Why was Japan more successful than China in maintaining its independence?

Introduction

Why was Japan the first non-Western nation to industrialize? Why did China take longer to modernize? At first glance, China’s later industrialization appears puzzling. Unified growth theory suggests that the transition from stagnation to growth is driven by the positive interaction between population expansion and technological progress and its impact on the demand for human capital and the onset of the demographic transition [Galor 2005, 2011]. All else equal, China, one of the most technologically advanced and certainly the most populous nation in the world throughout most of history, should be an early industrializer. While differences in geographical endowment, institutions, culture, and diversity could help explain the Great Divergence between China and Europe [Jones 1981; Landes 1998; Pomeranz 2000; Ashraf and Galor 2013], these differences seem unsatisfactory in explaining the economic divergence between China and Japan.

Traditional accounts typically attribute Japan’s earlier industrialization to the Meiji Restoration. According to this view, Qing China [1644–1911] and Tokugawa Japan [1600–1868] were both governed by despotic rulers who were uninterested in promoting economic growth.Footnote 1 Their paths diverged only after 1868, when the Tokugawa regime was overthrown and the new Meiji government introduced drastic reforms that transformed Japan. As Beasley [1972] put it,

During the middle decades of the nineteenth century China and Japan both faced pressure from an intrusive, expanding West. ... Emotionally and intellectually, Chinese and Japanese reacted to the threat in similar ways. ... Yet they differed greatly in the kind of actions that this response induced. ... The Meiji Restoration is at the heart of this contrast, since it was the process by which Japan acquired a leadership committed to reform and able to enforce it. For Japan, therefore, the Restoration has something of the significance that the English Revolution has for England or the French Revolution for France; it is the point from which modern history can be said to begin.

Recent reassessments have put the Chinese and Japanese economies on the eve of the modern age in better standing. It has been shown that, like Western Europe, China and Japan experienced widespread commercialization and proto-industrialization during the early modern period [Pomeranz 2000]. However, the revisionist view, too, tends to play down the differences between pre-1850 China and Japan, and focus instead on their similarities [Pomeranz 2000; He 2013].

Indeed, early modern China and Japan had much in common. Both depended heavily on small-scale, labor-intensive, and rice-based agriculture. Both were ruled by stable and sophisticated governments long before the arrival of the West. Furthermore, they shared a common cultural, institutional, and technological heritage. As a result of active cultural borrowing from China, Tokugawa Japan was also deeply influenced by Confucianism. Chinese administrative codes played an important role in shaping the way that the Tokugawa shogunate was run [Jansen 1992]. Existing evidence suggests that living standards in China and Japan were comparable during this period [Maddison 2001; Baten et al. 2010; Allen et al. 2011; Broadberry 2013].Footnote 2

We point to an important empirical observation that fits neither traditional nor revisionist perspectives, however. As Figure 1 illustrates, from 1650 to 1850, tax revenue per capita was significantly higher in Tokugawa Japan than in Qing China, and the gap widened over time.Footnote 3 In our estimates, the Chinese state’s annual revenue on the eve of the Opium War [1839–1842] was equivalent to 2 % of its national income at the maximum, while the comparable number for the Tokugawa shogunate was more than 15 %.Footnote 4

Fig. 1

Per Capita Tax Revenue in China and Japan. Sources: Shogunate’s land tax from Ohno [1996]; Japan’s population estimates from Hayami and Miyamoto [1988]; China’s tax revenues from Sng [2014]; China’s population estimates from Perkins [1969]

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What were the reasons for these diverging revenue trends? The existing literature offers two hypotheses for China’s low tax revenue in general: the absence of warfare and the ideology of benevolence. Economic historians have shown that warfare was a major driver for European states to expand fiscal capacity [Hoffman and Rosenthal 1997; O’Brien 2005; Dincecco et al. 2011; Gennaioli and Voth 2011]. In this view, the absence of interstate competition in China and the resulting low fiscal demand were the primary reasons for low taxation in China [Rosenthal and Wong 2011]. Alternatively, China historians have argued that low taxation was mainly a reflection of the Confucian ideology of “benevolent rule” [Elliott 2009; Rowe 2009]. However, neither of these hypotheses can fully explain the diverging trends in China and Japan because Tokugawa Japan, too, experienced no interstate competition and shared the Confucian ideology of benevolence.Footnote 5 If anything, the European experience suggests that China, not Japan, should have developed a higher fiscal capacity because it was exposed to military threats from Inner Asia.

In this paper we focus on geography as a primary factor. China was a sprawling land empire with vast inner frontiers, while Japan was a small island nation. We propose that the difference in their geographical size and heterogeneity led to a much more acute problem of political control in the former than in the latter.Footnote 6 In pursuing our research, we follow the methodology of comparative and historical institutional analysis proposed by Greif [1998]. That is, we first develop a context-specific model based on historical details to theoretically examine the nature of the problems that the rulers in China and Japan faced and then check its implications using comparative historical evidence.

Between 1650 and 1850, both nations were ruled by stable dictatorships. Following Olson [1993], we model stable dictators as “stationary bandits” who understand that excessive exaction in the short run would be counterproductive in the long run.Footnote 7 However, the ruler’s encompassing interest is by itself insufficient to guarantee good governance. Because dictators cannot rule alone and have to rely on agents to govern, a principal–agent problem is inherent in dictatorships.Footnote 8 Unless the interests of the ruler and the agents are well-aligned, in the absence of perfect monitoring, the agents tend to pursue their self-interest at the ruler’s expense. For example, they may extort the taxpayers and thereby increase the likelihood of rebellion.

We hypothesize that in a stable dictatorship, agency problems increase with its geographical size and heterogeneity. Given premodern information technologies, it is costly for the ruler of a large domain to monitor the agents closely. This gives the agents strong incentives to extort the taxpayers. To prevent overexploitation that could foment rebellion, the ruler has to keep taxes low and government small. By contrast, in a smaller domain, lower monitoring costs allow the ruler to impose heavier taxes without risking popular resistance.

If the sole purpose of taxation is to support the consumption of the ruling class, whether it enriches the ruler or his agents will not matter to the taxpayers. However, unlike corruption, taxation is rarely a pure rent-seeking activity. The ruler, as the owner of his domain, may use the tax receipts to invest in public goods to keep his property productive. If so, the competition between the ruler and the agents over the economic surplus may have an impact on social welfare, especially in the long run.

To formalize our hypothesis, we build a dynamic principal-agent model and analyze optimal taxation and public goods provision in a stable dictatorship. The ruler taxes the peasants through agents and invests part of the tax revenue in a local public good that protects the economy from exogenous shocks [e.g., natural disasters]. If the ruler under-invests in the public good, the risk of a large shock destroying the economy increases.

The static predictions of the model are straightforward: holding monitoring technology constant, as the geographical size of the ruler’s domain increases, bureaucratic expropriation worsens and per capita tax revenue falls due to managerial diseconomies of scale.

New insights come from the dynamic implications. While one may expect economic expansion to generate more tax revenues and higher public good investments, this is not always the case. The model predicts that economic expansion could actually hurt the ruler because it also exacerbates agency problems. When monitoring cost is sufficiently high, bureaucratic expropriation will outpace economic expansion. It is only when monitoring cost is low that economic change is likely to bring net benefits to the ruler as well as the population.

Our model provides a potential explanation for the tax revenue dynamics in China and Japan documented in Fig. 1. To further check its implications, we examine the provision of local public goods [coinage, transportation network, urban management, forest protection, famine relief] in the two regimes. In line with the model’s prediction, we find that, compared to the Chinese emperor, the Tokugawa shogun displayed a greater capability to provide these public goods over a longer period of time.

We take the size of domains in China and Japan as exogenous in our analysis. Given the high agency costs, one may ask if China’s vast size was ever optimal. In a broader framework, such as Alesina and Spolaore [1997], the ruler determines the size of his domain by balancing the accompanying costs and benefits, where agency costs are just one such factor. In the case of China, we conjecture that the benefits of political integration—peace and risk sharing among contiguous regions—outweighed high agency costs, thereby justifying its size. We do not model this, however, to keep the scope of our analysis manageable.Footnote 9

To our knowledge, this study is the first comparative analysis of state capacity in preindustrial Asia.Footnote 10 The European experience indicates that most states had a strong fiscal system in place before industrializing [Dincecco 2011; Johnson and Koyama 2014a, b]. Indeed, there is a growing body of theoretical and empirical research highlighting the importance of state capacity in facilitating modern economic growth [Acemoglu 2005; Besley and Persson 2009, 2013; Dincecco and Prado 2012; Dincecco and Katz 2014]. Studies also show that a proactive state could accelerate the transition from stagnation to growth by implementing policies that promote human capital formation [Doepke 2004; Doepke and Zilibotti 2005; Galor and Moav 2006; Galor et al. 2009].Footnote 11 In light of these works, our finding of an increasingly weak state in China in contrast to Japan might help explain the puzzle of China’s late industrialization.

This paper builds directly upon Sng [2014], who studies the impact of geographical size on the principal–agent problem in late imperial China. Our work significantly extends his model by incorporating public goods provision and offers new comparative empirical evidence by bringing Japan into the picture. Our approach is complementary to Brandt et al. [2014], who provide a comprehensive survey of the long-run evolution of the Chinese political economy since the tenth century.

Importantly, four recent contributions also explore the impact of geographical proximity and size on the quality of political and corporate governance. Stasavage [2011] finds that in preindustrial Europe, high communication and travel costs prevented representative assemblies in large polities from convening regularly and functioning effectively. Using contemporary data from 127 countries, Olsson and Hansson [2011] detect strong negative effects of territorial size on the rule of law. Giroud [2013] shows that a reduction of travel time [a proxy for monitoring costs] between company headquarters and plants has positive effects on plant-level productivity and profits. Campante and Do [2014] provide strong evidence that isolated capital cities in US states are associated with lower accountability, greater levels of corruption, and worse public goods provision. These studies show that distance and size are a challenge to good governance not only in premodern regimes in Asia and Europe, but also in modern states and corporations.

The rest of the paper is organized as follows: Section. 2 provides the historical background. Section 3 presents the model and derives predictions. Section 4 provides comparative historical evidence. Section 5 concludes.

Historical background

In this section, we compare the geography, administrative structure, and system of tax collection in Qing China and Tokugawa Japan to motivate our theoretical model.

Geography

Tokugawa Japan was an archipelago comprising three main islands,Footnote 12 while China was a continental empire [Fig. 2]. At its peak, China under the Qing dynasty [1644–1911] controlled a landmass larger than China or the United States today. Even if we disregard the thinly populated Inner Asian borderlands, the region known as China proper—the 18 provinces between the Great Wall and the South China Sea, which accounted for about 98 % of the empire’s population—was still 12 times the size of Tokugawa Japan.

Fig. 2

Early modern China and Japan. Source: CHGIS, Version 4, Cambridge: Harvard Yenching Institute, January 2007

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If information transmission posed any challenge to effective public administration, this challenge was clearly more acute in China than in Japan. To send a high-priority official document from Beijing 1000 kilometers down south to Shanghai would take up to 10 days [Xie 2002]. By contrast, a similar trip between Japan’s two biggest cities, Edo [Tokyo] and Osaka, about 520 km apart, would only require 4 days [Nakane and Oishi 1990]. It is also worth noting that no one in Japan lived more than 120 kilometers from the sea, which offered a cheap mode of transportation in an age before railroads.

Administrative structure

Both China and Japan were ruled by a succession of stable dictators between 1650 and 1850. However, while China was ruled by one dictator—the emperor of the Qing dynasty—during this period, multiple dictatorships coexisted in Japan.

Nominally, Japan was led by the shogun of the Tokugawa house, who controlled 15 % of the arable land [Fig. 3]. The bulk of the remaining land was divided into 260-odd mutually exclusive domains, each headed by a daimyo [local lord].Footnote 13 While a daimyo had to swear allegiance to the shogun and subject himself to a system of controls aimed to prevent dissent, he retained virtually complete autonomy over his domain.Footnote 14 As such, instead of treating Tokugawa Japan as a unified but decentralized empire, we interpret it as a league of dictatorships and treat each daimyo as a dictator.Footnote 15 We focus primarily on the shogunate, for which historical records are most abundant, and compare it with China proper.Footnote 16

Fig. 3

Tokugawa Japan in 1664. Source: China Historical GIS Project,“Tokugawa Japan GIS, Demo Version.” Feb 2004

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The systems of territorial administration in China proper and the shogunate were broadly similar. To administer his domain, the Qing emperor structured his bureaucracy into four layers [center–province–prefecture–local]. China proper was organized into 18 provinces; each province was then divided into several prefectures, and each prefecture into several counties. The responsibility of local administration fell on the county, which sat at the bottom of the bureaucratic hierarchy. Each county was headed by a magistrate, whose term was usually limited to 3 years [Ch’u 1962].

In the Tokugawa shogunate, local administration was also carried out by nonhereditary magistrates [daikan]. Like his Chinese counterpart, the shogunate magistrate was subjected to rotation.Footnote 17 They also shared a similar scope of responsibilities. In both regimes, the magistrate was expected to focus on two tasks: collection of taxes and adjudication of disputes [Wang 1985; Totman 1967].

There were only two layers of government [center–local] in the shogunate. At any one time, 40–50 magistrates reported directly to the shogun’s cabinet [Totman 1967]. By contrast, there were about 1,500 county-level jurisdictions and hence 1,500 magistrates in Qing China. A shogunate magistrate typically governed 50,000–100,000 people, while the size of an average Chinese county ranged from 100,000 [in 1700] to 300,000 [in 1850].

Monitoring system

Because China proper was almost 90 times bigger than the shogunate domain, it had a greater number of administrative officials and a longer bureaucratic chain of command. This implies that unless the Chinese emperor possessed superior monitoring technologies, it would be more difficult for him than for the shogun to monitor local officials. There is little evidence to suggest that monitoring technologies were better in China, however. In fact, the two regimes instituted similar monitoring systems that combined top-down, parallel, and bottom-up monitoring.

In top-down monitoring, local officials were supervised by higher-ranking officials within the same bureaucratic hierarchy. In the shogunate, the magistrate’s office was periodically audited by the Finance Office in Edo [Totman 1967, p. 76]. In China, the administration conducted a grand review once every three years to evaluate the magistrate’s performance and mete out reward or punishment accordingly [Watt 1977].

Top-down monitoring, however, could be ineffective in the presence of bureaucratic patronage networks. To prevent this, the Chinese emperor established an independent surveillance agency known as the Censorate to detect bureaucratic malpractices and report them to the emperor [Feuerwerker 1976]. Likewise, the shogun sent out censors to keep an eye on the local administration [Totman 1967; Nakane and Oishi 1990].

Finally, to carry out bottom-up monitoring, both regimes adopted petition systems. The system had a long tradition in China, where it had been in place since the seventh century [Ocko 1988; Fang 2009]. In Japan, it was not until 1720 that the shogun set up petition boxes in major cities and permitted the public to make suggestions for better governance or to report misconduct and abuse of power by shogunate officials. The petitions were sent directly to the shogun for his review. Over 75 % of large local domains instituted similar systems [Ohira 2003].

In both cases, the petition system was costly to implement, as it typically generated a large number of petitions including irrelevant requests and false accusations. In the Tokugawa shogunate, each petition was investigated and petitioners were punished for misstatements. The system functioned reasonably well and was maintained until the end of the Tokugawa period [Ohira 2003].Footnote 18

By contrast, the sheer size of the Chinese population made it extremely costly for the Qing rulers to verify the authenticity of every petition. Both the emperors Qianlong [r. 1736–1795] and Jiaqing [r. 1796–1820] initially encouraged petitions from their subjects but quickly reversed their policies after receiving a flood of complaints that they could not possibly deal with [Fang 2009]. The system did not function as intended, and some complainants resorted to extreme measures, such as committing suicide outside the palace gates, to attract the emperor’s attention to their grievances. In other words, although both China and Japan used similar systems of bottom-up monitoring to check corruption, it was less effective in China due to its much greater size and population.

The rulers in China and Japan were concerned about the well-being and grievances of their subjects for both ideological and practical reasons. Because Confucianism demanded rulers to treat their subjects benevolently, it legitimized popular resistance against an oppressive ruler.Footnote 19 This fear of a violent rebellion served as a constraint on dictators in both China and Japan and gave them an incentive to prevent the overexploitation of their subjects.

The system of tax collection

Land taxation was the most important source of government revenue in both Qing China and Tokugawa Japan. Both economies depended heavily on small-scale, labor-intensive agriculture. Every land-holding household was obligated to pay the land tax, the amount of which was determined based on the size and quality of the land the family held [Ch’u 1962; Nakane and Oishi 1990]. In the case of Japan, the fiscal base was measured in rice, the primary staple crop nationwide. Fields, forests, residential lands, mines, and fishing grounds were also assessed and taxed in terms of rice [Nishikawa 1985, pp. 23–24]. If rice were not the main crop cultivated, then part of the tax would be levied in cash at a conversion rate set by the lord.

By contrast, regional diversity necessitated the denomination and collection of taxes in a variety of crops and metals in China. Although most taxes had been monetized by the seventeen century, the peasants still had to pay part of their land taxes in kind, which, depending on the region, could be rice, wheat, millet, barley, sorghum, beans, or other staple crops. Furthermore, it was common for the portion of the land tax denominated in silver to be paid in copper coins when and where silver was scarce [Ch’u 1962]. In such cases, commutation rates were set by magistrates based on local conditions. This high heterogeneity created great difficulties for the imperial court to monitor the over-collection of taxes by the county administration [Ch’u 1962; Zelin 1984].

In Qing China, the primary unit of taxation was the household, whereas in Tokugawa Japan, it was the village instead of the household. Under the village contract system [murauke], the Japanese rulers levied the land tax on each village based on its total assessed yield. Village leaders were in charge of assigning and collecting taxes from individual households and transferring the sum to the magistrate. Moreover, households in the same village were made collectively responsible for the payment of taxes. This arrangement reduced the frequency of contact between the magistrate and individual peasants and, therefore, limited the opportunities for tax officials to abuse power. Indeed, the magistrate rarely showed up in the villages except for annual inspections, and villages retained a high degree of autonomy in running their affairs in Japan [Walthall 1991].

For this system to work, it was necessary that village communities remained tightly knit to facilitate mutual monitoring and discourage free riding. To restrict geographical mobility, the shogunate and local lords mandated every village to keep a household registry and required their subjects to obtain permission before changing residency or traveling.

We do not model the village contract system in Japan in the next section as doing so would further reduce the monitoring costs for the Japanese rulers and strengthen our main results. It should be noted, however, that the village contract system was not a uniquely Japanese system. In fact, China had instituted a similar system during the Ming dynasty [1368–1644]. The system eventually unraveled, however, as the potential for migration given China’s vast inner frontiers made it difficult to maintain tightly knit communities that were necessary to implement collective responsibility.Footnote 20 By contrast, the village contract system was firmly institutionalized in Tokugawa Japan. Even though it was abolished by the Meiji government with the introduction of a new land tax system, tax collection was delegated to local communities that continued to use a collective responsibility system well into the 1930s [Sakane 2011b].

We also do not incorporate taxpayer heterogeneity in our model. In China, taxpayers could be broadly classified into two groups: the gentry and the peasants. Historical studies suggest that unlike ordinary peasants, the gentry were rarely subjected to bureaucratic extortion because of their political connections [Ch’u 1962; Watt 1977].Footnote 21 Some gentry took advantage of their sheltered position to act as tax farmers, earning extralegal income by paying taxes on the peasants’ behalf and charging for the service.

There was heterogeneity among taxpayers in Japan too, although not to the extent observed in China. Under the village contract system, wealthy peasants were typically appointed as village leaders. Some village leaders took advantage of their position and colluded with the magistrate to extort the villagers [Nishizawa 2004].

Based on these observations, we consider local elites [the gentry and the village leaders] as tax intermediaries instead of taxpayers and incorporate them as a constituent of the tax agent.Footnote 22

The model

Motivated by the historical observations, in this section we develop a formal model to study the impact of geographical size on a ruler’s capacity to collect taxes and provide public goods.

Consider a discrete-time, infinite-horizon game with three types of players: ruler, tax agents, and peasants. As a stable dictator with dynastic succession, the ruler is assumed to live infinitely long, while the agents and the peasants are assumed to be short-lived.

For analytical simplicity, we assume that the dictatorship consists of \[S\] homogenous regions and that \[S\] is exogenously given to the Ruler.Footnote 23 We let the number of regions \[S\] represent the geographical size of the dictatorship and take a region as the unit of analysis. In other words, when comparing large and small dictatorships, we assume that the two regimes differ only in the number of regions they encompass and that all regions in the two regimes are “identical.”

The basic setup

We first describe a basic, single-period game in a representative region consisting of a fixed number of jurisdictions.Footnote 24 Assume that the region is populated by \[N\] Peasants who engage in agricultural production.Footnote 25 Let \[Y\] denote the agricultural output in the region and assume that it increases with labor inputs at a diminishing rate: \[Y=Y[N]\], where \[N>0\], \[Y[0]=0\], \[Y'[\cdot ]>0\], and \[Y''[\cdot ]0\], \[D''[\cdot ]>0\], and \[D'''[\cdot ]\le 0\].Footnote 30 A simple example would be a quadratic function: \[D[\theta ]=\theta ^2\].

To summarize, the timing of events in the basic, single-period game in the representative region is as follows: [1] The ruler sets a tax rate \[\tau \] to maximize tax revenue. [2] The representative Agent selects \[\theta \] to maximize his expected payoff and proceeds to collect taxes. [3] The peasants pay \[\tau +\theta \] of their outputs to the agents and decide whether or not to revolt. [4] The ruler conducts randomized audits and punishes the agents if the audits uncover misconduct.

The representative agent To provide benchmark results, we derive the equilibrium of the single-period game. First, consider the optimization problem of the representative agent. The Agent chooses a rate of extralegal surcharge \[\theta \] to maximize his expected payoff, given the monitoring mechanism, \[A[\cdot ], D[\cdot ]\], and \[X\]:

$$\begin{aligned} \mathop {\max }_{0\le \theta \le 1}\text { }v^{A}=\theta \cdot Y[N] - A[S]\cdot D[\theta ] \cdot X \end{aligned}$$

[1]

The optimal rate of surcharge \[\theta ^{*}\] is given by the following condition:

$$\begin{aligned} Y[N]=A[S]\cdot D'[\theta ^{*}]\cdot X \end{aligned}$$

[2]

The ruler The ruler chooses a tax rate to maximize tax revenue. In doing so, however, we assume that, unlike the agents, the ruler is deeply concerned about peasant rebellion and thus constrained by the no-revolt condition: \[\tau + \theta \le r\].

Formally, the ruler’s maximization problem can be written as:

$$\begin{aligned}&\mathop {\max }_{0\le \tau \le 1}v^{R}=\tau \cdot Y[N]\nonumber \\ s.t.&\tau + \theta \le r \end{aligned}$$

[3]

Anticipating the responses by the agents and the peasants, the ruler sets a tax rate given the optimality condition [2] and the no-revolt condition. It is simple to show that there is a unique equilibrium in the single-period game in which \[\tau ^{*}\] and \[\theta ^{*}\] are determined by \[Y[N]=A[S]\cdot D'[\theta ^{*}]\cdot X\] and \[\tau ^{*}+ \theta ^{*}=r\].

Comparative statics To examine the effects of the size of a dictatorship on the optimal tax and corruption rates, we perform comparative statics with respect to the number of regions \[S\]. From the optimality condition \[Y[N]=A[S]\cdot D'[\theta ^{*}]\cdot X\] and the assumptions \[A'[S]0\], we obtain the following result:

Result 1

The equilibrium corruption rate \[\theta ^{*}\] is higher in a larger dictatorship: \[\frac{d\theta ^{*}}{dS}>0\].

From \[\tau ^{*}+ \theta ^{*}=r\], it also follows that:

Result 2

The equilibrium tax rate \[\tau ^{*}\] is lower in a larger dictatorship: \[\frac{d\tau ^{*}}{dS}0\], and \[G''[\cdot ]0\], \[u_{2}[.]>0\], \[u_{11}[.]r\]. [4] The ruler conducts randomized audits and fines the agents if misconduct is detected. [5] Exogenous shock hits the region and destroys the economy unless \[\gamma _{t}\] is sufficiently large; the game continues to the next period with probability \[G[\gamma _{t}]\].

The representative peasant We derive an equilibrium of the dynamic game by backward induction.

First, the optimization problem of the representative Peasant in period \[t\] is given by:

$$\begin{aligned}&\underset{c_{t}, n_{t+1}>0}{\max }u_{t}=u[c_{t},n_{t+1}]\end{aligned}$$

[4]

$$\begin{aligned} s.t.&c_{t}+n_{t+1}\le [1-\tau _{t}-\theta _{t}]\cdot y_{t} \end{aligned}$$

[5]

where individual income is defined by \[y_{t}=\frac{Y[N_{t}]}{N_{t}}\]. Note that \[y_{t}\] is exogenous to the peasant even though \[N_{t}=N_{t-1}\cdot n_{t}\] because \[n_{t}\] is a decision variable of the previous generation. From the first order condition and the assumption \[u_{12}[.]>0\], it can be shown that the optimal number of offspring \[n_{t+1}^{*}\] is an increasing function of net individual income \[[1-\tau _{t}-\theta _{t}] \cdot y_{t}\].

The representative agent The representative Agent is assumed to be short-lived. As a result, the maximization problem of the representative Agent is essentially the same as in the single-period game, and thus the optimal rate of extralegal expropriation in period \[t\] is given by:

$$\begin{aligned} Y[N_{t}]=A[S]\cdot D'[\theta ^{*}_{t}]\cdot X \end{aligned}$$

[6]

The ruler The ruler is assumed to live for infinitely many periods. He sets the current and future values of \[[\tau , \gamma ]\] to maximize the expected discounted value of the tax revenue stream. In doing so, we again assume that the ruler is bound by the no-revolt condition in every period. Let \[V^{R}_{t}\] represent the ruler’s present value of the future revenue stream in period \[t\]. His maximization problem in period \[t\] is given by:

$$\begin{aligned} \mathop {\max }_{0\le \tau _{t+j} \le 1, \gamma _{t+j}\ge 0}V^{R}_{t}=&\tau _{t} \cdot Y[N_{t}]-\gamma _{t}+G[\gamma _{t}]\cdot V^{R}_{t+1}\\ s.t.&\tau _{t+j} + \theta _{t+j} \le r \quad \forall j=0, 1, 2 \ldots \nonumber \end{aligned}$$

[7]

The optimal level of public good investment \[\gamma _{t}\] is given by the following condition:

$$\begin{aligned} G'[\gamma ^{*}_{t}]\cdot V^{R*}_{t+1}=1 \end{aligned}$$

[8]

In other words, the ruler invests in the public good up to the level where the marginal return from the investment equals its marginal cost. The higher the present value of his future revenue stream \[V^{R*}_{t+1}\], the more willing the ruler is to invest in the public good to increase the continuation probability.

The ruler sets an optimal tax rate, taking the agent’s optimality condition [6] and the Peasant’s no-revolt condition as given. Because these conditions are the same as before, the equilibrium tax and corruption rates [\[\tau _{t}^{*}, \theta _{t}^{*}\]] in the dynamic game are again determined by \[Y[N]=A[S]\cdot D'[\theta _{t}^{*}]\cdot X\] and \[\tau _{t}^{*}+ \theta _{t}^{*}=r\] [\[t=1, 2, 3 \ldots \]].

Population dynamics We now turn to equilibrium population dynamics. Because the Peasant’s net income is \[[1-r]\cdot y_{t}\] in the equilibrium and \[r\] is a constant, the optimal number of offspring can be expressed as \[n^{*}_{t+1}=n^{*}_{t+1}[y_{t}]\], where \[n^{*}_{t+1}[\cdot ]\] is strictly increasing in \[y_{t}\]. This, in turn, provides the population dynamics because by definition:

$$\begin{aligned} n^{*}_{t+1}[y_{t}]=\frac{N^{*}_{t}\cdot n^{*}_{t+1}}{N^{*}_{t}}=\frac{N^{*}_{t+1}}{N^{*}_{t}} \end{aligned}$$

[9]

In the spirit of Malthus, Condition [9] implies that the direction and rate of population growth depends on the peasant’s per capita income. Let \[\underline{y}\] denote the level of income defined by \[n^{*}_{t+1}[\underline{y}]=\frac{N^{*}_{t+1}}{N^{*}_{t}}=1\]. If \[y_{t}>\underline{y}\], then \[N_{t+1}>N_{t}\] or population will expand; if \[y_{t}V^{R*}_{t+1}[S_{large}]\]. Therefore, \[V^{R*}_{t+1}[S_{small}]>V^{R*}_{t+1}[S_{large}]\] must hold. The ruler’s optimality condition [8] and the assumption \[G''[\cdot ]0\] if population is below \[\hat{N}[S]\], and \[\frac{dv^{R*}}{dN}0\], \[D'''[\cdot ]\le 0\], and \[A'[\cdot ]