在總要素生產力的領域，Cave et al. (1982) 的Malmquist生產力指數 (MPI) 受到廣泛的重視；然而在應用上，當生產單位非屬同一技術集合，而面對不同的技術邊界，跨群組MPI的分析，將受限於共同比較基礎的缺乏，而無法進行。為突破此限制，Rao (2006) 引進共同邊界 (metafrontier) 的概念，在非參數法下，提出共同邊界Malmquist生產力指數 (MMPI) 的估計架構。本文針對Rao (2006) 的MMPI，進行三個層面的延伸：首先，考量生產力指數衡量的完全性，嚐試透過距離函數的推導，在非參數法下，將規模效率調整項 (SEC) 納入MMPI的構成項目中；其次，並在參數法下，提供一般化共同邊界Malmquist生產力指數 (gMMPI) 的拆解，與可行估計架構；另外，亦針對MMPI構成項目中的「追趕」，進一步再拆解為純粹技術追趕 (PTCU) 與邊界追趕 (FCU) 二項，其中，PTCU所捕捉的，為個別國家的技術追趕，而FCU則反應出整個群組的技術追趕，其意涵與可應用的範圍不同。
在實證的應用上，本文利用1980-2003年間，35個高所得國家 (HICs)、73個中所得國家 (MICs)，以及48個低所得國家 (LICs) 群組，進行參數法的生產力推估與拆解。結果發現，在MMPI的架構下，平均而言，中所得國家的生產力成長高於高、低所得國家群組，而在gMMPI的架構下，高所得國家的生產力成長則高於中、低所得國家群組，此反轉的結果，說明了未考量SEC的MMPI，所潛藏的衡量不完全的風險。其次，再將焦點轉向個別國家的比較，可以發現各個國家生產力成長的排序，也有類似的現象。此外，本文並發現，在樣本期間中，台灣的生產力指數呈現每年衰退0.3173%的現象，相較其他亞洲四小龍國家、金磚四國，以及美、日而言，位居最後。探究其原因，發現台灣在生產效率與規模效率上，持續以來的表現並不理想，分別呈現平均每年2.1679%與1.1594%惡化的現象；然而，在此同時也可以發現，在樣本期間內，各國的技術多呈現進步的結果，其中，台灣的技術進步則以年成長幅度3.0470%，位居亞洲四小龍國家、金磚四國，以及美、日之首。對此極端的結果，本文認為台灣過去的生產力，存在著對技術進步高度的依賴，但卻忽視了效率的提昇；因此，如何維持技術的成就，同時有效地進行在生產與規模上效率的調整，使欠缺天然資源稟賦的台灣，能獲得進一步經濟發展的動力，是當前重要的課題。最後，本文亦利用所推估的PTCU項，進行技術收斂假說的實證，觀察是否技術水準較低的國家，技術追趕的速度較快。結果發現，在絶對收斂的設定下，技術收斂並不存在，然而在條件收斂的設定下，技術將呈現收斂的現象。推論原因，主要可能與技術取得的差異有關。

This study extends Rao’s (2006) Metafrontier Malmquist Productivity Index (MMPI) to three dimensions. First, given the consideration to the completeness of a productivity index measure, we attempt to subsume a scale efficiency change term (SEC) in the MMPI with inductions of distance function under a non-parametric context. Second, a feasible structure of generalized MMPI (gMMPI) is also proposed under a parametric context. Furthermore, the ‘catching-up’ term in MMPI is disintegrated into two sub-components: pure technological catching-up (PTCU) and frontier catching-up (FCU), for achieving a more meaningful decomposition. Where, the PTCU is more suitable for observing the catching-up dynamics of a specific country, while the FCU is more suitable for observing the catching-up dynamics of the group frontier. For demonstrating the feasibility of this gMMPI framework, an empirical study is then conducted using data based on three country groups: 35 high income level countries (HICs), 73 middle income level countries, and 48 low income level countries (LICs) during the period from 1980-2003 with the parametric context. Consequently, the results reveal the risk embedded in the MMPI if the effect of the SEC is not taken into account. When shifting the focus to specific countries, the inconsistency between the rankings of MMPI and gMMPI is also found to be existent. Furthermore, the empirical results also reveal that the technical efficiency and scale efficiency of Taiwan exhibit annual averages of 2.1679% and 1.1594% declines, respectively. However, on average the margin of technical progress of Taiwan is about 3.0470% annually, which is ranked as the first when compared to the other four Asian countries, the BRIC countries and Japan and the US. Ultimately, this study also inspects the hypothesis of convergence in technology by using the pure technological catching-up term to observe whether a country with a lower level of technology would exhibit a higher catching-up rate. The results show that the technological convergence does not exist in the specifications of absolute convergence but exists in the situation of conditional convergence.

CONTENTS
1.	RESEARCH BACKGROUND	1
			
	1.1	Research Motivation	1
	1.2	Research Purpose and Framework	3
		
2.	TFP, MMPI AND CONVERGENCE HYPOTHESIS	7
			
	2.1	Introduction to the Total Factors Productivity (TFP) Index	7
	2.2	The Metafrontier Malmquist Productivity Index (MMPI)	12
	2.3	Convergence Hypotheses	16
		
3.	GENERALIZED MMPI AND MODEL SPECIFICATIONS	21
			
	3.1	Extensions of Rao’s Technological Catching-up	21
	3.2	Extensions of the Metafrontier Malmquist Productivity Index	25
		3.2.1	Non-Parametric Extension of MMPI	30
		3.2.2	Parametric Extension of MMPI	32
	3.3	Model Specifications and variable constructions	36
		3.3.1	Stochastic and Metafrontier Production Function Models	36
		3.3.2	Models for Convergence in Technology	39
		
4.	EMPIRICAL APPLICATIONS	45
			
	4.1	Data and Its Arrangements	45
	4.2	Estimations of the Stochastic and the Metafrontier Production Function	49
	4.3	Estimations of MPI, MMPI and gMMPI	53
	4.4	Tests for Hypothesis of Convergence in Technology	73
		
5.	CONCLUDING REMARKS AND RESEARCH LIMITATIONS	84
			
	5.1	Concluding Remarks	84
	5.2	Research Limitations and Further Research Suggestions	89
		
6.	APPENDIX	92
		
	Appendix 1: Rao’s (2006) Decomposition of MMPI	92
	Appendix 2: The Inductions of Decompositions of CRS Based GMPI	93
	Appendix 3: The Inductions and Decompositions of CRS Based MMPI	94
	Appendix 4: The Inductions of gMMPI under the Parametric Context	97
	Appendix 5: The Further Decompositions of gMMPI	98
		
7.	REFERENCES	102

 

LIST OF TABLES	
Table 1.	Average annual growth rates: gross domestic product, capital and labor	47
Table 2.	MLE estimation of the group frontiers - stochastic frontier analysis model	50
Table 3.	Estimation of metafrontier - linear programming approach	52
Table 4.	Decomposition of the generalized metafrontier Malmquist productivity index: 1980-2003 (HICs, MICs and LICs)	54
Table 5.	Decomposition of generalized metafrontier Malmquist productivity index: 1980-2003 (BRICs, Asian four countries, Japan and the US)	60
Table 6.	Inspections of scale efficiency changes (BRICs, Asian four countries, Japan and the US)	67
Table 7.	Inspection to hypothesis of absolute β convergence in technology	77
Table 8.	Inspection to the hypothesis of conditional β convergence in technology	78
Table 9.	Inspection to hypothesis of conditional β convergence in technology (some extensions for checking robustness)	83
Table A1.	Country list of high, middle and low income countries	100
Table A2.	Difference tests of the MMPI, gMMPI and its components of HICs, MICs and LICs	101

 

LIST OF FIGURES	
Figure 1.	Illustrations of the concept of Rao’s catching-up and the pure technological catching-up and frontier catching-up of this study	23
Figure 2.	Illustrations of the importance of scale efficiency in a productivity measure and the concept of CRS and VRS technology	27
Figure 3.	The average annual growth rates of output, capital and labor for the HICs, MICs and LICs	48
Figure 4.	The dynamics of gMMPI and MMPI for the HICs, MICs and LICs	56
Figure 5.	The dynamics of the TEC*, TC* and SEC* for the HICs, MICs and LICs	57
Figure 6.	The dynamics of the PTCU, FCU and RCU decomposed from the gMMPI in this study for the HICs, MICs and LICs	59
Figure 7.	The dynamics of the TEC* for the BRIC countries, Asian four countries, Japan and the US	64
Figure 8.	The dynamics of the TC* for the BRIC countries, Asian four countries, Japan and US	65
Figure 9.	The dynamics of the gMMPI for the BRIC countries, Asian four countries, Japan and US	69
Figure 10.	The dynamics of the MMPI for the BRIC countries, Asian four countries, Japan and US	70
Figure 11.	The dynamics of the SEC* for the BRIC countries, Asian four countries, Japan and US	72
Figure 12.	The dynamics of the PTCU for the BRIC countries, Asian four countries, Japan and US	74
Figure 13.	The dynamics of FCU for the BRIC countries, Asian four countries, Japan and US	75
Figure 14.	The relation ship of technological level and technological catching-up for the HICs, MICs and LICs	81

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