To close this discussion, to the best of our knowledge, the existing literature has paid no attention to theoretically analyzing regional economic **growth** when the underlying creative capital using production function displays increasing returns to scale. Given this lacuna in the literature, in this note we study aspects of economic **growth** in a region that produces a final consumption good with creative and physical capital. This consumption good is manufactured with a production function that exhibits increasing returns to scale. Our analysis leads to three results delineated in sections 3 through 5 below. Section 2 describes the theoretical framework which is adapted from Lucas (1988). Section 3 computes the **growth** rate of creative capital in our regional economy. Section 4 shows that despite the presence of increasing returns, our regional economy converges to a **balanced** **growth** **path** (BGP). Section 5 computes the **growth** rates of physical capital and output on the BGP. Section 6 concludes and then suggests two ways in which the research described in this note might be extended.

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Economic **growth** measures the inflation-adjusted increase in the market value of goods and services in an economy. A common measure is the gross-domestic product (GDP). The GDP of the most developed countries has grown about by two percent per year since World War II. This sustained **growth** supports the idea of so called **balanced** **growth** **path** (BGP), which correspond to trajectories along which certain functions grow exponentially in time. Understanding what conditions initiate **balanced** **growth** in the long-run has been an active area of research. Different mathematical models have been proposed to describe substantial **growth**, which can be roughly grouped into exogenous and endogenous **growth** models. Endogenous **growth** theory is primary based on the assumption that economic **growth** is related to ingenuous forces, such as human capital, internal policies or innova- tion. In these models investments in ingenuous factors may lead to substantial **growth**. This is in contrast to exogenous **growth** models, which are based on the assumption that economic prosperity is primarily determined by external rather than internal factors. In this paper we study the existence of BGP solutions for an endogenous **growth** model proposed by Lucas and Moll, see [13]. It is based on the assumption that knowledge **growth** in an economy is promoted by ’imitation’ and ’innovation’. Imitation corresponds to learn- ing from others, innovation to developing new ideas through experimentation. Luttmer [15, 14] assumed that agents are characterized by their knowledge level and exchange knowledge in meetings. Innovation is incorporated via additional Brownian motion. His model serves as a starting point for the model proposed by Lucas et al [13], who linked the meeting/interaction frequency of agents to an optimal choice. Hence agents decide how much time they spend on learning and how much on producing goods. Their model corresponds to a Boltzmann mean-field game (BMFG), which we shall detail below. In all these models meetings between agents are described using mathematical tools and methods from statistical mechanics, in particular kinetic theory. Kinetic theory was ini- tially developed by Ludwig Boltzmann to analyze the statistical behavior of a system not in equilibrium [3], for example to describe the thermodynamics of dilute gases, and has led to extensive research on its mathematical properties (cf. [7, 9, 20] and references therein). The Boltzmann equation describes the evolution of the probability distribution

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leading regions in different parts of the world. Second, many of these studies have pointed to the significance of R&D in particular and to innovation more generally in augmenting the economic prospects of the lagging regions being studied. Finally, the above two points notwithstanding, to the best of our knowledge, there are no theoretical studies that have analyzed how small initial differences in the physical capital and the R&D stocks between stylized lagging and leading regions lead to substantially magnified differences on the **balanced** **growth** **path** (BGP) for these two regions. The reader should note that this is the lacuna in the extant literature that we seek to fill with our analysis in the present paper. 4 We now proceed to discuss the specific contributions

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PROPOSITION 1. In the model of endogenous technological change with government spending, there exists a minimum overall government spending proportion to ensure the economy having an endogenous **growth** solution. Giv- en a constant government spending ratio strictly larger than this minimum ratio, the government can choose either a higher spending ratio on research or a lower one. A government with the goal of high **growth** rate on the **balanced** **growth** **path** will prefer to spend more on the research sector. Then for a consumer, consumption propensity to income will be low, and for the government, the scale of the government spending tends to be larger.

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This work shows an application of Leontief’s closed model that, as far as I am aware, has not been explored previously. Such application is the study of income distribution between wages and profits when the rate of profit is the same in all industries. The results are consistent with those of Sraffa’s model, except for the fact that in Leontief’s model it is possible to build a standard system for each level of income distribution. This system, except for the scale of production and the units of measure employed, is equal to any whole-industry production process taking place within the **balanced**-**growth** **path** corresponding to Leontief’s closed model for the given level of income distribution. Furthermore, in the **balanced**-**growth** **path**, for each good, the amounts of invest- ment and profit in the industry producing the good are equal to the value, respectively, of the quantity of that good consumed and the surplus of that good produced in the whole industry. For this reason, for the set of households, included in the model as a particular industrial branch, the common profit rate measures the **growth** of the labor force. Unlike von Neumann’s model, the **balanced** **growth**-**path** corresponding to Leontief’s closed model shows explicitly the quantities of labor used in each industry, the quantities of goods consumed by work- ers and the **growth** of the labor force. Under a weak assumption, the **growth** rate is independent of worker’s choice.

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This paper provides an important contribution to the analysis of optimal tourism taxation, as it extends the models used in the literature so far in several ways. First, we consider intersecting externalities in a general equilibrium framework, taking account of all repercussions. As we will see, the optimal policy to handle intersecting externalities and to replicate the social (first-best) optimum can be a tax or a subsidy, depending on the sign of the “multiple externality”. Second, we explicitly introduce capital accumula- tion, necessary to increase the production potential of tourism services. We thus add an intertemporal dimension to the problem of internalizing externalities. Third, in contrast to previous work 7 , we allow for international borrowing (lending) to finance investment of domestic tourism firms and domestic res- idents’ consumption. This is an important extension, as, referring to international capital mobility, it realistically relaxes a country’s or region’s resource constraint. Thus, investments in tourism are not constrained period by period by earnings from tourism. 8 Rather, an intertemporal budget constraint has to be met. Fourth, we do not restrict our attention on the economy’s steady state (or **balanced** **growth** **path**), an analysis which has its own merits, but also pay close attention on the transitional dynamics. 9

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We focus on the steady-state economic **growth** and employment. For sufficiently high money **growth** rates, there is a unique **balanced** **growth** **path**, and the economy exhibits sustained **growth** based on sustained R & D. Faster money **growth** causes greater employment and faster economic **growth** along the **balanced** **growth** **path**. Furthermore, under some parameter restrictions, there is no **balanced** **growth** **path** for low money **growth** rates, and the economy is trapped in a steady state without long-run **growth**. These results suggest that money **growth** may be an important factor for long- run economic **growth**. That is, financial authorities are required to maintain high money **growth** rates to achieve sustained and faster economic **growth**.

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term. Assuming that the effective capital per labor is k * and effective consumption per labor is c * when it achieves **balanced** **growth** **path**, after implementing the market-oriented environmental policy instruments, k * and c * will change. However, the mechanism of change is different. Because the capital stock is decided by the past investment decisions, the capital does not change at the moment when governments announce that they begin to use market-based environmental policy instruments, and will still stay in the original **balanced** **growth** **path** k * . Then due to the decline of return on capital, capital stock decreases, eventually reaching a new equili- brium. This is a continuous process. However the dynamic change mechanism of effective consumption per la- bor is different. As environmental policy reduces the rate of return on savings and capital accumulation, families will choose to save less and consume more. Then, in order to achieve a new equilibrium **growth** state, the economy will move along the **path** of saddle point, and finally reach a new equilibrium. Compared with the changing process of effective capital per labor, the process of the effective consumption per labor is not conti- nuous. So we can get a two-sided conclusion about the impact of market-oriented environmental policy instru- ments on production and consumption in the short term. On one hand, as mentioned above, under the new equi- librium **growth** **path**, the **balanced** capital stock and effective consumption per labor are decreased, and the ef- fective output per labor also falls. Therefore, the implementation of market-oriented environmental policy in- struments has a certain impact on the economy. On the other hand, although the final effective consumption per labor drops, this is not a continuous process, but a process of rising sharply and declining slowly and conti- nuously. Because consumers pursue the utility maximization of lifetime, after the implementation of environ- mental policy instruments, the sum of discounted consumption in each period may not be less than the original level. When households pay more attention in the current consumption, the discount factor ρ is greater. Under this situation, the leap **growth** of consumption caused by the use of environmental policy instruments will in- crease the utility of household.

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This paper constructs a model which is more close to the reality to describe individual saving behavior accurately and explain the long-term **growth** of economy. The model constructed in this paper is the improvement and inno- vation of neoclassical **growth** models including Solow model, RCK model and OLG model. Starting from the micro decision making of individuals and firms, there exists **balanced** **growth** **path** (BGP) and capital stock in steady state satisfies the golden rule in general equilibrium. Total saving rate equals the elasticity of output with respect to capital. The long-term economic **growth** depends on exogenous technological progress.

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It is worth emphasizing that in our setting each country may attain a diﬀerent rate of **balanced** **growth**. In particular, in the case of Figure 1-b, the **balanced**-**growth** rate of each country may diﬀer from each other even though both countries have the identical production and preference structures. This is because the **balanced** **growth** **path** where country 1 (country 2) grows faster than country 2 (country 1) may be attained as a steady state equilibriums the world economy even in the presence of free trade and a well-organized international ﬁnancial market.

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We present a continuous time overlapping generations model for an endowment Arrow-Debreu economy with an age-structured popu- lation. For an economy with a **balanced** **growth** **path**, we prove that Arrow-Debreu equilibrium prices exist, and their dynamic properties are age-dependent. Our model allows for an explicit dependence of prices on critical age-specific endowment parameters. We show that, if endowments are distributed earlier than some critical age, then spec- ulative bubbles for prices do exist.

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technology advances are not exogenous, but they happen due to innovations by entrepreneurs. These entrepreneurs innovate in an attempt to replace the current monopolists that populate the diﬀerent economic sectors. To fund the innovation process, entrepreneurs should rely on the ﬁnancial sector, whose role is to allocate resources in the most proﬁtable way.The model exhibits a **balanced** **growth** **path** in its two statae variables: eﬀective capital per worker and productivity relative to the technology frontier. In section 3 this theoretical model is taken to the data. To do so, I estimate a subset of the model structural parameters by bayesian methods, as in Smets and Wouters (2003). Bayesian estimation allows to combine the a priori information that could be used in calibration with a full information approach. Calibration is not feasible as some of the parameters have not been estimated before neither in macroeconomic nor in microeconomic studies. Maximum likelihood estimation does not seem advisable as such a simpliﬁed model can hardly stand as the true data generating process. To explore the relevance of the new proposed mechanism, the model is compared with a simple RBC one with exogenous technology shocks. In section 4 I summarize the main ﬁndings of the paper and conclude by proposing future lines of research.

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of ω. In Case 5, the long run fertility rates in the efficient solution differ across all four values of ω. The equilibrium time series **path** of total fertility reproduces the long classical-Malthusian world, interrupted by a brief rapid Demographic Transition, followed by a stationary total fertility rate of near zero population **growth**. In contrast, each of the five Cases produces efficient fertility time series that typically have an intermediate phase of fertility, lower than the classical-Malthusian fertility, but different from the long run **balanced** **growth** **path** fertility. This intermediate phase, unlike the short Demographic Transition in the equilibrium solution, is much longer lasting. Roughly lasting 19000 years, whereas the equilibrium Demographic Transition lasts on the order of 700 years. Figure ?? contains the mortality rate for these solutions. Here the standard equilibrium demographic transition via the mortality revolution is evident. In this parameterization, the mortality decline is identical across all cases. This occurs because when the value of A changes in Cases 4 and 5, coincides with a decline of mortality to 0. If the mortality function required a higher value of human capital before it declines to 0, then there would be a divergence between Cases 1-3 and Cases 4 and 5. For all Cases, the equilibrium mortality decline begins 1320, but does not drop below .70 until 1680. From 1680 until 1880 mortality declines only from .58 to .57, before dropping to .38 in 1920, .38 in 1960 and .18 in 2000 and then 0 in 2040. The mortality decline in the efficient solution does differ by ω for each case. However the mortality decline occurs between -17640 and -17320 for Case 1. For Case 2, the efficient solution has mortality decline between -17640 and -17360. The efficient mortality decline occurs between -17640 and -17280 for Case 3. For Case 4 again the decline begins in -17640 and ends by -17320. Finally in Case 5 decline begins and ends in -17640 and -17280. If we date the end of the mortality decline in the year when mortality is less than 1%, the ending date for ω = 1.25 varies between -17480 for Cases 1 and 2, -17440 for Cases 3, 4 and 5, At the other end, for ω = 2.50, the ending date of the mortality decline, defined as mortality below 1%, occurs in year -17520 for Case 1, -17480 for Cases 2 and 4, -17440 for Case 3 and 5.

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This paper studies social welfare in a heterogeneous population under the criteria of efficiency and sustainability. Becker (1980) showed that, if time preference is heterogeneous, the most patient household eventually will own all capital and substantial inequality emerges. Although this state is Pareto efficient, less patient households cannot achieve optimality. The endogenous **growth** models in this paper indicate that a multilateral **balanced** **growth** **path** exists on which all the optimality conditions of all heterogeneous households are indefinitely satisfied, and that heterogeneity is sustainable on this **path**. However, sustainable heterogeneity is socially fragile and is not necessarily naturally obtained, because a unilateral **balanced** **growth** **path** also exists that is not sustainable and causes political conflicts. An advantaged economy can achieve optimality on both the unilateral and multilateral paths, whereas less advantaged economies can only do so on the multilateral **path**. In this paper, **path** selection is modeled using a political loss function. If less advantaged economies unite and the authority utilizes various measures such as progressive taxes, financial transfers, and affirmative actions, the multilateral **path** is secured. Voluntary donations are also effective in this regard.

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In this paper we want to move further and investigate not only the balanced growth path properties and the transitional dynamics Asea and Zak [1], and Bambi [3] but also the consumption [r]

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only successful if the initial values entered are close to the true steady state. Since this can prove di¢ cult even for simple models I normally prefer to enter the steady state solution by hand in Dynare or to write a Matlab program to …nd the model’s steady state. In the case of the Real Business Cycle model described in this section it is simple to …nd the model’s steady state or **balanced** **growth** **path** of the economy (in which output, capital, investment, marginal productivity of labor, consumption and government expenditures all grow at a constant common rate, the exogenous **growth** rate of the labour augmenting technological process).

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There are six spokes/parameters to a **balanced** life. The physical, mental, social, spiritual, family and economic. One needs to balance these spokes so that the wheel of life runs smooth. Gorakh sutras as mentioned supra could provide a balancing act. No studies actually have been carried out to exactly find out their effect. The present study can act as a catalyst to probe, apply and implement this wisdom in times to come which are going to be with more stress, more strain, more tension, more tightness, more taxing, more demanding, more anxiety and nervousness leading to breakouts and burnouts.

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In this paper, the choice of the most advantaged household is examined based on a non-scale endogenous **growth** model. I show that choosing the unilateral behavior is not always better because choosing unilateral behavior has an important negative side effect. Overall innovation activities (generation of new technologies) are severely constrained as a result of the unilateral behavior of the most advantaged household. Unilateral behavior severely restricts the opportunities of less advantaged households to receive higher education, and education is indispensable to generate innovations (Becker, 1964; Weisbroad, 1966; Lynch, 1991). Because of this negative side effect, economies’ capacity to generate new technologies will be severely constrained, and the rate of **growth** will be notably lower. As a result, unilateral behavior is not always in the best interest of the most advantaged household. Furthermore, at the present time, it is highly likely that multilateral behavior is better for the most advantaged households as well as all other households.

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Another branch of this literature emphasizes the importance of no-Ponzi- game conditions (constraints on debt accumulation) for the existence of bubbles in in…nite-horizon models. Kocherlakota (1992) …rst pointed this out by showing that an individual cannnot reduce his asset position permanently when facing constraints on debt accumulation. Technically, these constraints help to guar- antee transversality conditions not to be violated when asset price has a bubble term. If this constraint is a wealth constraint, the su¢cient and necessary con- ditions for the existence of a bubble is zero net supply of the asset. On the other hand, if this constraint is an exogenous short sales constraint, bubbles can arise if and only if the **growth** rate of individual’s income is not less than the real interest rate. As Kocherlakota (2008) stressed, with short sales con- straints, bubbles can arise even if the present value of aggregate consumption is …nite.

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A BSTRACT This paper presents a discrete-time **growth** model to describe the dynamics of a multi- agent economy, and the model consists of production process, exchange process, price and technology adjustment processes etc. Technologies of agents in each period are represented by a technology matrix pair, and some properties of Perron-Frobenius eigenvalues and eigenvectors of technology matrix pairs are discussed. An exchange model is also developed to serve as the exchange part of the **growth** model. And equilibrium paths of the **growth** model are proved to be **balanced** **growth** paths sharing a unique normalized price vector. Though this paper focuses mainly on the case of n agents and n goods, the **growth** model can also deal with the case of m agents and n goods. A numerical example with 6 agents and 4 goods is given, which describes the dynamics of a two-country economy and has endogenous price fluctuations and business cycles.

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