Harutaka Takahashi

- Address:
- Department of Economics

Meiji Gakuin University

1-2-37 Shirokanedai,

Minato-ku,Tokyo

108-8636, Japan - Electric Mail:
- haru@eco.meijigakuin.ac.jp
- Telephone:
- +81 3 5421-5641 ( Office )

+81 3 5421 5207 ( FAX )

## 1.Research Interest

As we will see from my review on the history of turnpikes, most of the Samuelson turnpike models were set up so that there are more than two-sectors, except the models rooted in the Ramsey model. Does the number of sectors matter?

To see this, let us compare a one-sector growth model and a two-sector growth model which consists of a consumption good sector and a capital good sector. In the one-sector model, a produced good will be used for a consumption good as well as a capital good without any cost. On the other hand, in a two-sector model, both goods are physically different.

Therefore when you want to use more consumption goods, you must transform capital goods into consumption goods as shown by a transformation curve, which is nonlinear. This nonlinearity will make dynamics of the multi-sector models very complicate as analyzed by a series of papers by Kazuo Nishimura, Makoto Yano and Alain Venditti.

Although we have witnessed fairly active research works on two-sector and multi-sector growth models in the 1990s and recent years, R. M. Solow (2005) has thrown doubt on the capital intensities, which take an important role in the above analysis:

*“In the early stages there was active exploration of two-sector models, culminating in the book by Duncan Foley and Miguel Sidrauski, but it petered out fairly soon. The reason was probably internal-intellectual rather than any feeling that the applications were unimportant. The usual, perfectly reasonable, choice was to distinguish between a consumption-good producing sector and an investment-good producing sector. I have the feeling that too much in those models turned out to depend on differences in factor-intensity between the sectors. We have very little in the way of facts or intuition about that issue, and there was no reason to expect or postulate any systematic pattern that could lead to exciting results ” (from p.4 in Handbook of Economic Growth (2005)).*

His main criticism is focused on the commonly assumed capital-intensity assumption - the consumption good sector is more capital intensive than the investment good sector - which is often postulated in two-sector models.

In the following paper, I have based discussions on two-sector growth models originally investigated in series of paper by Uzawa and will try to measure the capital intensities of the consumption good and the investment good sectors mainly in the post-war Japanese economy. By so doing we would like to give firm empirical evidence of the capital-intensities and to reply to the questions raised by Solow.

- “Does the capital-intensity Matter?: Evidence from the Post-war Japanese Economy and Other OECD

Countries ” forthcoming in Macroeconomic Dynamics.

My first project is to integrate the Samuelson turnpike models and the Ramsey turnpike models. To do so, I set up a multi-sector Ramsey turnpike model and study the behavior of an optimal path by applying the turnpike theory developed by the Scheinkman and McKenzie explained in the history of turnpikes, where the “Neumann-Mckenzie facet” took an important role in the turnpike proof. One important result is that any optimal path of a sector will converges to its own optimal steady state with a common growth rate without depending on their initial stocks. Let me call this result the **“generalized Ramsey turnpike.”** Most of this is done in my unpublished dissertation submitted to the University of Rochester and is further elaborated as the following unpublished book chapters:

- “Global Analysis of the Growth and Cycle of Multi-sector Economies with Constant Return Technologies”

However, the result derived above is not consistent with the recent empirical results; growth in an individual industry’s per capita capital stock and output grow at industry’s own growth rate, which is closely related to its own technical progress measured by total factor productivity of the industry as reported by many researchers, among others by Syverson (2011).

McKenzie (1998) has articulated this point: “Almost all the attention to asymptotic convergence has been concentrated on convergence to balanced path, although it is not clear that optimal balanced growth path will exists. This type of path is virtually impossible to believe in, if the model is disaggregated beyond the division into human capital and physical capital, and new goods and new methods of production appear from time to time.”

By applying the theoretical method developed in the first project, I will fill the gap between the results derived by the theoretical research explained as the first project and the empirical evidence from recent studies as the industry level among countries. This project is partially accomplished as the following paper:

- “An Unbalanced Multi-sector Growth Model with Constant Returns: A Turnpike Approach”

## 2.Publications in English

“Does the Capital-intensity Matter?: Evidence from the Post-war Japanese Economy and Other OECD Countries,” (with K. Mashiyama and T. Sakagami) forthcoming in

*Macroeconomic Dynamics.*

*Summary:*The capital-intensity takes an important role in two-sector and multi-sector growth models. Surprisingly very few empirical studies have been conducted so far except by Kuga (1967). This fact implies that few people ever tried to perform any empirical research to studywhether the two-sector and multi-sector optimal growth models could explain properly the economic development based on the empirical data. Although we have witnessed fairly active theoretical research on two-sector and multi-sector growth models in the 1990s and recent years, R. M. Solow has thrown doubt on the capital-intensities in Solow (2005). Our purpose is to measure the capital-intensities of the consumption good and the investment good sectors mainly in the post-war Japanese economy, and also in other OECD countries. By so doing, we will demonstrate that the capital-intensity does matter and our empirical evidence will strongly support the commonly assumed capital-intensity assumption: the consumption good sector is more capital-intensive than the capital good sector.

“Optimal Balanced Growth in a General Multi-sector Endogenous Growth Model with Constant Returns,”

, p31-49, 2008.*Economic Theory Vol.37#1*

*Summary:*I will study a multi-sector endogenous growth model with general constant returns to scale technologies and demonstrate the existence, uniqueness and the saddle-path stability of the balanced growth equilibrium. I will first demonstrate the existence of abalanced growth equilibrium, by showing that the balanced growth rate associated with the balanced growth equilibrium is solely determined by solving a Frobenius root problem of the price equations derived from the Euler equations and the property of the nonsubstitution theorem. Then I will show the saddle-path stability of the balanced growth equilibrium without any capital intensity conditions, which is a generalized property proved in the two-sector endogenous growth models by de Guevara et al. (J Econ Dyn Control 21, 115–143, 1997), Bond et al. (J Econ Theory 68, 149–173 1996) and Mino (Int Eco Rev 37, 227–251 1996). The theorem clearly implies that the balanced growth equilibrium has a transition path in the neighborhood of the balanced growth equilibrium

“Endogenous Fluctuations in Two-sector models: Role of Preferences,” (with K. Nishimura and A. Venditti)

*Journal of Optimization Theory and Applications Vol.128#2, p.309-331*

*Summary:*We consider a discrete-time two-sector CES (constant elasticity of substitution) economy with sector specific external effects and nonlinear preferences. Our goal is to examine carefully the influence of the utility curvature on the occurrence of multiple equilibria.We show that local indeterminacy depends on an interplay between factor substitutability and the elasticity of intertemporal substitution in consumption. Moreover, considering that, when the external effects are set equal to zero, we get a two-sector optimal growth model, we study also the role of the utility curvature on the occurrence of competitive equilibrium cycles. We show that persistent endogenous fluctuations and macroeconomic volatility require a strong enough elasticity of intertemporal substitution in consumption.

“Stable Optimal Cycles with Small Discounting in A Two-Sector Discrete-Time Model: A Non-bifurcation Approach,

*”*, p.328-338, 2001.*Japanese Economic Review Vol52, No.3*

*Summary:*This paper presents a standard two-sector optimal growth model with general neoclassical production functions: strictly quasi-concave, twice continuously differentiable homogeneous of degree 1 functions. Instead of applying the standard local bifurcation theory,I exploit two well established properties in Turnpike Theory—“simple dynamics” and the Neighbourhood Turnpike— and, combining both results, I demonstrate that there exists an interval of the discount factor near 1 such that a corresponding optimal steady state is totally unstable and an optimal path converges asymptotically to a two-period cycle for a chosen discount factor in it.

“Comparing Open-Loop with Markov Equilibria in a Class of Differential Games,” (with N. V. Long and K. Shimomura)

*Japanese Economic ReviewVol.50 No.4*

*Summary:*We consider a class of differential games with transition equations that are homogeneous of degree one. For any game G with a discount rate r, consider a Markov-perfect equilibrium (MPE) with strategies that are linear in the state variables. We show thatthe time paths of the control variables of this equilibrium constitute an open-loop equilibrium of a corresponding game G, which differs from G only in that its rate of discount r is equal to r plus a suitably chosen constant. In the context of a resource depletion game, this implies that the open-loop solution is more conservationist.

“Transitional Dynamics of Economic Integration and endogenous Growth,” (with T. Sakagami),

*Journal of Economic**Behavior and Organization vol. 33*

*Summary:*Rivera-Batiz and Romer have demonstrated some comparative dynamic analysis, namely comparing pre- and post-economic integration balanced growth paths and have derived interesting economic results. Even though these results are interesting, their analysishas a serious shortcoming. Namely, the balanced growth path studied by them exhibits a knife-edge property and is unstable. This clearly implies that their results are not robust. In this paper, we propose a modified knowledge driven model based on their model. Then we give a comparative dynamic analysis and show different results of economic integration from theirs.

“ Capital Movements, Intersectoral Resource Shifts, and The Trade Balance with Recursive Preferences,”

*Pacific Economic Review Vol.2 #1, pp.73-86, 1997.*

*Summary:*This paper examines interactions between internal resource shifts and the external balance. It sets up a simple infinite-period model with recursive preferences. It shows simple patterns between capital movements and trade balances and clarifies theresource shifts between traded goods and nontraded goods sectors when world interest rate changes and government spending changes are taking place.

“Stability and Entry in a Dynamic Cournot Market,” (with E. Dockner)

*The Economic Studies Quarterly Vol.45 #2*

*Summary:*In this paper we deal with dynamic Cournot competition and explore the relationship between the stability properties of an equilibrium path and quasi-competitive behavior of the market. More precisely we show that the stability conditions introduced for thestatic case ensure global convergence of a unique dynamic equilibrium path towards the steady state and quasi-competitiveness of the entire equilibrium path in the neighborhood of the steady state as well as the stationary state itself.

“The von Neumann Facet and a Global Asymptotic Stability,”

, pp.273-282, 1992.*Annals of Operations Research Vol.37*

*Summary:*We will study a multi-sector discrete-time optimal growth model with neoclassical non-joint technology and show that any path on ann-dimensional flat supported by the optimal steady state price will converge to the optimal steady state and is optimal. Burmeister andGraham have proved a similar result in a continuous-time setting. Although their result is limited, it is a first challenge to generalize the global stability result obtained by Uzawa and Srinivasan in a two-sector optimal growth model. One prominent advantage of our approach is that due to the discrete-time model setting, we can apply the duality approach and introduce the so called "von Neumann facet" intensively studied by McKenzie, which plays a very important role in proving the saddle point stability.

“On the Saddle-Point Stability For a Class of Dynamic Games,” ( with E. Dockner)

*Journal of Optimization Theory and Applications Vol. 62 #2*

*Summary:*In this paper, we introduce an iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping.The iterative process is based on the so-called extragradient method. We obtain a weak convergence theorem for two sequences generated by this process

*“Further Turnpike Properties For General Capital Accumulation Games,” (with E. Dockner)*pp.321-325, 1988.*Economics Letters Vol.28 #4*,

*Summary:*We consider a class of finite and infinite horizon discrete-time N-player capital accumulation games.Three turnpike results for the case of open-loop Nash equilibrium solutions are derived under fairly weak conditions.

, pp.190-199, 2000.*"A Turnpike Theorem with Public Capital," in Optimization, Dynamics, and Economic Analysis, Physica-Verlag*

*Summary:*Most of the literature on global stability of optimal public capital often treats private and public capital as physically identical goods. As a result, one important characteristics of public capital, “non-rivalness,”will be neglected. This is clearly a serious analytical defect.To tackle the problem, a three-sector optimal growth model, where one of the sectors is a public sector and produces public capital, is set up and the global stability of the optimal path of public capital will be demonstrated.

”Factor Intensity and Hopf Bifurcations,” (with K. Nishimura) in Gustav Feichtinger ed.,

*Dynamic Economic Models And Optimal Growth*

*Summary:*The following is a study of the relationship between factor intensity conditions and the local-saddle point stability of a steady state in multi-sector optimal growth models.The authors first provide the sufficient condition to guarantee the saddle-point stability, and then proceed to study the factor intensity condition which produces the Hopf Bifurcation.

“Turnpike Properties, and Comparative Dynamics of General Capital Accumulation Games,” (with E. Dockner) in R. Becker, M. Boldrin and W. Thomson eds.,

*General Equilibrium, Growth, and Trade II*

*Summary:*In this paper we introduce a discrete time N-player generalization of the Freshtman-Mueller (1984) game and state a sufficient condition for the global convergenceof the capital stocks that relates to theirs and has a nice economic interpretation.

## 3.Work in Progress

“Global Analysis of the Growth and Cycles of Multi-sector Economies with Constant Returns,”

*Meiji Gakuin Working Paper 2010*.*Summary:*The purpose of this research is to, based on the results of the above empirical analysis, use constant returns to scale technology to construct an exoginous optimal growth model and to investigate the global nature of that optimal path (equilibrium path) from theview of multi -capital-goods economy. Also, these results will be applied to the global analysis of a multi-sector endogenous growth model. Ever since the 1970s, optimimal growth models that include multi capital goods have been reported in research results as consumption turnpike theories.

“An Unbalanced Multi-sector Growth Model with Constant Returns: A Turnpike Approach”

*Meiji Gakuin Working Paper 2011.**Summary:*Recent industry-based empirical studies among countries demonstrate that individual industry's per capita capital stock and output grow at industry's own steady state growth rate. The industry growth rate is highly correlated to industry's technical progressmeasured by total factor productivity TFP) of the industry. Let us refer to this phenomenon as "unbalanced growth among industries." Very few research concerned with this phenomenon has been done yet. Some exceptions are Echevarria (1997), Kongsamut, Rebelo and Xie (2001), and Acemoglu and Guerrieri (2008) among others. However their models and analytical methods are different from mine. Applying the theoretical method developed by McKenzie and Scheinkman in turnpike theory, I now construct a multi-sector optimal growth model with a industry specific Hicks-neutral technical progress and show that each sector's per capita capital stock and output grow at the rate of the sector's technical progress

“The Core Involving Public Goods Revisited: A Diagrammatical Analysis”

*submitting to the journal.**Summary:*Using a diagram called a “Kolm triangle” adopted in Kolm (1970), the important issues of 1) Pareto efficiency and the core, and 2) Lindahl equilibrium and the core in the resource allocation problem involving public goods analyzed by Foley (1970) andNikaido (1976) can be illustrated merely using plane figures. One advantage of using such a diagrammatical method is that it allows an intuitive understanding of the salient problems of the core without using highly complicated mathematics to any degree.

## 4.Publications in Japanese (Refereed)

*Reconsideration of the Median Voter Hypothesis on Local Government Finance” (2004)*

*in Nihon Keizai Kenkyu Vol.50.**Summary:*We have empirically examined the median-voter hypothesis based on the Japanese prefectural financial panel data. Our empirical model is the one developed by Wyckoff (1988), where he has generalized the model by Bergstrome andGoodman (1973) so that it includes a bureaucratic behavior. Applying the panel-data analysis to the public investment expenditure as well as the total expenditure of the Japanese prefectural government data, we have found that the median-voter hypothesis will be established in the Japanese prefecture government level.

*“Diagrammatical Analysis of Resource Allocation Problem with Public Goods” (2008)*

*in Zaisei Kenkyu Vol.4 (Yuhikaku).**Summary:*Using a diagram called a “Kolm triangle” adopted in Kolm (1970), the important issues of 1) Pareto efficiency and the core, and 2) Lindahl equilibrium and the core in the resource allocation problem involving public goods analyzed byFoley (1970) and Nikaido (1976) can be illustrated solely using plane figures. One advantage of using such a diagrammatical method is that it allows an intuitive understanding of the core of the problem without using highly complicated mathematics to any degree.