the dictator will lose for sure. d can be seen as
a measure of incumbent advantage in power struggle, with the larger the size of
d; the lower the incumbent advantage.
In addition to the problem of power transfer, the dictator also has to delegate (some of) his power to the bureaucrats. Due to the nature of dictatorship,
there can not be any source of independent check and balance of the bureausuccessor when alive dominates any other method.
19
crats’power since this means the erosion of the dictator’s power15 . Moreover,
the asymmetric information between the dictator and bureaucrats create the
opportunities for corruption. The unbalanced power plus the asymmetric information between the dictator and the bureaucrats makes corruption hard to
be eradicated in dictatorship. In the model economy, bureaucratic corruption
is re‡ected as the surcharge of tax by the bureaucrats. That is, a bureaucrat
can say a citizen, who actually has paid the tax, has not paid; or a bureaucrat
can say a citizen, who actually has not paid the tax, has paid. In equilibrium,
the bureaucrats can charge more than the tax rate announced by the dictator.
Since the bureaucrats’surcharge distorts the citizens’investment decision and
decreases the tax base of the dictator16 , it is not in the interest of the dictator.
The size of the surcharge depends on the strength of the dictator in regulating
the bureaucrats17 . Note that the ability for a dictator (successor) to …ght in
the power struggle with a successor (dictator) and the ability to regulate the
bureaucrats are in fact the same thing or at least positively correlated, since
these two abilities both re‡ects of the leader’s political skills.
Technically, I assume if the tax rate announced by the dictator
ante; the bureaucrats can surcharge (n
any risk. This means for given
d
t
i)
d
t,
and
i
is
d
t
ex
ex post on the citizens without
the real tax rate
r
t
that the citizens
face ex post, is
r
t
with n
=
d
t
+n
d
t
m: As can be seen from the above expression, for given
d
t;
the stronger
the dictator, the lower the tax burden on the citizens.
4
Political Equilibrium
1 5 See
Yi(2007) for a detailed discussion.
shows corruption is negatively related to growth and investment, and cor-
1 6 Mauro(1995)
ruption a¤ects growth through investment. See also Fisman and Svensson (2007) for a study
about corruption and growth in the …rm level.
1 7 pp.153 of Feng(1985) documented the dramatic decrease of bureaucrats’ surcharge soon
after a strong dictator took power in China. In some provinces, for example, Henan and
Shandong, the surcharge rate went down from 80% to 13% and 18%, respectively.
20
The purpose of this paper is to explore the impact of interest con‡ict between the
incumbent dictator and his successor on the strength of dictator generation after
generation, which a¤ects the extent of bureaucratic corruption over time and
the evolution of dictatorship. More speci…cally, can a regime with continuous
interest con‡ict between current and future ruler, which a¤ects the distortion on
investment caused by bureaucratic corruption, be sustainable in the long run?
In order to answer this question, I start to solve an equilibrium without crown
prince problem as a benchmark. which can help to characterize the equilibrium
with crown prince problem.
4.1
Equilibrium without Crown Prince problem
In this case, I assume the successor’s moral concerns always dominate his economic concerns. That is, the successor never tries to get the power one period
earlier from the incumbent dictator. The timing of the game is as following:
1. At the beginning of period t; the old incumbent dictator chooses his successor, who gets the power in the end of period t when the incumbent
dies;
2. The successor candidates other than the one chosen by the dictator as the
successor are eradicated;
d
t;
3. The incumbent sets the tax rate
4. The young citizens born at period t make their investment it ;
5. The bureaucrats surcharge and collect the tax for the old incumbent;
6. The incumbent transfers his power to the successor at the end of period t:
Given the assumption about the game, the indirect utility functions of the
living agents are as follows
V oc = (1
21
r
t ) it 1
V yc = (1
r
t ) it
V od =
d
t
+
(it
r
t+1
1
1
+ it )
i2t
2
it
(1)
2wt ;
where V oc ; V yc ; Vtod denote the objective of the old citizen, the young citizen,
d
t,
and the incumbent old dictator, respectively.
r
t;
d
t;
it ; wt denote the tax
rate imposed by the dictator, the real tax rate that the citizens face, the strength
of the incumbent dictator, the investment made by young agent and the wage
to the bureaucrats at period t; respectively. Simple maximization in (1) shows
that wt = 0 and the solution to the optimal investment problem of the young
r
t
citizen, given the real rates in his two periods of life,
r
t)
it = (1
+
r
t+1 ,
and
is
r
t+1
1
(2)
De…nition 1 A (Markov Perfect) political equilibrium is de…ned as a triplet
of functions hA; T; Ii ; where A : [0; m]
[0; 1
decision rule on the strength of his successor,
= T adt ; it
and I : [0; m]
1
] ! [0; m] is the dictator’s
= A adt ; it
1
, T : [0; m]
n + adt is the dictator’s policy decision rule on the tax rate,
[0; 1 + ] ! 0; 1
d
t
adt+1
[0; 1] ! [0; 1 + ] is the young citizens’private
d
t+1 ;
investment decision rule it = I
r
t
, such that the following functional
equations hold:
1.
A adt ; it
subject to
d
t+1
d
t+1 ;
2. I
3. V od
d
t;
; T adt ; it
1
= T A adt ; it
1
1
r
t
=1
d
t+1 ;
r
t+
d
t+1 ;
= arg maxadt+1 ;
d
t+1 ;
;I
1
d
t ; it 1
r
t
d
t
it
V od
d
t;
d
t+1 ;
d
t+1 ;
d
t ; it 1
adt
+n
d
t+1
:
T adt+1 ; I
=
d
t
1
d
t+1 ;
+I
d
t
+n
d
t+1 ;
r
t
:
According to De…nition 1, the state of the model economy at period t is
captured by two state variables, adt and it
1.
The …rst equilibrium condition
requires that the incumbent old dictator chooses adt+1 and
d
t
to maximize his
indirect utility function, taking into account that future dictator’s decisions
about tax rate and the successor’s strength depend on the current dictator’s
choice via the equilibrium decision rules. Furthermore, it requires A adt ; it
and T
adt ; it 1
1
are both …xed points in the functional equation in part 1 of
22
the de…nition. The second equilibrium condition implies that all young citizens
choose their investment optimally, given adt and
r
t,
and that these agents hold
rational expectations about how future tax rate and dictator’s strength are
determined. The third equilibrium condition means the old incumbent does not
need to worry about his safety since by assumption, the successor never tries
d
t
to seize the power one period earlier. The constraint that
r
t
is equivalent to
2 0; 1
n + adt
2 [0; 1] ; which means the real tax rate that the citizens face
can not be larger than one as there is no saving in the economy.
Proposition 2 If m
n
1
2m
1
; in the equilibrium without crown prince
problem, hA; T; Ii is characterized as follows:
T adt ; it
d
t+1 ;
I
=
1
a
t
=
A adt ; it
(
1
2
(
m
it
1
+1+
n n
2+
+ 12 adt + 2
1 + n adt ;
)
a
t
)(4
(2
+ 4 adt+1 +
2
;
if
if
r
t;
1
(2
=m
1
2n
m
4( +2)
it
it
2
1 2 [0; {t 1 ]
2 ({t 1 ; 1 + ]
1
+4)
;
if
r
t
if
r
t
2 [0;
r
t]
2 ( rt ; 1]
for given ad0 and all t;where
2
{t
+2
1
1
1 d
m + adt+1 +
a
2
2 t+1
n
and
(
2)
(2 + (1
r
t
d
t+1
1
(2n + 2
+2
+
d))
n )
Furthermore,
(1) With any ad0 2 [0; m] and i
1
2 [0; 1 + ], hA; T; Ii converges to the
following equilibrium in one period with
A adt ; it
T adt ; it
I
d
t+1 ;
r
t
=
1
1
(
it
2
=
1
1
+ +1
n + 1) ; if
+2 (m
1 + n m;
if
1
(2
)
2
r
t
+
=m
(2
4
)
m+
23
r
t;
(2
)(4
2n
m
4( +2)
it
it
2
1
1 2 [0; {]
2 ({; 1 + ]
+4)
;
if
r
t
if
r
t
2 [0;
r
]
2 ( r ; 1]
where
{
2
(1
+2
n + m)
and
2
r
2
m+
4n + 4 + 2m
2n
2 ( + 2)
m
2
(2) The equilibrium law of motion of dt is as follows
(
1 n + m;
if
d
d
d
+2
8
4n+2m
6n
m 2 +8
t+1 =
t +n at
+ 4 m+
; if
2
4( +2)
d
t
d
t
2
if
r
t
if
r
t
2 [0;
2 0;
d
t;1
d
t
n + adt
where
d
t
r
t
(3) The equilibrium law of motion of
(
1;
r
r
(4n+8 +2m
t+1 =
2
t
2 + 4 m+
4(
(4) Starting with any ad0 2 [0; m] and i
and
then
r
t
2 (0; 1) for all t
d
t
=1
n + ad0 ,
r
0
n + adt
r
t
is
2n
+2)
1
m
2 [0; {
2
+8)
1] ;
;
then
d
t
0: Starting with any ad0 2 [0; m] and i
= 1 and
d
t
2 0; 1
n + adt ;
r
t
2 ( rt ; 1]
2 0; 1
1
r
t]
2 ({
n + adt
1; 1
+ ],
2 (0; 1) for all t > 0.
In either of the above two cases, the economy converges asymptotically with an
oscillatory pattern to the following steady state with
ass = m
d
ss
=
1
(4m
3 +6
r
ss
=
2n
2m + 4 + m
3 ( + 2)
2
iss =
4n + 4 + 4m
+
+ 2 (m
3 ( + 2)
4n + 4)
n +4
n + 1)
Figure 1 here
Figure 1 represents the equilibrium decision rules of the incumbent dictator
and the citizens when there is no Crown Prince problem. Panel a shows that any
incumbent will choose the strongest successor. Panel b shows that for given adt ,
24
d
t
the equilibrium
some threshold {t
increases linearly with it
1
1;
which is sunk at period t; before
d
t
and then achieves a corner solution with
and a corresponding
r
t
= 1+n
adt
= 1 henceforth. Panel c shows that for given adt+1 , the
r
t.
citizens’investment decreases with
The discontinuity at
r
t
=
fact that to the left of this point, the next period real tax rate,
r
t
re‡ects the
r
t+1 ,
will get
a corner solution of one and the citizens’investment rule is di¤erent than that
to the right. Intuitively, without Crown Prince problem, an incumbent with
any strength will choose the strongest successor, who distorts least in it , since
the citizens’investment increases with adt+1 . Given the choice of the strongest
d
t
successor, the incumbent chooses a
that makes the tax income at the peak
of the La¤er curve, taking into account that how the future dictator makes
decisions about tax rate and successor’s strength. Therefore, in this case, the
tax base e¤ect dominates the safety e¤ect.
Figure 2 here
Figure 2 represents the equilibrium law of motion of tax rates. Panel a
shows that if
d
t
is lower than some threshold level
corner solution with
is higher than
d
t,
d
t+1
then
= 1+n
d
t+1
d
t,
then
m and a corresponding
will decrease linearly with
following, other things given, a lower
d
t
d
t:
r
t+1
d
t+1
will get a
= 1, while if
d
t
The intuition is as
will lead to a higher it , which is sunk
seen at period t + 1. This increases the period t + 1 incumbent dictator’s tax
base and will be taxed more heavily. This will generate an oscillatory pattern
of equilibrium
r
t:
d
t
across time. Panel b shows the equilibrium law of motion of
The shape and the mechanism is similar as the equilibrium law of motion of
d
t:
Figure 3 here
Figure 3 represents the time series of the tax rates. Panel a and b show that
if i
1
2 ({
1; 1
+ ], then
c and d show that if i
1
d
t
and
2 [0; {
1 ],
r
t
get a corner solution only at t = 0. Panel
then
25
d
t
and
r
t
never get corner solution. In
both cases,
d
t
r
t
and
converge asymptotically with an oscillatory pattern and
without any trend to their steady states, respectively.
4.2
Equilibrium with Crown Prince problem
This equilibrium can be analyzed in three steps. Firstly, I solve the Markov Perfect Equilibrium where all the incumbent dictators chooses a su¢ ciently weak
successor
d
t
d
t+1
d , and derive the indirect utility of the old incumbent
d
t
d
t+1
dictator as a function of
for given it
1
d
t.
and
Secondly, I analyze the
case in which the old incumbent dictator at period t chooses a non-su¢ ciently
weak successor
d
i
j
i
< d , given that all the past and future dictators
choose a su¢ ciently weak successor, and derive the indirect utility of the old
incumbent dictator as a function of
d
t+1
for given it
1
and
d
t:
Thirdly, I derive
the condition under which the indirect utility of the old incumbent in the …rst
case is always higher than that in the second case for any it
1
and
d
t.
If this
condition holds, then by one-stage deviation principle, the Markov Perfect Equilibrium where all the incumbent dictators chooses a su¢ ciently weak successor
is a Subgame Perfect Nash Equilibrium without retrictions on the successor’s
strength.
4.2.1
Equilibrium without threat from the successor
In this case, the safety e¤ect still dominates the tax base e¤ect. This means
d
t
d
t+1
d
t
d for all t. The timings of the game and the indirect utility functions
of living agents at period t are the same as in the equilibrium without Crown
Price problem as there is no threat from the successor.
De…nition 3 A (Markov Perfect) political equilibrium is de…ned as a triplet
of functions hA; T; Ii ; where A : [0; m]
[0; 1
decision rule on the strength of his successor,
[0; 1 + ] ! 0; 1
adt+1
] ! [0; m] is the dictator’s
= A adt ; it
1
, T : [0; m]
n + adt is the dictator’s policy decision rule on the tax rate,
26
d
t
= T adt ; it
and I : [0; m]
1
[0; 1] ! [0; 1 + ] is the young citizens’private
d
t+1 ;
investment decision rule it = I
r
t
, such that the following functional
equations hold:
A adt ; it
1.
d
t+1
subject to
d
t+1 ;
2. I
3. V od
r
t
=1
d
t+1 ;
d
t+1 ;
= arg maxadt+1 ;
1
= T A adt ; it
r
t
d
t;
; T adt ; it
1
1
+
;I
d
t+1 ;
1
T adt+1 ; I
d
t ; it 1
=
d
t
r
t
it
V od
d
t
and
1
d
t+1 ;
d
t+1
r
t
d
t+1 ;
d
t+1 ;
d
t ; it 1
d:
d
t
d
t+1 ;
+I
d
t;
d
t
d
t+1
+n
r
t
:
According to De…nition 3, the state of the model economy at period t is
captured by two state variables, adt and it
requires that adt+1 and
d
t
1.
The …rst equilibrium condition
maximize the indirect utility function of the old in-
cumbent dictator, taking into account that future dictators’decisions about tax
rate and the successor’s strength depend on the current dictator’s choice via the
equilibrium decision rules. Also, it requires A adt ; it
1
and T adt ; it
1
are both
…xed points in the functional equation in part 1 of the de…nition. Furthermore,
the constraint
d
t
d
t+1
d needs to be satis…ed as all the dictators secure their
d
t
power by choosing a su¢ ciently weak successor. The second equilibrium condition implies that all young citizens choose their investment optimally, given adt
and
r
t,
and that these agents hold rational expectations about how future tax
rate and dictator’s strength are determined. The third equilibrium condition
means the old incumbent does not need to worry about his safety since in this
case, the su¢ ciently weak successor has no chance to win the power struggle.
Proposition 4 If m
n
1 and 0 < d <
1
2
, in the equilibrium with crown
prince problem but without threat from the successor, hA; T; Ii is characterized
as follows:
T adt ; it
I
d
t+1 ;
1
r
t
=
=
(
(
A adt ; it
1
2
it
1
(2
)
2
+
r
t
1
= (1
d) adt
1+ (1 d) d
1+
n n
2+ (1 d) at +
2+
1 + n adt ;
+
r
1
t;
(2
) d
(2
t+1
2(2+ (1 d)) +
27
;
if
if
it
it
1
if
)(2
n +2)
;
2( +2)
if
2 [0; {t 1 ]
2 ({t 1 ; 1 + ]
1
r
t
r
t
2 [0;
r
t]
2 ( rt ; 1]
for given ad0 and all t; where
{t
(2
1
d
t+1
(
2)
(2 + (1
r
t
(1 n)
adt
+
2+
2 + (1
)
d))
d)
1
(2n + 2
+2
+
n )
Furthermore,
1. The equilibrium law of motion of dt is as follows
(
1i n + adt ;
h
d
d
d
2+2 (1 d)+
1
d
t+1 =
t +n at
+ 2(2+ (1 d))
t+1 + 2( +2) (4
2
d
t
if
2n
3n + 4) ;
if
d
t
where
d
t
r
t
n + adt
2. The equilibrium law of motion of rt ; is as follows
(
1;
if
h
i
r
r
2
1
d
t+1 =
t
n + 4) ; if
t+1 + 2( +2) (2n + 4
2
2(2+ (1 d))
3. Starting with any ad0 2 [0; m] and i
all t
0<
r
t
1
0: Starting with any ad0 2 [0; m] and i
2 [0; {
1
1] ;
then 0 <
2 [{; 1 + ], then
r
t
r
t
r
t
r
t
2 [0;
2 ( rt ; 1]
< 1 for
= 1 and
< 1 for all t > 0, where
{
1
= (2
)
(1 n)
ad0
+
2+
2 + (1
d)
In either of the above two cases, the economy converges asymptotically with an
oscillatory pattern to the following steady state with
ass = 0
d
ss
r
ss
=
iss =
=
4 ( + 1) (1 n)
3 ( + 2)
1
(2n + 4
3 ( + 2)
1
3 ( + 2)
2
28
+
n + 4)
+ 2 (1
n)
r
t]
2
2 0;
d
t;1
d
t
n + adt
Figure 4 here
Figure 4 represents the equilibrium decision rules of the incumbent dictator
and the citizens when there is no Crown Prince problem. Panel a shows that the
successor’s strength increases linearly with the incumbent’s strength. Panel b
shows that for given adt , the equilibrium
sunk at period t; before some threshold {t
with
d
t
adt
= 1+n
d
t
increases linearly with it
1
=
r
t
tax rate,
which is
and then achieves a corner solution
r
t
and a corresponding
= 1 henceforth. Panel c shows
that for given adt+1 , the citizens’ investment decreases with
r
t
1;
r
t.
The kink at
re‡ects the fact that to the left of this point, the next period real
r
t+1 ,
will get a corner solution of one and the citizens’ investment
rule is di¤erent than that to the right. Intuitively, when there is Crown Prince
problem, the dictator’s choice of adt+1 and
d
t
can be separate, given the model’s
assumption about agents’preferences and how the winner of the power struggle
being determined. That is, …rstly, to ensure his safety, an incumbent with any
strength will choose a successor as strong as possible to keep the distortions
on investment as low as possible, given the constraint
d
t
d
t+1
d is satis…ed.
d
t
Secondly, given the choice of the successor, the incumbent chooses a
d
t
that
makes the total taxation on the peak of the La¤er curve, taking into account
that how the future dictator makes decisions about tax rate and successor’s
strength.
Figure 5 here
Figure 5 represents the equilibrium law of motion of tax rates. Panel a
shows that if
d
t
is lower than some threshold level
corner solution with
if
d
t
is higher than
d
t+1
d
t,
= 1+n
then
d
t+1
adt
d
t,
and a corresponding
will decrease linearly with
is as following, other things given, a lower
d
t
d
t+1
then
r
t+1
d
t:
will get a
= 1, while
The intuition
will lead to a higher it , which is
sunk seen at period t + 1. This increases the period t + 1 incumbent dictator’s
tax base and will be taxed more heavily. This will generate an oscillatory
29
pattern of
d
t.
Panel b shows the equilibrium law of motion of
is similar as the equilibrium law of motion of
d
t
and
r
t
d
t.
r
t:
The pattern
The oscillatory pattern of
has three important implications: (i) Growth-enhancing economic
reforms in dictatorial regime will probably to be reversed with the change of
the ruler, if there is no institutional reform that balances the power of the ruler,
because without institutional reform, the power to set the policies stays on
the dictator, and as the tax base becomes larger due to the growth-enhancing
economic reforms, the new dictator will tax heavily on the sunk investment. This
will reverse the growth-enhancing economic reform; (ii) Bureaucratic corruption
and economic growth can be positively correlated in dictatorial regime. The
intuition is as following. When the tax base is low due to less sunk investment,
the dictator has an incentive to lower the tax rate, which is growth-enhancing
to increase the tax base. However, the lower tax rate itself can not put any
constraint on bureaucratic corruption. On the contrary, this increases the rent
base of the bureaucrats to get corrupt income. Thus, bureaucratic corruption
and growth can be positively correlated. This explains the high corruption and
high growth puzzle in east Asia after Second World War after which not much
capital is left. (iii) As the oscillatory tax rates between generations can be seen
as the variations of economic policies that are growth-enhancing or growthretarding and can be controlled by dictators, it is wrong to use variables that
re‡ect economic institutions as an indicator of political institutions in empirical
analysis. This supports the view of Gleaser et al.(2004).
Figure 6 here
Figure 6 represents the time series of the tax rates. Panel a and b show
that if i
1
2 [0; {
1 ],
then
Panel c and d show that if i
solution. In both cases,
d
t
d
t
1
and
2 ({
r
t
1; 1
get a corner solution only at t = 0.
+ ], then
d
t
and
r
t
never get corner
converges asymptotically with an oscillatory pattern
and a downward trend to the steady steady state. The downward trend is
h
i
(1 d)+
d
re‡ected in the term 2+2
t+1 in the equilibrium law of motion of
2(2+ (1 d))
30
d
t
as this term is decreasing period by period due to decreasing
in both cases,
r
t
d
t+1 .
Also
converges asymptotically with an oscillatory pattern and an
upward trend to the steady steady state. The upward trend is re‡ected in the
h
i
2
d
r
term
t+1 in the equilibrium law of motion of t as this term is
2(2+ (1 d))
increasing period by period due to decreasing
d
t+1 .
The mechanism to generate
the trends is as follows. Other things given, the weaker the dictator, the worse
is he in controlling his bureaucrats and the higher the bureaucrats’ surcharge
will be. This will increase real tax rate that the citizens face and shift La¤er
curve to the left, which means tax rate set by the dictator will be lower. As
the dictator becomes weaker and weaker within one dictatorial dynasty, the real
tax rate faced by the citizens tends to increase and the tax rate charged by the
dictator will be lower and lower. This means dictatorial government’s revenue
will be lower and lower because on the one hand, the increasing real tax burden
will reduce the citizens’investment, which decrease the dictator’s tax base and
on the other hand, the dictator’s share of the pie becomes lower and lower.
As we can see, in presence of the crown prince problem, if all the dictator
wants to be safe, the evolution of dictatorship can be summarized as following:
1. The dictator will become weaker and weaker period by period.
2. Bureaucratic corruption, which is measured by the fraction of tax income
that goes to the bureaucrats, will become higher and higher.
3. The real tax rate that the citizens face,
r
t,
will become higher and higher,
which makes the tax base to be smaller and smaller.
4. The fraction of tax income that goes to the dictator,
d
t,
will become lower
and lower.
5. Dictatorial can hardly survive in the long run due to the decreasing …scal
revenue.
31
4.2.2
Equilibrium with threat from the successor
Now I explore the following question: given all the past and future dictators
choose a su¢ ciently weak successor, is it optimal for the incumbent dictator at
period t to deviate for one period from choosing a su¢ ciently weak successor or
equivalently, to choose an insu¢ cient weak successor ( d
d
t
d
t+1
d
t
d)? If
the answer is no, then by the one-stage deviation principle, the Markov Perfect
Equilibrium where all the incumbent dictators choose su¢ ciently weak successors is a Subgame Perfect Nash Equilibrium without retrictions on the successor’s strength
As there is now threat from the successor and the result of the political
struggle is probablistic, the timing of the game at period t is modi…ed as following:
1. At the beginning of period t; the old incumbent dictator chooses his successor with strength
d
t+1 ;
2. The successor candidates other than the one chosen by the dictator as
successor and the one with strength
3. The old incumbent sets the tax rate
d
t+1 + ";with
" ! 0; are eradicated18 ;
d
t;
4. The young citizens born at period t make their investment it ;
5. The bureaucrats surcharge n
d
t
and collect the tax;
6. The power struggle between the incumbent and the successor takes place;
7. If the old incumbent wins, the successor is replaced with the candidate
with strength
1 8 If
d
t+1
+ " at the end of period t:
there is a power struggle between the incumbent and the successor at period t, then a
potential question is, who will be the ruler in period t + 1 if the successor loses in the power
struggle at period t: For simplicity, I assume the dictator keeps a candidate with almost the
same strength as the successor and if the successor loses in the power struggle, then the
incumbent transfers his power to the candidate with strength
s
t
+ " at the end of period t:
With this assumption, the equilibrium tax rate and the young citizens’investment will not be
a¤ected by the result of the political struggle.
32
8. If the successor wins, he gets the tax income at period t and also rules in
period t + 1: In this case, the utility of the old incumbent is
:
Giving the timing of the game, the indirect utility function of the old incumbent at period t is
od
Vnsw
=
1
2
d
t
(it
1
+ it ) +
1
2
This indirect utility function consists of two terms: with probability 21 ; the old
incumbent can maintain his power and get the tax at period t; and with probability 21 , he loses the power and the utility of being removed is
. Furthermore,
as the power struggle at period t takes place after the strength of successor (or
equivalently, the strength of period t + 1 dictator), the tax rate
tax rate
r
t
d
t
and the real
are determined, no matter who wins the power struggle at period
t, the citizens’ investment decision rule will be the same as in the case when
all the incumbents choose su¢ ciently weak successors, given that all the future
successors choose su¢ ciently weak successors.
Proposition 5 If
8
>
< (2
< min
>
:
3+ (1
)
2(1 n)2 (1+ )2 ( +2)2 ( 4+2(1
d)
(1
m+
d)
n)(1+ )
+2
2
)
2( +2)2
2(2
)(1 n)2 (1+ )2 ( +2)2 (2+
2( +2)2
n+m)2
9
>
; =
>
;
then all the dictators will choose a su¢ ciently weak successor and the Markov
Perfect Political Equilibrium de…ned in De…nition 3 is a Subgame Perfect Nash
Equilibrium without the constraint
d
t
d
t+1
d
t
d:
The intuition of Proposition 5 is that, if the utility
of the old incumbent
from being replaced by the successor is su¢ ciently low, then any dictator will
concern more about his own safety than his rent. Therefore, all the dictators
will choose a su¢ ciently weak successor. Figure 7 illustrates the relationship
between the incumbent’s utility and adt+1 for given adt and it
V od increases with adt+1 for all adt+1 2 (1
1:
In panel a,
d) adt ; min (1 + d) adt ; m
, and if
is su¢ ciently low, the incumbent’s indirect utility of choosing a su¢ ciently
33
od
weak successor(Vsw
) is higher than that of choosing a non-su¢ ciently weak
od
: In panel b, V od increases with adt+1 for all adt+1 2 (1
successor Vnsw
and gets a corner solution henceforth because
tion of one for adt+1 2
low
adt+1 ; min (1 + d) adt ; m
r
t+1
d) adt ; adt+1
will get a corner solu-
: In this case, su¢ ciently
also ensures the incumbent’s indirect utility of choosing a su¢ ciently
od
weak successor(Vsw
) is higher than that of choosing a non-su¢ ciently weak
od
successor Vnsw
:
5
Discussion and conclusion
Many problems in dictatorship are dynamic and can not be analyzed without
exploring the internal organization of such regime. In this paper, I construct
a positive theory on the evolution of dictatorship, The main contribution of
the analysis consists in showing that the demise of any dictatorial regime is
inevitable if there are discontinuity of power caused by dictator’s physical death
and the delegation of the dictator’s unbalanced power, which are two common
properties shared by all dictatorial regimes. More speci…cally, I have identi…ed two opposing e¤ects that the incumbent concerns in the determination of
successor. The …rst is tax base e¤ ect. Since the functions of dictatorship depend a lot on the quality of the leader, a stronger future dictator will increase
the investment of forward looking citizens. This increases the incumbent’s tax
base. The second is safety e¤ ect. That is, a stronger successor is always more
dangerous to the incumbent, as the former always has an incentive to take the
place of the latter and enjoy the power earlier. Under the assumption that every
incumbent puts primary concerns on his own safety rather than tax base, the
safety e¤ ect will dominate the tax base e¤ ect and the quality of successor, or
future dictator will be lower and lower. The unnatural selection of successor
is not costless, because weaker dictators are worse in regulating the agents and
bureaucratic corruption, which is modeled as bureaucrats’surcharge of tax, will
tend to increase generation by generation. Therefore, the overall pattern of the
evolution of dictatorial regime is increasing burden on the citizen caused by
34
increasing bureaucrats’ tax surcharge due to weaking dictator, and the …scal
revenue of the dictator is decreasing due to the decreasing of tax base, as will
cause the demise of dictatorship in the long run.
The analysis so far is subject to some potential caveats:
Firstly, I ignore the con‡ict among the successor candidates for the Crown
Prince by assuming the incumbent dictator can pave the way for the successor
by eradicating all the other successor candidates once designating the successor.
However, the main result of this paper, that dictatorship must fall down in the
long run due to increasing corruption caused by weakering dictator, is robust
when adding this con‡ict because with more power centers, bureaucratic corruption will be uncoordinated and this distorts investment more than monopolistic
corruption (Shleifer and Vishny 1993).
Secondly, I assume the incumbent dictator has perfect information about the
successor candidates’strength. Adding information asymmetries to the model
does not change the prediction about the long run demise of dictatorship because
if all the incumbents choose weaker successors based on their imperfect information about the successors’strength, then dictators will tend to be weaker in the
long run, although in the short run, there could be equilibrium in which a strong
successor was chosen by sending a signal of weakness. Therefore, Information
asymmetries may delay, but not prevent the long run demise of dictatorship.
Thirdly, to make the model as simple as possible in order to gain tractability,
I ignore the fact about the heterogeneity of citizens and the transfer of production factor ownership. According to the analysis in section 2.2, some citizens are
connected to the privileged class and the members in the privileged class may
also own land. The possibility of tax evasion will shift the tax burden to the
remaining citizens, forcing them to sell the land to the two groups and work for
them. This will lead to an over concentration of land in the privileged class and
shrink the tax base. Although the simpli…cation in my model gains tractability
without sacri…cing the main fact of the long run decreasing of the dictator’tax,
modelling the heterogeneity and ownership are interesting per se. This is left as
a future research.
35
6
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7
Technical Appendix
7.1
Proof of Proposition 2
7.1.1
The decision rules
I will use the strategy of “guess and verify” to derive the incumbent’s decision rules about successor’s strength and tax rate in this case and the citizens’
decision rule about investment.
d
t
Start by guessing
that
d
t
2 0; 1
n+
d
1 +B t +C
= Ait
adt
for all t and ignoring the constraint
: Given this guess, we must have
d
t+1
= Ait + B
d
t+1
d
t+1
d
t+1
+C
and
r
t+1
=
+n
= Ait + (B
r
t+1
Plug the expression of
d
t
it = 1
n+
d
t
1)
d
t+1
+C +n
in (2), we get
+
1
Ait
(B
d
t+1
1)
C
n
(3)
Solve for it in (3), we have
it =
d
t
1
n+
d
t
+
1 (B
1+ A
d
t+1
1)
C
n
(4)
Plug (4) in the indirect utility function of the old incumbent and rearrange,
we have
V od =
d
t
it
1
+
1
d
t
n + dt + (1
1+ A
C
n)
+
(1 B) dt+1
1+ A
As we can see from (5), given the guess about the expression of
(5)
d
t,
the
incumbent’s decisions of the successor’s strength and the tax rate can be separate now. That is, if
will choose
d
t+1
(1 B)
1+ A
> 0;then for any it
1;
d
t ;and
= m, since this maximizes his tax base; If
40
d
t,
the incumbent
(1 B)
1+ A
0; the
incumbent will choose
d
t+1
= 0; for any it
d
t
incumbent just chooses a
d
t ;and
1;
d
t.
d
t+1 ,
For given
the
to ensure his tax revenue is on the peak of La¤er
curve.
(1 B)
1+ A
My following strategy is to suppose
d
t+1
> 0; plug in
= m in (5),
get a solution candidate fA1 ; B1 ; C1 g of fA; B; Cg ; and then to verify in this
case,
(1 B1 )
1+ A1
(1 B)
1+ A
> 0: Then I suppose
0 and plug in
d
t+1
= 0 in (5), get
a solution candidate fA2 ; B2 ; C2 g of fA; B; Cg and then to verify in this case,
(1 B2 )
1+ A2
0 does not hold. With this strategy, I can show that fA1 ; B1 ; C1 g is
the solution of fA; B; Cg :
Suppose
V od
(1 B)
1+ A
=
> 0, then
d
t
d
t
A1 + 1
+
d
t+1
+
d
t+1
= m: Plug
(1
= m into (5), we get
B1 ) m + 1 +
=
(1 + A1 ) it
C1
n
d 2
t
d
t it 1
(6)
A1 + 1
Take …rst order condition in (6) with respect to
d
t
n
1
d
t
+
+
(1
d
t,
B1 ) m + 1 +
2
we get
n
C1
n
(7)
As we have guessed
d
t
= Ait
for all t and we get an expression of
d
t
1
+B
+C
d
t
in (7), then the following equality must
hold for all t if the guess is correct
(1 + A1 ) it
1
+
d
t
+
(1
B1 ) m + 1 +
2
n
C1
n
= A1 it
d
1 +B1 t +C1
(8)
If (8) holds for all t; the following equation system must hold
8
1+A1
= A1
<
2
B1 = 21
: (1 B1 )m+1+ n C1 n
= C1
2
Solving the above equation system, we get
8
A1 = 2 1
>
<
B1 = 21
>
: C = 2m +1+ n
1
2+
41
n
(1 B1 )
1+ A1
and B1 = 12 ; then
1
Since A1 =
2
=
(2
4
)
> 0 for
2 (0; 1). This
means fA1 ; B1 ; C1 g is one solution of fA; B; Cg :
Suppose
(1 B)
1+ A
d
t+1
0 instead, then
= 0: Performing exactly the same
steps above as in deriving fA1 ; B1 ; C1 g ; we get
8
<
However, in this case,
suppose
(1 B)
1+ A
A2 = 2 1
B2 = 21
:
C2 = 1+ 2+n
(1 B2 )
1+ A2
(2
4
=
)
n
> 0 for
2 (0; 1). This contradicts our
0 : Therefore, fA2 ; B2 ; C2 g is not the solution of fA; B; Cg
and we conclude that
8
>
<
A = A1 = 2 1
B = B1 = 21
>
: C=C =
1
d
t
Now consider the constraint
m
2 +1+
n n
2+
2 0; 1 + n
adt : This can be done in two
steps. Firstly, consider the constraint
d
t
0
(9)
With the solution of fA; B; Cg ; (9) can be rewritten and simpli…ed as
n
2+
it
) (1 + )
(2
1
In order for (10) to hold for all it
hold when evaluating at it
1
+
1
2+
m
ad +
+1
2 (1 + ) t
2 (1 + )
(10)
2 [0; 1 + ] and adt 2 [0; m] ; (10) must
= 0 and adt = 0 since the RHS of (10) achieves its
minimum in this case. With this …ndings, the necessary and su¢ cient condition
for (10) to hold is
m
+1
2 (1 + )
n
(11)
Secondly, consider the constraint
d
t
n + adt
1
(12)
With the solution of fA; B; Cg ; (12) can be rewritten as
1
2
it
1
1
+ adt +
2
m
2
+1+
2+
42
n
n
1
n + adt
(13)
Simplifying (13), we get
adt
n
2
m
2
adt
2+
it
2
For given adt ; in order for (14) to hold for all it
when evaluating at it
(14) at it
1
1
= 1+
2+
2
because
1
1
+1
(14)
2 [0; 1 + ], (14) must hold
< 0 for
2 (0; 1) : Evaluating
= 1 + ; we get
adt
n
2
m
2
adt
(2 + ) (1 + )
+1
2
(15)
Simplifying (15), we get
adt
n
adt
2
(4 + ) (1 + )
2
m
The RHS of (16) must be negative because the term
for
2 (0; 1) and the term
2
(16)
(4+ )(1+ )
2
is negative
m is larger or equal to zero for adt 2 [0; m] :
adt
The LHS of (16) is bureaucrats’ surcharge, which must be larger or equal to
zero by assumption. Therefore, there is a contradiction and (14) can not hold
for all it
1
2 [0; 1 + ] : This means for given adt ;
d
t
when it
1
d
t
gets a corner solution with
n + adt
=1
is larger than some threshold value {t
1,
which can be derived by
equalizing the two sides of (13), with
{t
1
=
2
1
+2
To avoid corner solution of
d
t
1
1 d
m + adt +
a
2
2 t
n
for all it
1
2 [0; 1 + ] ; {t
(17)
1
should be positive
and this can be transferred to the following condition
n
1 d
1
m + adt +
a
2
2 t
1
(18)
for all adt 2 [0; m] : This condition is equivalent to
n
1
m
2
1
43
(19)
where the RHS of (19) is derived by evaluating the RHS of (18) at adt = 0:
Comparing (11) and (19), we can …nd that (11) must hold if (19) holds
because of positive
1
2m
1
and m. At this moment, we can conjecture that if n
and if there is no Crown Prince problem, the incumbent dictator’s
decision rule about successor is
A adt ; it
1
=m
and the decision rule about tax rate is
(
m
1
1 d
2 +1+
+
i
+
a
d
t
1
t
2
2
2+
T at ; it 1 =
1 + n adt ;
n n
;
if
if
it
it
1
2 [0; {t 1 ]
2 ({t 1 ; 1 + ]
(20)
1
With the conjecture in (20), the citizens’decision rule of investment can be
derived in two steps.
Firstly, if 0
it
{t due to a high
r
t
2 ( rt ; 1] ; where {t is derived by moving
one period forward in (17):
{t =
2
+2
1
1
1 d
m + adt+1 +
a
2
2 t+1
n
(21)
and citizens’decision rule of investment in this case can be derived by plugging
in the values of A; B and C in (4):
d
t+1 ;
I
r
t
(2
=
r
t
)
2
(2
+
)
4
adt+1 +
(2
) 4
2n
m
4 ( + 2)
2
+4
(22)
r
t
The threshold level
can be derived by plugging the (21) into (22) for
investment and solving for the corresponding real tax rate with
r
t
=
2
2
Secondly, if {t < it
adt+1 +
4n + 4 + 2m
2n
2 ( + 2)
1+ due to
r
t
2 [0;
r
t];
rule of tax rate in (20),
d
t+1
=1+n
and
r
t+1
=1
44
adt
m
2
(23)
then according to incumbent’s
The citizens’decision rule of investment in this case can be derived by plugging
r
t+1
= 1 in (2) with
d
t+1 ;
I
r
t
r
t
=1
(24)
From (22) and (24), we can see the citizens’ decision rule of investment is
stepwise. Due to this, there can be a problem with the conjecture in (20). That
r
t
is, if the real tax rate at period t is smaller than
due to a low it
1,
then
expecting the next period actural rate will be a corner solution that equals to
one, the citizens’ decision rule will be the expression in (24). If we plug (24)
in the indirect utility function of the old incumbent and redo the guessing and
verifying, we can get another decision rule about tax rate of the incumbent,
which will be di¤erent than what we get (20). This will make the problem
complictated. In the following step, we will show that given n
r
t.
real tax rate at period t can never be smaller than
rule out of the possibility that there is a low it
= 0: If we can show the minimum of
is higher than
r min
t
d
t
d
t,
From (20), we can see that for given
1
, the
With this result, we can
which makes a real tax rate
r
t:
rate lower than
when it
1
1
2m
1
r
t;
r
t
then we can conclude that
can be solved by plugging it
1
=n
m
2
1 d
a +
2 t
r
t
=
(1
1
2m
) 1
which is denoted as
r min
t
n
+2
r min
,
t
will always be higher than
+1+
2+
By (23) and (25), the di¤erence between
r min
t
r
t;
achieve their minimum
n
and
n
r
t
1 d
a
2 t
r
t:
adt on both
= 0 into (20) and adding n
sides:
r min
t
r
t
and
(25)
is
2
2
adt+1
(26)
From our conjecture about the incumbent’s decision rule about sucessor’s
strength, we know adt+1 = m for all t
adt+1
0: Inspecting (26), we can see that given
= m; the minimum di¤erence between
r min
t
and
r
t
is obtained when
adt = m, and this minimum value is
r min
t
r
t min
=
(1
) 1
1
2m
+2
45
n
1
+ m (1
2
)
As we can see, given n
1
2m
1
,
r min
t
2 (0; 1) and m > 0;
r
t
must be positive and we can conclude that
>
r
t
r
t min
for all t. Therefore, the
conjecture about the incumbent’s decision rules in (20) is correct.
Given the dictator’s decision rules, the citizens’decision rule of investment
is
I
d
t+1 ;
r
t
=
(
r
t;
1
(2
)
r
t
2
+ 4 adt+1 +
)(4
(2
2n
m
4( +2)
2
+4)
where
r
t
7.1.2
The equilibrium law of motion of tax rates
;
if
r
t
if
r
t
2 [0;
r
t]
2 ( rt ; 1]
(27)
is de…ned in (23)
The equilibrium law of motion of
r
t
Firstly, we know that if
d
t
then it 2 ({t ; 1 + ],
r
t
Secondly, if
2(
d
t
and
2 [0;
r
t]
can be derived in two steps.
d
t
or equivalently
r
t
n + adt and
=1
r
t ; 1]
r
t
n + adt ;
= 1:
d
t
or equivalently
r
t
2 0;
2
n + adt ; 1
r
t
n + adt ; then
it 2 [0; {t ]. From the citizens’decision rule of investment, we have
it =
(2
)
r
t
2
+
(2
)
4
adt+1 +
(2
) 4
2n
m
4 ( + 2)
2
+4
(28)
From the incumbent’s decision rule about tax rate, we have
d
t+1
1
=
2
1
it + adt+1 +
2
m
2
+1+
2+
n
n
(29)
Plug (28) in (29) and rearrange, we have
d
t+1
Plug
r
t
=
=
d
t
r
t
2
+n
+
+2 d
8
at+1 +
4
4n + 2m
6n
4 ( + 2)
m
2
+8
adt in (30), we get the equilibrium law of motion of
this case
d
t+1
=
r
t
2
+
+2 d
8
at+1 +
4
4n + 2m
6n
4 ( + 2)
46
m
2
+8
(30)
d
t
in
The equilibrium law of motion
r
t
adt+1 on both
can be derived by adding n
sides of (30):
r
t+1
d
t+1
=
r
t
=
2
adt+1
+n
2
+
4
4n + 8 + 2m
2n
4 ( + 2)
adt+1 +
2
m
d
t
Now we can conclude that the equilibrium law of motion of
(
1 n + adt+1 ;
if
d
d
d
2
+n
a
+2
8
4n+2m
6n
m
+8
t+1 =
d
t
t
+ 4 at+1 +
; if
2
4( +2)
+8
is
d
t
d
t
d
t
2 0;
2
d
t;1
n + adt
(31)
2 [0;
r
t]
where
d
t
r
t
and the equilibrium law of motion of
(
1;
r
r
(4n+8
t+1 =
2 d
t
2 + 4 at+1 +
7.1.3
n + adt
r
t+1
is
+2m
2n
4( +2)
m
2
+8)
;
if
r
t
if
a
t
2 ( rt ; 1]
(32)
The dynamics of the economy
From the incumbent’s decision about the successor’s strength, we know that for
any ad0 2 [0; m] and i
1
1: Replacing adt and adt+1
2 [0; 1 + ] ; adt = m for all t
with m in (20), (27), (31) and (32) respectively, we get the decision rules of the
incumbent and the citizens, and the equilibrium laws of motion of
d
t
r
t
and
of
the equilibrium where the model economy in one period:
T adt ; it
I
d
t+1 ;
d
t+1
=
(
r
t
1
2
=
1
=
(
it
1
+1
+ +2
(m n + 1) ; if
1 + n m;
if
1
(2
)
2
a
t
+
(2
4
1
d
t +n
2
m
+
+2
4 m
)
m+
r
t;
(2
)(4
it
it
2n
m
4( +2)
n + m;
+
8
4n+2m
6n
4( +2)
47
1
2
1 2 [0; {]
2 ({; 1 + ]
+4)
;
m
+8
;
if
if
r
t
if
r
t
d
t
if
2
d
t
(33)
2(
2 [0;
]
r
2 ( ; 1]
(34)
2 [0;
r
r
r
n + m]
n + m; 1
(35)
n + m]
and
r
t+1
=
(
1;
r
t
+
2
where { and
r
2
m+
4
(4n+8 +2m 2n
4( +2)
m
2
+8)
;
if
r
t
if
r
t
2 [0;
r
]
(36)
r
2 ( ; 1]
are obtained by replacing adt and adt+1 with m in (21) and (23),
respectively:
{=
r
2
=
2
m+
2
+2
(1
n + m)
4n + 4 + 2m
2n
2 ( + 2)
From section 7.1.1, we know that
r
t
r
t
>
m
2
r
t
for all t; where
is the threshold
r
t+1
level of real tax rate at period t below which the next period real tax rate
achieves the corner solution of one. This result can help to characterize the
evolution of
r
t,
Firstly, if i
r
1
which can be done by three steps.
1
r
2
2 [0; {] ; then
>
r
, then
[0; {] ;
r
t
< 1 for all t
Secondly, if i
r
1
>
r
, then
({; 1 + ] ;
r
0
<
r
0
r
0
< 1: Since
>
r
r
1
,
< 1. Since
< 1: If we do this recurssively, we can know that for i
1
r
2
r
1
2
0:
2 ({; 1 + ] ; then
r
0
r
0
= 1: Since
>
r
r
1
,
< 1. Since
< 1: If we do this recurssively, we can know that for i
r
t
= 1 and
1
2
< 1 for all t > 0.
Thirdly, since the slope of the equilibrium law of motion of
r
t
is negative and smaller than one in absolute value, this means
r
t
is
1
2;
which
converges in
an oscillatory pattern to the steady state.
With exactly the same three steps, we can get the the evolution of
If i
1
2 [0; {] ; then
d
0
n + adt and
=1
d
t
to the steady state with
< 1 for all t
d
t
The steady state of
1: (ii) If i
1
2 ({; 1 + ] ; then
=
d
t+1
can be derived by setting
second part of (35) and solving the corresponding
d
ss
1
(4m
3 +6
4n + 4 + 4m
48
: (i)
converges in an oscillatory pattern
converges in an oscillatory pattern to the steady state with
d
t
d
t
d
ss
d
t
< 1 for all t
=
d
t
=
d
ss
d
t
0:
in the
:
4n + 4)
(37)
r
t
The steady state of
r
ss
can be derived by adding n
m on both sides of (37):
d
ss
=
+n m
1
(2n
3 ( + 2)
=
2m + 4 + m
n + 4)
The steady state of investment can be derived by plugging
r
ss
in the second
part of (34):
iss =
7.2
1
3 ( + 2)
2
+
+ 2 (m
n + 1)
Proof of Proposition 4
7.2.1
The decision rules
Like in the proof of Proposition 2, I will also use the strategy of “guess and
verify” to derive the incumbent’s decision rules about successor’s strength and
tax rate, and the citizens’decision rule about investment.
Start by guessing
that
d
t
2 0; 1
n+
d
t
= Dit
adt
1 +E
d
t +F
for all t and ignoring the constraint
: Given this guess, we have
d
t+1
= Dit + E
d
t+1
d
t+1
d
t+1
+F
and
r
t+1
=
+n
= Dit + (E
r
t+1
Plug the expression of
it
=
(1
=
1
r
t)
d
t
1)
d
t+1
+F +n
in (2), we have
+
r
t+1
1
n+
d
t
(38)
+
1
Dit
(E
1)
d
t+1
F
d
t
1 (E
1+ D
1)
d
t+1
F
n
n
Solve for it in (38), we have
it =
1
d
t
n+
+
49
(39)
Plug (39) in the indirect utility function of the old incumbent and rearrange,
we have
V od =
d
t
it
+
1
d
t
1
n + dt + (1
1+ D
F
n)
+
(1 E) dt+1
1+ D
(40)
d
t,
As we can see from (40), given the guess about the expression of
the
incumbent’s decisions of the successor’s strength and the tax rate can be separate
now. That is, if
d
t+1
choose
(1 E)
1+ D
= (1
d)
> 0; for any it
d
t,
the incumbent will choose
d
t ;and
1;
d
t,
the incumbent will …rstly
(1 E)
1+ D
since this maximizes his tax base; If
d
t+1
0;
= 0; since this maximizes his tax base. After
d
t
choosing the strength of the successor, the incumbent just chooses a
to ensure
his tax revenue is on the peak of La¤er curve.
(1 E)
1+ D
My following strategy is to suppose
> 0; plug in
d
t+1
= (1
d)
d
t
in (40), get a solution candidate fD1 ; E1 ; F1 g of fD; E; F g ; and then to verify
in this case,
(1 E1 )
1+ D1
(1 E)
1+ D
> 0:Then I suppose
d
t+1
0 and plug in
= 0 in
(40), get a solution candidate fD2 ; E2 ; F2 g of fD; E; F g and then to verify in
(1 E2 )
1+ D2
(1 E)
Suppose 1+
D
0 does not hold.
this case,
> 0; then
d
t+1
= (1
d
t
d)
(41)
Plug (41) in (40), we have
V od =
d
t
D1 + 1
(1 + D1 ) it
d
t
1
+ (1 +
First order condition with respect to
d
t
=
(1 + D1 ) it
1
+ (1 +
(1
d
t
E1 ) (1
2
(1
E1 ) (1
d))
d
t
+1+
n
F1
n
, we have
d))
d
t
+1+
n
F1
n
Since we guess
d
t
= Dit
1
+E
d
t
+F
Then the following equality must hold for all t
(1 + D1 ) it
1
+ (1 +
(1
E1 ) (1
2
d))
d
t
+1+
n
F1
n
= D1 it
d
1 +E1 t +F1
(42)
50
If (42) holds for all t, then the following equation system must hold
8
<
:
1+D1
= D1
2
1+ (1 E1 )(1 d)
= E1
2
1+
n F1
n
= F1
2
Solve the equation system, we get
8
D1 = 2 1
>
<
(1
E1 = 1+
2+ (1
>
: F = 1+ n
1
2+
In this case,
(1 E1 )
1+ D1
=
(2
)
2 (1 d)+4
> 0 for
d)
d)
n
2 (0; 1) and d 2 (0; 1): Therefore,
fD1 ; E1 ; F1 g is a solution of fD; E; F g :
Suppose
(1 E)
1+ D
d
t+1
0; then
= 0: Performing exactly the same steps
above as in deriving fD1 ; E1 ; F1 g ; we get
8
<
In this case,
that
(1 E2 )
1+ D2
(1 E2 )
1+ D2
=
1
2
D2 = 2 1
E2 = 12
:
n n
F2 =
2+
1
4
> 0 for
+1
2 (0; 1) : This contradicts our guess
0. Therefore, fD2 ; E2 ; F2 g is not the solution of fD; E; F g and
we conclude that
8
>
<
D = D1 = 2 1
1+ (1
E = E1 = 2+
(1
>
: F = F = 1+ n
1
2+
Now consider the constraint
d
t
2 0; 1 + n
d)
d)
n
adt : This can be done in two
steps. Firstly, consider the constraint
d
t
0
(43)
With the solution of fD; E; F g ; (43) can be rewritten as
1
2
it
1
+
1+
2+
(1
(1
d) d 1 +
n
at +
d)
2+
n
0
(44)
Simplifying (44), we get:
n
(2
2+
it
) (1 + )
1
+
(1 + (1 d)) (2 + ) d
a +1
(2 + (1 d)) ((1 + )) t
51
(45)
In order for (45) to hold for all it
hold when evaluating at it
1
2 [0; 1 + ] and adt 2 [0; m] ; (45) must
1
= 0 and adt = 0 since the RHS of (45) achieves its
minimum in this case. With this …ndings, the necessary and su¢ cient condition
for (45) to hold is
n
1
(46)
1
n + adt
(47)
Secondly, consider the constraint
d
t
With the solution of fD; E; F g ; (12) can be rewritten as
1
2
it
1
+
1+
2+
(1
(1
d) d 1 +
n
at +
d)
2+
n
1
n + adt
(48)
Simplifying (48), we get
+2
n
2
it
1
+2
ad + 1
2 + (1 d) t
For given adt ; in order for (49) to hold for all it
when evaluating at it
(49) at it
1
1
= 1+
because
2+
2
1
2 [0; 1 + ], (49) must hold
< 0 for
2 (0; 1) : Evaluating
= 1 + ; we get
n
2
+2
ad
2 + (1 d) t
( + 4)
The RHS of (50) must be negative because the term
for
(49)
2 (0; 1) and the term
+2
d
2+ (1 d) at
2
(50)
( + 4) is negative
is larger or equal to zero for adt 2
[0; m] : The LHS of (50) must be positive by assumption. Therefore, there is
a contradiction and (50) can not hold for all it
given
adt ;
d
t
1
2 [0; 1 + ] : This means for
gets a corner solution with
d
t
when it
1
n + adt
=1
is larger than some threshold value {t
equalizing the two sides of (48), with
52
1,
which can be derived by
{t
= (2
1
)
(1 n)
adt
+
2+
2 + (1
From (51), we can easily see that {t
adt
1
(51)
d)
> 0 because
2 (0; 1), n < 1 and
2 [0; m] by assumption.
At this moment, we can conjecture that if n < 1 and all the incumbents
choose a su¢ cient weak successor, the incumbent dictator’s decision rule about
successor is
A adt ; it
1
= (1
and the decision rule about tax rate is
(
(1 d) d
1
it 1 + 1+
d
2
2+ (1 d) at +
T at ; it 1 =
1 + n adt ;
1+
d)
d
t
n n
2+
;
if
if
it
it
1
2 [0; {t 1 ]
2 ({t 1 ; 1 + ]
(52)
1
With the conjecture in (52), the citizens’decision rule of investment can be
derived in two steps.
Firstly, if 0
it
{t due to a high
r
t
2 ( rt ; 1] ; where {t is derived by moving
one period forward in (51):
{t = (2
)
adt+1
(1 n)
+
2+
2 + (1 d)
(53)
and citizens’decision rule of investment in this case can be derived by plugging
in the values of D; E and F in (39):
I
d
t+1 ;
r
t
=
(2
)
2
The threshold level
r
t
r
t
+
(2
2 (2 +
)
(1
d
t+1
d))
+
(2
) (2
n + 2)
2 ( + 2)
(54)
can be derived by plugging the (53) into (54) for
investment and solving for the corresponding actural tax rate with
r
t
=
(
2)
(2 + (1
d
t+1
d))
+
53
1
(2n + 2
+2
n )
(55)
Secondly, if {t < it
1+
due to
r
t
r
t];
2 [0;
then according to the
conjecture of incumbent’s rule of tax rate in (52),
d
t+1
=1+n
adt
and
r
t+1
=1
The citizens’decision rule of investment in this case can be derived by plugging
r
t+1
= 1 in (2) with
d
t+1 ;
I
r
t
r
t
=1
(56)
From (54) and (56), we can see the citizens’ decision rule of investment is
stepwise. Due to this, there can be a problem with the conjecture in (52). That
r
t
is, if the real tax rate at period t is smaller than
due to a low it
1,
then
expecting the next period real rate will be a corner solution that equals to one,
the citizens’ decision rule will be the expression in (56). If we plug (56) in
the indirect utility function of the old incumbent and redo the guessing and
verifying, we can get another decision rule about tax rate of the incumbent,
which will be di¤erent than what we get in the …rst part of (52). This will make
the problem complictated.
r
t
In the following step, we will …gure out the condition under which
>
r
t
for all t. This can greatly simplify the anylysis.
d
t,
From the …rst part of (52), we can see that for given
their minimum when it
denoted as
r min
,
t
be higher than
r
t:
1
r
t;
then we can conclude that
can be solved by plugging it
r min
t
=
2+
1
(1
d)
adt +
By (55) and (57), the di¤erence between
r
t
=
r
t;
r
t
achieve
which is
will always
= 0 into the …rst part
adt on both sides:
of (52) and adding n
r min
t
1
r
t
and
= 0: If we can show the minimum of
is higher than
r min
t
d
t
2+
1
(1
d)
adt
1
(n +
+2
r min
t
(
2)
(2 + (1
54
and
d
t+1
d))
+
+ 1)
r
t
(57)
is
1
+2
(1
n)
(58)
From our conjecture about the incumbent’s decision rule about sucessor’s
strength, we know adt+1 = (1
d) adt for all t
0: Plug adt+1 = (1
d) adt into
(58), we get
r min
t
r
t
(2
) (1 d) 1 d 1
at +
(1
(2 + (1 d))
+2
=
Examing (59), we can see that if (2
1
2
d <
r
t
>
r min
t
, then
r
t
r
t
) (1
d)
n)
(59)
1 > 0 or equivalently,
1
2
must be positive. This means if d <
, then
for all t and we can conclude that the incumbent’s decision rules are
A adt ; it
= (1
1
d
t
d)
and
T
adt ; it 1
=
(
1
it
2
1
1+ (1 d) d
1+
n n
2+ (1 d) at +
2+
1 + n adt ;
+
;
if
if
it
it
1
2 [0; {t 1 ]
2 ({t 1 ; 1 + ]
1
where
{t
= (2
1
(1 n)
adt
+
2+
2 + (1
)
d)
Given the dictator’s decision rules, the citizens’decision rule of investment
is
I
d
t+1 ;
r
t
=
(
1
(2
)
r
t
2
+
d
t+1
r
t;
(2
)
2(2+ (1 d))
r
t
r
t
if
+
(2
)(2
n +2)
;
2( +2)
if
2 [0;
r
t]
2 ( rt ; 1]
where
r
t
7.2.2
=
(
2)
(2 + (1
d
t+1
d))
1
(2n + 2
+2
+
n )
The equilibrium law of motion of tax rates
The equilibrium law of motion of
r
t
Firstly, we know that if
d
t+1
then it 2 ({t ; 1 + ],
Secondly, if
r
t
=1
r
t
2 [0;
d
t
and
r
t]
can be derived in two steps.
or equivalently
n + adt+1 and
2 ( rt ; 1] or equivalently
d
t
r
t+1
2
d
t
r
t
2 0;
n + adt ;
= 1:
r
t
n + adt ; 1
n + adt ; then
it 2 [0; {t ]. From the citizens’decision rule of investment, we have
it =
(2
)
2
r
t
+
(2
2 (2 +
)
(1
55
d
t+1
d))
+
(2
) (2
n + 2)
2 ( + 2)
(60)
From the incumbent’s decision rule about tax rate, we have
d
t+1
=
1
2
it +
1+
2+
(1
(1
d) d
1+
n
at+1 +
d)
2+
n
(61)
Plug (60) in (61) and rearrange, we have
d
t+1
r
t
=
2
r
t
Plug
2 + 2 (1 d) +
2 (2 + (1 d))
+
=
d
t
+n
d
t+1 +
1
(4
2 ( + 2)
2n
3n + 4) (62)
adt in (62), we get the equilibrium law of motion of
d
t
in
this case
d
t+1
d
t
=
+n
2
adt
+
2 + 2 (1 d) +
2 (2 + (1 d))
d
t+1 +
1
(4
2 ( + 2)
2n
3n + 4)
(63)
The equilibrium law of motion
a
t
adt+1 on both
can be derived by adding n
sides of (63):
r
t+1
=
r
t
2
2
2 (2 + (1
d
t+1
d))
+
1
(2n + 4
2 ( + 2)
n + 4)
(64)
Now we can conclude that the equilibrium law of motion of dt is
(
if
1i n + adt ;
h
d
d
d
=
+n
a
2+2
(1
d)+
1
d
t+1
t
t
+ 2(2+ (1 d))
2n 3n + 4) ; if
t+1 + 2( +2) (4
2
(65)
d
t
d
t
where
d
t
r
t
n + adt
and the equilibrium law of motion of rt+1 is
(
1;
h
i
r
r
2
1
d
t+1 =
t
t+1 + 2( +2) (2n + 4
2
2(2+ (1 d))
7.2.3
if
n + 4) ;
if
r
t
r
t
2 [0;
2 ( rt ; 1]
(66)
The dynamics of the economy
From section 7.2.1, we know that
r
t
>
r
t
for all t; where
r
t
is the threshold
level of real tax rate at period t below which the next period real tax rate
56
r
t]
r
t+1
2
2 0;
d
t;1
d
t
n + adt
achieves the corner solution of one. This result can help to characterize the
evolution of
r
t,
Firstly, if i
r
1
which can be done by three steps.
1
r
2
>
r
, then
[0; {] ;
r
t
< 1 for all t
Secondly, if i
r
1
r
>
, then
({; 1 + ] ;
r
0
r
2
r
2 [0; {] ; then
<
r
0
< 1: Since
r
0
r
>
,
r
1
< 1. Since
< 1: If we do this recurssively, we can know that for i
1
1
2
0:
2 ({; 1 + ] ; then
r
0
= 1. Since
r
0
r
>
r
1
,
< 1. Since
< 1: If we do this recurssively, we can know that for i
r
t
= 1 and
1
2
< 1 for all t > 0.
Thirdly, since the slope of the equilibrium law of motion of
r
t
is negative and smaller than one in absolute value, this means
r
t
is
1
2;
which
converges in
an oscillatory pattern to the steady state.
d
t
With exactly the same three steps, we can get the the evolution of
If i
1
2 [0; {] ; then
d
0
n + adt and
=1
to the steady state with
d
t
d
t
< 1 for all t
: (i)
converges in an oscillatory pattern
1: (ii) If i
1
2 ({; 1 + ] ; then
converges in an oscillatory pattern to the steady state with
d
t
< 1 for all t
d
t
0:
The steady state of the model economy can be solved by the following steps:
Firstly, since the incumbent’s decision rule about successor’s strength is
A adt ; it
then when t ! 1;
d
t
! 0:
d
t
Secondly, the steady state of
d
t
and
d
ss
=
d
t+1
= (1
d)
d
t;
can be derived by setting
d
t+1
=
d
t
=
d
ss
= 0 in the second part of (65) and solving the corresponding
:
d
ss
Thirdly, the steady state of
d
t
and
r
ss
1
=
d
t+1
=
r
t
4 ( + 1) (1 n)
3 ( + 2)
can be derived by setting
r
t+1
=
r
t
=
r
ss
= 0 in the second part of (66) and solving the corresponding
:
r
ss
=
1
(2n + 4
3 ( + 2)
57
n + 4)
(67)
The steady state of investment can be derived by plugging (67) and
d
t+1
d
t
=
= 0 in the second part of (34):
1
3 ( + 2)
iss =
7.3
2
+
+ 2 (1
n)
Proof of Proposition 5
If the incumbent at period t does not choose a su¢ ciently weak successor, obviously, he will choose a successor with strength
because if he chooses a successor with
d
t+1
d
t+1
2 (1
d) adt ; min (1 + d) adt ; m
> (1 + d) adt ; the incumbent will lose
for sure.
As we already know, given the timing of the game, the citizens’investment
rule in this case is the same as in the case where all the incumbents choose suf…ciently weak successor. From Proposition 4, we know the citizens’investment
rule is
I
d
t+1 ;
r
t
=
(
(2
)
2
r
t
+
r
1
t;
(2
) d
(2
t+1
2(2+ (1 d)) +
r
t
r
t
if
)(2
n +2)
;
2( +2)
if
2 [0;
r
t]
2 ( rt ; 1]
(68)
where
r
t
=
(
2)
(2 + (1
d
t+1
d))
+
1
(2n + 2
+2
n )
Also, since there is only one period deviation at period t, all the future
dictators will choose a su¢ ciently weak successor. This means at period t + 1,
(
m
+1+
n n
1
it + 12 adt+1 + 2
; if
it 2 [0; {t ]
d
2
2+
=
(69)
t+1
if it 2 ({t ; 1 + ]
1 + n adt+1 ;
where (69) is derived by moving one period forward in the incumbent’s decision
rule about tax rate in Proposition 4. Note that since in this case, the incumbent
may choose a successor with strength higher than (1
d) adt ; this implies the
it can be larger than in the case where the successor is su¢ ciently weak as it
increases when
than
r
t.
d
t+1
goes up (see the second part of (68), and
This can make
r
t
r
t
can be higher
have a corner solution of one. Therefore, unlike
58
in the case where all the incumbents choose a su¢ ciently weak successor and
r
t
>
r
t;
given d <
[0; {t ] for all
d
t+1
1
2
; there can be the following two possibilities: (i) it 2
2 (1
it 2 [{t ; 1 + ] for some
d) adt ; min (1 + d) adt ; m
d
t+1
d) adt ; min (1 + d) adt ; m
2 (1
d
t+1
it 2 [0; {t ] for all
7.3.1
: This means
r
t+1
< 1:(ii)
.
d) adt ; min (1 + d) adt ; m
2 (1
In this case,
r
t+1
<1
and
d
t+1
=
m
2
1
it + adt+1 +
2
1
2
+1+
2+
n
n
(70)
Plug (70) in the indirect utility function of the incumbent, we have
od
Vnsw
=
1
2
d
t
it
(2
1
d
t
)
2
+
) adt
(2
2
+
(2
2 (2 +
)
(1
d
t+1
d))
1
(n
+2
1)
(71)
From (71), we can see the dictator will choose
d
t+1
= min (1 + d) adt ; m
(72)
since this maximizes his tax base.
Plug (72) in (72), we get
od
Vnsw
=
1
2
d
t
(2
it
)
1
2
d
t
+N
+
1
2
(73)
where
) adt
(2
N
2
+
(2
) min (1 + d) adt ; m
2 (2 + (1 d))
d
t
First order condition with respect to
d
t
=
it
1
1
(n
+2
1)
2
+
+2
in (73), we get
+N
2
and
2
od
Vnsw
=
(it 1 + N )
1
+
4 (2
)
2
59
(74)
2
+
+2
+
1
2
7.3.2
d
t+1
it 2 [{t ; 1 + ] for some
d) adt ; min (1 + d) adt ; m
2 (1
As we can see from the citizens’investment rule, it increases as
can be that at some thresold level
which it
{t .
successor with
d
t
d
t+1
increases. It
d) adt ; min (1 + d) adt ; m
2 (1
above
In this case, the incumbent will be indi¤erent in choosing a
d
t+1
2
d
t ; min
(1 + d) adt ; m
: According to the incumbents’
decision rule about tax rate,
d
t+1
adt+1
=1+n
and
r
t+1
=1
In this case, the citizens’investment at period t is
r
t
it = 1
(75)
Pliu (75) into the indirect utility function of the incumbent, we have
od
Vnsw
=
1
2
d
t
it
1
d
t
+Q
+
1
2
(76)
where
Q
n + adt
1
d
t
First order condition with respect to
d
t
=
it
+Q
2
1
and
od
Vnsw
=
7.3.3
(it
in (76), we have
2
1
+ Q)
1
+
4
2
(77)
The SPNE condition
The condition that ensures the Markov Perfect Equilibrium where all the incumbent dictators choose su¢ cient weak successors is a Subgame Perfect Equilibrium without retrictions on the successor’s strength is the condition that makes
the incumbent’s indirect utility when choosing an su¢ ciently weak sussessor,
od
higher than Vnsw
for any it
1
and
d
t
in (74) and (77).
60
To get the indirect utility of the old incumbent dictator when he chooses
a su¢ ciently weak successor and all the past and future dictators choose su¢ ciently weak successor, we …rstly have
od
Vsw
=
d
t
r
t
From Proposition 4, we know that
is not the …rst period, then
(2
it =
)
2
r
t
+
r
t+1
(it
1
+ it )
(78)
< 1 for all t > 0:Therefore, if period t + 1
< 1 and
(2
2 (2 +
d
t+1
)
(1
d))
2
(2
2 +4
+
n + 2)
(79)
and
d
t+1
= (1
d
t
d)
(80)
Plug (79) and (80) in (78), and use the fact that
r
t
=
d
t +n
d
t,
we get the
indirect utility function of the old incumbent if he chooses a su¢ ciently weak
successor:
od
Vsw
=
d
t
it
(2
d
t
)
1
2
+M
(81)
where
M
(2
)
2
1+
1
2+
d
(1 d)
d
t
First order condition with respect to
d
t
=
1
(n
+2
adt
it
1
1)
2
+
+2
in (81), we have
+M
2
and
2
od
Vsw
=
(it 1 + M )
2 (2
)
od
od
If we want to derive the condition to ensure Vsw
> Vnsw
; we must show
under what conditions,
2
2
(it 1 + M )
(it 1 + N )
1
>
+
2 (2
)
4 (2
)
2
and
2
(it 1 + M )
(it
>
2 (2
)
61
(82)
2
1
+ Q)
1
+
4
2
(83)
hold for all it
1
2 [0; 1 + ] and adt 2 [0; m].
In the following steps, we just derive the conditions that ensure the lower
bound of the LHS of (82) and (83) are larger than the RHS of (82) and (83)
respectively.
Firstly, if we evaluate
(it 1 +M )2
2(2
)
at it
= 0 and adt = 0; we get the lower
1
bound of the LHS of (82):
(2
2
) (1
2
n) (1 + )
(84)
2
2 ( + 2)
Secondly, if we evaluate
(it 1 +N )2
4(2
)
at it
and adt = m, we get the
= 1+
1
upper bound of the RHS of (82):
(2
)
3+ (1 d)
4+2(1 d) m
+
(1 n)(1+ )
+2
2
+
4
Thirdly, if we evaluate
(it
1 +Q)
4
2
+
1
2
at it
1
= 1+
1
2
(85)
and adt = m, we get
the upper bound of the RHS of (83):
(2 +
2
n + m)
1
+
4
2
(86)
Foruthly, with some simple calsulation, we can see if
0
2
2
2
(1 d)
( + 2) 3+
4+2(1 d) m +
B 2 (1 n) (1 + )
< (2
)@
2
2 ( + 2)
(1 n)(1+ )
+2
2
holds, then (84) must be larger than (85) and if
<
2 (2
) (1
2
2
2
n) (1 + )
1
C
A
2
( + 2) (2 +
n + m)
2
2 ( + 2)
holds, then (84) must be larger than (86).
Now, we can conclude that if
8
2(1 n)2 (1+ )2 (
>
< (2
)
< min
>
2(2
)(1 n)2 (1+
:
3+ (1
+2)2 ( 4+2(1
d)
(1
m+
d)
n)(1+ )
+2
2
)
2( +2)2
2
) ( +2)2 (2+
2( +2)2
n+m)2
9
>
; =
>
;
holds, then it is optimal for the incumbent at period to choose a su¢ ciently
weak successor, giving all the past and future dictators do the same.
62