R&D經濟成長模型：資本稅之探討 - 政大學術集成

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(1)國立政治大學經濟學系博士論文. R&D 經濟成長模型：資本稅之探討 Capital Taxation政 in R&D 治 Growth Models. 大. 立. ‧. ‧ 國. 學. io. er. n. al. sit. y. Nat 指導教授：賴景昌教授. Ch. e n朱智豪 gchi. 研究生：陳平河. iv n 教授 U 撰. 中華民國一百零五年七月.

(2) 謝辭首先，感謝我的指導教授賴景昌老師及朱智豪老師。在漫長的博士生涯中，兩位老師不但提升我的學術視野，並且也嚴格訓練了我對學術研究的嚴謹及敏感性。認真地說，老師們對我的影響絕對是一生中，最全面性的一次巨大的改變。因為兩位老師，除了學術上各種功力的傳授、對學術的高標準、研究態度的嚴謹、學術上視野、甚至待人接物都為我提供了非常棒的學習標準。同時，我也感謝口試委員陳明郎老師、洪福聲老師、與蕭明福老師，在百忙中願意抽空審閱我的論文並指正疏漏且提供許多寶貴建議，令我的論文更加. 政治大攜與照顧，我也非常感謝。當然同時，也感謝政大經濟系其他老師的照顧。立. 完整，在此特別致謝。另外學校老師黃俞寧老師與李文傑老師，在校時的提. ‧ 國. 學. 在這裡我要特別感謝朱巡學長熱心的建議及經驗教授，從與學長的討論中，讓我在學術研究上少走許多冤枉路，而學長在我論文撰寫中也扮演非常重要的角. ‧. 色，可以說沒有他，我可能真的畢不了業，所以真的非常、非常感謝朱巡學長的. sit. y. Nat. 提拔。另外，我也要感謝這些曾經幫助過我的人，志興學長、欣榮、韻文、BOBO、. al. er. io. 亦帆、偉奇、冠任、玫英、士夫、立文、振維學長以及中正大鳥團的各位們，. v. n. 你們的加油打氣，為我這漫長的博士生涯增添不少生活色彩。. Ch. engchi. i n U. 最後，我還要感謝老爸、老媽、老姐、老妹、裕翔及宏恩，在我的生活提供相當程度的便利。還有懶貓霸卜及跑酷貓歐比，在我壓力扛不住的時候，帶給我舒壓及歡笑，你們這些小動物總是讓我心情愉悅，讓我將煩惱拋諸腦後。當然也要謝謝阿綠仔一路走來的陪伴與支持，生命有了你們是多麼美好，在此將論文獻給你們。. 平. 河. 2016 年 7 月.

(3) 摘要本論文有系統性地利用不同的 R&D 模型，分析了資本所得稅對經濟成長及社會福利的效果。第二章利用第一代 R&D 模型，發現資本所得稅對經濟成長於長短期下具有顯著不同的效果。第三章建立半內生成長( semi-endogenous ) R&D 模型，重新檢驗 Chamely-Judd 命題是否成立。我們發現最適的資本所得稅應大於零，並且檢驗最適資本所得稅在對應不同的 R&D 外部性程度下，正資本所得稅率結論是否仍然成立。第四章利用第二代 R&D 成長模型，在具有廠商家數內生化的特性下，重新檢視第二章的結論，並發現其資本所得稅對經濟成長. 政治大. 的效果，在長短期下仍然具有顯著不同的效果。. 立. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i. i n U. v.

(4) Abstract This dissertation has provided a systematic analysis on the growth and welfare effects of capital taxation within distinct R&D-based growth models. In Chapter 2, we employ a first-generation R&D-based growth model to examine the effcts of capital taxation on innovation and economic growth, and find that capital taxation has drastically different effects in the short run and in the long run. In Chapter 3, we set up a semi-endogenous growth model, and examine whether the Chamley-Judd result of a zero optimal capital income tax is valid. We find that the optimal capital income tax is. 政治大. positive, and this result is robust with respect to varying the degrees of. 立. various types of R&D externalities. In Chapter 4, we build up a. ‧ 國. 學. second-generation R&D-based growth model which features endogenous market structure. In line with Chapter 2, we also find that capital taxation has. ‧. drastically different effects in the short run and in the long run.. n. er. io. sit. y. Nat. al. Ch. engchi. ii. i n U. v.

(5) 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. iii. i n U. v.

(6) TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction. iii iv 1. 2 Short-run and long-run e¤ects of capital taxation on innovation and economic growth 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Household . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Final goods . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Intermediate goods . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 R&D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Government . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.7 Decentralized equilibrium and the balanced-growth path . . . 2.3 Long-run growth e¤ects of capital taxation . . . . . . . . . . . . . . 2.3.1 Tax shifting from lump-sum tax to capital income tax . . . . 2.3.2 Tax shifting from labor income tax to capital income tax . . 2.4 Quantitative analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 立. 政治大. ‧. ‧ 國. 學. Nat. y. sit. n. al. er. io. 3 Optimal capital taxation and R&D externalities 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Households . . . . . . . . . . . . . . . . . . . . . 3.2.2 The …nal-goods sector . . . . . . . . . . . . . . . 3.2.3 The intermediate-goods sector . . . . . . . . . . . 3.2.4 The R&D sector . . . . . . . . . . . . . . . . . . 3.2.5 The government and aggregation . . . . . . . . . 3.2.6 The decentralized equilibrium . . . . . . . . . . . 3.3 Steady-state properties . . . . . . . . . . . . . . . . . . . 3.3.1 Comparative static analysis . . . . . . . . . . . . 3.3.2 Optimal capital income tax . . . . . . . . . . . . . 3.4 Quantitative results . . . . . . . . . . . . . . . . . . . . . 3.4.1 Calibration . . . . . . . . . . . . . . . . . . . . . 3.4.2 The optimal capital tax with transitional dynamics 3.4.3 Policy implications of R&D externalities . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .. Ch. engchi. iv. i n U. v. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. 5 5 9 9 11 12 13 13 14 14 15 15 19 21 28 41 41 45 46 47 48 49 52 53 53 55 60 62 63 65 66 70.

(7) 4 Short-run and long-run e¤ects of capital taxation on economic growth in a R&D-based model with endogenous market structure 88 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.2.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.2.2 The …nal goods sector . . . . . . . . . . . . . . . . . . . . . 93 4.2.3 The intermediate goods sector . . . . . . . . . . . . . . . . . 94 4.2.4 Entrants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2.5 Government . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.2.6 Markets clearing and aggregation . . . . . . . . . . . . . . . 96 4.2.7 Decentralized equilibrium . . . . . . . . . . . . . . . . . . . 97 4.3 Long-run e¤ects of capital taxation . . . . . . . . . . . . . . . . . . 97 4.4 Quantitative analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 109. 立. 5 Conclusion. 政治大. 118. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. v. i n U. v.

(8) CHAPTER 1 INTRODUCTION. The linkage among capital taxation, economic growth, and social welfare has been one of the central issues in the literature on growth economics. As far as we know, in the real world the estimated e¤ective average tax rates on capital income are around 40% in the United States and 30% in EU countries. In some countries, such as the United Kingdom and Japan, the capital income tax rates. 政治大 is another important issue that will never cease being debated by economists and 立 are even up to near 60%. However, how much should the capital income be taxed. policymakers.. ‧ 國. 學. One of the major topics in the literature on growth economics is whether capi-. ‧. tal taxation boosts or impedes economic growth. The answer is hardly conclusive from both empirical and theoretical perspectives. On the empirical side, a number. y. Nat. io. sit. of studies have found that capital taxation, such as corporate pro…t tax and capital. n. al. er. gains tax, is harmful to economic growth (see e.g., Lee and Gordon, 2005; Hunger-. i n U. v. ford, 2010; Arnold et al., 2011; Dahlby and Ferede, 2012), whereas other studies. Ch. engchi. have found a neutral or positive growth e¤ect of capital taxation (see e.g., Mendoza et al., 1997; Angelopoulos et al., 2007; ten Kate and Milionis, 2015). On the theoretical side, earlier studies employing an AK-type endogenous growth model show that the impact of raising the capital tax rate on long-run economic growth is negative (Judd, 1985; Chamley, 1986; King and Rebelo, 1990; Rebelo, 1991; Jones et al.,1993; Pecorino, 1993, 1994; Devereux and Love, 1994; Milesi-Ferretti and Roubini, 1998), although the quantitative magnitude could be negligibly small (Lucas, 1990; Stokey and Rebelo, 1995). However, the point to observe is that these theoretical studies are unanimously centering on capital taxation in capital-driven. 1.

(9) growth models. Another one of the major topics is optimal capital taxation. The pioneering work by Judd (1985) and Chamley (1986) proposes that the government should only tax labor income and leave capital income untaxed in the long run. The idea of a zero optimal capital tax has then been dubbed the Chamley-Judd result, which turns out to be one of the most well-established and important benchmarks in the optimal taxation literature. A number of subsequent studies, including Chari et al. (1994), Jones et al. (1997), Atkinson et al. (1999), and Chari and Kehoe. 政治大. (1999), relax key assumptions in Judd (1985) and Chamley (1986), and …nd that. 立. their result to be quite robust. A common fact in these studies is that they focus. ‧ 國. 學. on AK-type (capital-based) growth models.. In general, the existing studies on the linkage among capital taxation, economic. ‧. growth, and social welfare can be classi…ed into two strands of literature. The …rst. sit. y. Nat. strand emphasizes the growth engine of capital accumulation. The relevant litera-. io. al. er. ture in this strand includes Judd (1985), Chamley (1986), King and Rebelo (1990),. n. Jones et al. (1993), Devereux and Love (1994), and Milesi-Ferretti and Roubini. Ch. i n U. v. (1998), just to mention a few. The second strand instead highlights the growth. engchi. engine of R&D investment. Up till now, to the best of our knowledge, only a few studies including Lin and Russo (1999), Zeng and Zhang (2002), Haruyama and Itaya (2006), Aghion et al. (2013), Yilmaz (2013), and Chen et al. (2016) falls into this strand. As a consequence, it is obvious that, within the context of R&Dbased growth models, the issue on the growth and welfare e¤ects of capital taxation is still an area that is less discussed. Moreover, as reported by Aghion and Howitt (2009), technological progress driven by innovation and R&D acts as a more important engine of economic growth compared to capital accumulation.1 In view of 1. Aghion and Howitt (2009, p.108) report that “TFP growth accounts for about two-thirds of. 2.

(10) the fact that innovation is a crucial factor to drive up economic growth, overlooking this facet may lead to inadequate design of tax policies. This dissertation thus aims to …ll this gap. The dissertation provides a systematic analysis regarding the growth and welfare e¤ects of capital taxation with distinct R&D-based growth models. The models this dissertation deal with include the …rst-generation R&D-based growth model developed by Romer (1990), the semi-endogenous growth model developed by Jones and Williams (2000), and the second-generation R&D-based growth model devel-. 政治大. oped by Dinopoulos and Thompson (1998) and Peretto (1998). This dissertation is. 立. composed of …ve chapters, including this Introduction. The main content of each. ‧ 國. 學. chapter can be brie‡y described as follows.. Chapter 2 makes an extension of the seminal workhorse R&D-based growth. ‧. model developed by Romer (1990), and discusses how capital taxation a¤ects eco-. sit. y. Nat. nomic growth in the short run and in the long run. We …nd that an increase in. io. al. er. the capital income tax rate has both a consumption e¤ect and a tax-shifting e¤ect. n. on the equilibrium growth rates of technology and output. In the long run, the. Ch. i n U. v. tax-shifting e¤ect dominates the consumption e¤ect, thereby yielding an overall. engchi. positive e¤ect of capital taxation on steady-state economic growth. However, in the short run, the consumption e¤ect becomes the dominant force causing an initial negative e¤ect of capital taxation on the equilibrium growth rates. These contrasting e¤ects of capital taxation at di¤erent time horizons may provide a plausible explanation for the mixed evidence in the empirical literature on capital taxation and economic growth. Chapter 3 sets up an innovation-based growth model (semi-endogenous growth economic growth in OECD countries, while capital deepening accounts for one third.”. 3.

(11) model) developed by Jones and Williams (2000), and uses it to examine whether the Chamley-Judd result of a zero optimal capital income tax is valid. It is found that the optimal capital income tax is more likely to be positive if labor supply is endogenous and the government size is relatively large. Moreover, by calibrating our model to the US economy, it is also found that the result of a positive optimal capital income tax is robust with respect to varying the degrees of various types of R&D externalities. Chapter 4 constructs a second-generation R&D-based growth model developed. 政治大. by Dinopoulos and Thompson (1998) and Peretto (1998). The main salient feature. 立. of the second-generation R&D-based growth model is that both vertical and hor-. ‧ 國. 學. izontal innovations are present simultaneously. In the vertical dimension, each of incumbent …rms engages in in-house R&D to improve the quality of their speci…c. ‧. product. In the horizontal dimension, …rms enter the market through the creation of new products. It is found that, in response to a change in the capital tax rate,. y. Nat. sit. the long-run and short-run responses of the economic growth rate exhibit distinct. er. io. patterns. In the short run where the number of …rms is …xed, a higher capital. al. n. iv n C U h eendogenously, with the number of …rms adjust growth keeps on rising as n g c h i economic income tax rate is harmful to economic growth. During the transitional process,. each of the in-house R&D …rms continues to expand its market size. In the long run, with the equal counteracting strength between the short run and the transition period, capital taxation is neutral with economic growth. As a result, the same as Chapter 2, this provides a plausible explanation for the mixed empirical observations between capital taxation and economic growth. Finally, some concluding remarks are provided in Chapter 5.. 4.

(12) CHAPTER 2 SHORT-RUN AND LONG-RUN EFFECTS OF CAPITAL TAXATION ON INNOVATION AND ECONOMIC GROWTH. 2.1. Introduction. In this chapter, we examine the e¤ects of capital taxation on innovation and economic growth. In the literature of endogenous growth, one of the major issues is. 政治大. whether capital taxation stimulates or impedes growth. Earlier studies employ-. 立. ing an AK-type endogenous growth model show that the impact of raising the. ‧ 國. 學. capital tax rate on long-run economic growth is negative (Judd, 1985; Chamley, 1986; King and Rebelo, 1990; Rebelo, 1991; Jones et al., 1993; Pecorino, 1993,. ‧. 1994; Devereux and Love, 1994; Milesi-Ferretti and Roubini, 1998), although the quantitative magnitude could be negligibly small (Lucas, 1990; Stokey and Re-. y. Nat. io. sit. belo, 1995).1 The intuition of this negative growth e¤ect of capital taxation is that. n. al. er. a higher capital tax rate discourages the accumulation of physical capital and is therefore detrimental to economic growth.. Ch. engchi. i n U. v. On the empirical side, the results are rather inconclusive. A number of empirical studies have found that capital taxation, such as corporate pro…t tax and capital gains tax, can be harmful to economic growth (see e.g., Lee and Gordon, 2005; Hungerford, 2010; Arnold et al., 2011; Dahlby and Ferede, 2012), whereas other empirical studies have found a neutral or even positive e¤ect of capital taxation on growth (see e.g., Mendoza et al., 1997; Angelopoulos et al., 2007; ten Kate and Milionis, 2015). Therefore, although the abovementioned theoretical prediction is 1. Other than focusing on the long-run growth e¤ect, Frankel (1998) studies the dynamics of capital taxation during the transition process.. 5.

(13) consistent with some of the empirical studies, it seems to contrast other empirical …ndings in the literature. While capital accumulation is undoubtedly an important engine of economic growth, technological progress driven by innovation and R&D also acts as an important driver for growth; see Aghion and Howitt (2009, p.109) for a discussion on data from OECD countries.2 Recently, R&D-based growth models pioneered by Romer (1990) and Aghion and Howitt (1992) have been used to explore the interrelation between capital taxation, innovation and economic growth; see e.g.,. 政治大. Lin and Russo (1999), Zeng and Zhang (2002), Haruyama and Itaya (2006) and. 立. Aghion et al. (2013). This chapter contributes to the literature by providing an. ‧ 國. 學. analysis of both the short-run and long-run e¤ects of capital taxation on innovation and economic growth within the seminal innovation-driven growth model in. ‧. Romer (1990), which is a workhorse model in R&D-based growth theory. In our analysis, we consider di¤erent tax-shifting schemes. Speci…cally, we examine the. y. Nat. sit. growth e¤ects of capital taxation with tax shifting from lump-sum tax and also. n. al. er. io. labor income tax to capital income tax.. Ch. i n U. v. In the case of tax shifting from lump-sum tax to capital income tax, an increase. engchi. in the capital tax rate leads to a decrease in the steady-state equilibrium growth rate via a consumption e¤ect of capital taxation. Intuitively, a higher capital tax rate causes households to decrease their saving rate and increase their consumption rate, which in turn leads to an increase in leisure and a decrease in labor supply. Given that labor is a factor input for R&D, a smaller labor supply gives rise to a lower growth rate of technology, which in turn determines the long-run growth rates of output and capital. 2. Aghion and Howitt (2009, p.108) report that “TFP growth accounts for about two-thirds of economic growth in OECD countries, while capital deepening accounts for one third.”. 6.

(14) In the case of tax shifting from labor income tax to capital income tax, an increase in the capital tax rate leads to an increase in the steady-state equilibrium growth rate via a tax-shifting e¤ect of capital taxation. Intuitively, an increase in the capital income tax rate allows the labor income tax rate to decrease, which in turn leads to a decrease in leisure and an increase in labor supply. The larger labor supply gives rise to higher growth rates of technology, output and even capital despite the lower capital-investment rate caused by the higher capital tax rate. Although the previously mentioned consumption e¤ect of capital taxation is. 政治大 we …nd that the relative magnitude of these two e¤ects becomes very di¤erent in 立 the short run. also present, it is dominated by the tax-shifting e¤ect in the long run. However,. ‧ 國. 學. We calibrate the model to aggregate data in the US to provide a quantita-. ‧. tive analysis on the dynamic e¤ects of capital taxation on economic growth. We consider the case of tax shifting from labor income tax to capital income tax. y. Nat. sit. and …nd that an increase in the capital tax rate leads to a short-run decrease. er. io. in the equilibrium growth rates of technology and output and a gradual conver-. al. n. iv n C U is that the consumption e¤ect hand for these contrasting short-run e nlong-run g c h ie¤ects. gence to the higher long-run growth rates of technology and output. The reason. of capital taxation is relatively strong in the short run. Intuitively, an increase in the capital income tax rate leads to a decrease in the steady-state equilibrium capital-technology ratio. Before the economy reaches this new steady-state capitaltechnology ratio, households drastically cut down their saving rate below its new steady-state level, which in turn increases their consumption rate substantially. This substantial increase in consumption leads to a substantial increase in leisure and a substantial decrease in labor supply, which in turn reduces temporarily the equilibrium growth rates of technology and output. In the long run, the e¤ect of. 7.

(15) a lower wage-income tax rate becomes the dominant force and instead raises the supply of labor, which in turn increases the steady-state equilibrium growth rates of technology and output. Our paper is most closely related to recent studies on taxation and economic growth in the R&D-based growth model. Zeng and Zhang (2002) show that the long-run growth rate is independent of labor income tax and consumption tax but decreasing in capital income tax. In contrast, Lin and Russo (1999) analyze how the taxation of di¤erent sources of capital income a¤ects long-run growth and. 政治大. …nd that a higher capital income tax rate for innovative …rms could be growth-. 立. enhancing if the tax system permits tax credits for R&D spending. Moreover, by. ‧ 國. 學. focusing on the stability analysis of equilibria, Haruyama and Itaya (2006) also show that the growth e¤ect of taxing capital income is positive when the economy. ‧. exhibits indeterminacy. Although these two papers …nd that capital taxation and economic growth may exhibit a positive relationship, our paper departs from them. y. Nat. sit. in highlighting the contrasting dynamic e¤ects of capital taxation on economic. er. io. growth. More recently, Aghion et al. (2013) and Hong (2014) adopt a quality-. al. n. iv n C primary focus, however, is on h theenormative i U with respect to the Chamleyn g c hanalysis. ladder R&D-based growth model to investigate optimal capital taxation. Their. Judd (Chamley 1986; Judd 1985) result (i.e., the optimal capital tax is zero), while the present paper focuses on the positive analysis regarding the growth e¤ect of capital taxation. Furthermore, their analysis does not deal with the case in which innovation is driven by R&D labor (e.g., scientists and engineers). When R&D uses labor as the factor input, we …nd that the e¤ects of capital taxation are drastically di¤erent at di¤erent time horizons. This …nding may provide a plausible explanation for the mixed evidence in the empirical literature on capital taxation and economic growth.. 8.

(16) The remainder of this chapter is organized as follows. In Section 2.2, we describe the basic model structure. In Section 2.3, we investigate the growth e¤ects of capital taxation. In Section 2.4, we calibrate the model to provide a quantitative analysis of capital taxation. Finally, some concluding remarks are discussed in Section 2.5.. 2.2. The model. 治政 The model that we consider is an extension of the 大seminal workhorse R&D-based 立 (1990). In the Romer model, R&D investment creates growth model from Romer 3. ‧ 國. 學. new varieties of intermediate goods. We extend the model by introducing endogenous labor supply and distortionary income taxes. In what follows, we describe. ‧. the model structure in turn.. sit. y. Nat. er. Household. io. 2.2.1. al. n. iv n C The economy is inhabited by h a representative household. Population is stationary engchi U. and normalized to unity. The household has one unit of time that can be allocated between leisure and production. The representative household’s lifetime utility is given as:4 U=. Z. 1. e. t. [ln C +. 0. 3. L)1. (1 1. ] dt,. (1). In the case of extending the model into a scale-invariant semi-endogenous growth model as in Jones (1995), the long-run growth e¤ect of capital taxation simply becomes a level e¤ect. In other words, instead of increasing (decreasing) the growth rate of technology, capital taxation increases (decreases) the level of technology in the long run. 4 For notational simplicity, we drop the time subscript.. 9.

(17) where the parameter parameter. > 0 is the household’s subjective discount rate and the. > 0 determines the disutility of labor supply.. 0 determines the. Frisch elasticity of labor supply. The utility is increasing in consumption C and decreasing in labor supply L 2 (0; 1). Two points regarding the utility function in eq. (1) should be noted. First, to make our analysis tractable, the household is speci…ed to have a quasi-linear utility function. In the quantitative analysis in Section 2.4, we will consider a more general utility function in order to examine the robustness of our results. Second,. 政治大. as pointed out by Hansen (1985) and Rogerson (1988), the linearity in work hours. 立. in the utility function can be justi…ed as capturing indivisible labor.. ‧ 國. 學. The representative household maximizes its lifetime utility subject to the budget constraint:. L )wL. io. sit. + (1. C. Z.. (2). er. K )rK K. y. ‧. Nat. K_ + a_ = (rA + V_ =V )a + (1. The variable K denotes the stock of physical capital. The variable a (= V A). n. al. Ch. i n U. v. denotes the value of equity shares of monopolistic …rms, in which A is the number. engchi. of monopolistic …rms and V is the market value of an invented variety, w is the wage rate. rA is the rate of dividends, V_ =V is the rate of gain or loss of an invented variety, and rK is the capital rental rate.5 The policy instrument Z is a lump-sum tax.6 The other policy instruments f. L;. Kg. < 1 are respectively the labor and. capital income tax rates.7 5. For simplicity, we assume zero capital depreciation rate. We allow for the presence of a lump tax simply to explore the implications of di¤erent tax-shifting schemes. Our main results focus on the more realistic case of Z = 0. 7 In our analysis, we focus on the case in which K > 0; see for example Zeng and Zhang (2007) and Chu et al. (2016), who examine the e¤ects of subsidy policies in the R&D-based growth model. 6. 10.

(18) The rates of return on the two assets, physical capital and equity shares, must follow a no-arbitrage condition at any time:. rA + V_ =V = (1. (3a). K )rK. We denote the common net return on both assets as r, i.e., r (1. rA + V_ =V =. K )rK .. By solving the household’s optimization problem, we can easily derive the typical Keynes-Ramsey rules:. 立. 政治大 _ C = (1 C. ,. K )rK. (3b). ‧ 國. 學. and also the optimality condition for labor supply, which is in the form of a hori-. ‧. zontal labor supply curve given the quasi-linear utility function in eq. (1): = (1. L )w.. n. al Final goods. (4). er. io. 2.2.2. y. L). sit. Nat. (1. Ch. engchi. i n U. v. There is a single …nal good Y , which is produced by combining labor and a continuum of intermediate goods, according to the following aggregator: Y =. L1Y. Z. A. xi di,. (5). 0. where LY is the labor input in …nal goods production, xi for i 2 [0; A] is the intermediate good of type i, and A is the number of varieties of intermediate goods. The …nal good is treated as the numeraire, and hence in what follows its price is normalized to unity. We assume that the …nal goods sector is perfectly. 11.

(19) competitive. Pro…t maximization of the …nal goods …rms yields the following conditional demand functions for labor input and intermediate goods: LY = (1. (6). )Y =w,. xi = LY ( =pi ) 1. 1. ,. (7). where pi is the price of xi relative to …nal goods.. 2.2.3. 政治大. Intermediate goods. 立. Each intermediate good is produced by a monopolist who owns a perpetually. ‧ 國. 學. protected patent for that good. Following Romer (1990), capital is the factor input for producing intermediate goods, and the technology is simply a linear one-to-one. ‧. function. That is, the production function is expressed as xi = ki , where ki is the. sit. y. Nat. capital input used by intermediate …rm i. Accordingly, the pro…t of intermediate. io. i. n. al. Ch. = p i xi. er. goods …rm i is:. rK ki .. i n U. v. (8). Pro…t maximization subject to the conditional demand function for intermediate. engchi. goods …rm i yields the following markup-pricing rule: pi =. rK. > rK .. (9). Equation (9) implies that the level of price is the same across intermediate goods …rms. Based on eq. (7) and the production function xi = ki , we have a symmetric equilibrium among intermediate …rms; i.e., xi = x and ki = k. Then, we can obtain the following pro…t function of intermediate goods …rms: i. =. =. (1. ) Y A. 12. .. (10).

(20) 2.2.4. R&D. In the R&D sector, the familiar no-arbitrage condition for the value of a variety V is: rV =. + V_ .. (11). Equation (11) states that, for each variety, the rate of return on an invention must be equal to the sum of the monopolistic pro…t and capital gain (or loss) . As in Romer (1990), labor is the factor input of R&D. The innovation function of new varieties is given by:. 立. (12). A. > 0 is the R&D productivity parameter and LA denotes R&D labor.. ‧ 國. 學. where. 政治大 A_ = AL ,. Given free entry into the R&D sector, the zero-pro…t condition of R&D is (13). sit. y. Nat. er. Government. io. 2.2.5. ‧. _ = wLA , AV = w. AV. al. n. iv n C h eincluding The government collects taxes, i U income tax, labor income tax, n g c hcapital and lump-sum tax, to …nance its public spending. At any instant of time, the government budget constraint can be expressed as: K rK K. +. L wL. + Z = G.. (14). The variable G denotes government spending, which is assumed to be a …xed proportion. 2 (0; 1) of …nal output such that G= Y.. 13. (15).

(21) 2.2.6. Aggregation. Since the intermediate …rms are symmetric, the total amount of capital is K = Aki = Ak. Given xi = ki , xi = x, ki = k, and K = Ak, the …nal output production function in eq. (5) can then be expressed as: K L1Y :. Y = A1. (16). After some calculations using eqs (2), (6), (7), (11)-(14), and (16), we can derive. 政治大 K_ = Y C G:. the resource constraint in this economy:. 立. ‧ 國. 學. 2.2.7. (17). Decentralized equilibrium and the balanced-growth path. ‧ K;. L;. Z g, such that at any. er. al. n. instant of time:. io. 1 x ; Gg1 t=0 , prices fw; r; rK ; pi ; V gt=0 , and policies f. sit. y. Nat. The decentralized equilibrium is a sequence of allocations fC; K ; A; Y ; L; LY ; LA ;. Ch. engchi. i n U. v. a. households maximize lifetime utility (1) taking prices and policies as given; b. competitive …nal goods …rms choose fx; LY g to maximize pro…t taking prices as given; c. monopolistic intermediate …rms i 2 [0; A] choose fki ; pi g to maximize pro…t taking rK as given; d. R&D …rms choose LA to maximize pro…t taking fV; wg as given; e. the market for …nal goods clears, i.e., K_ = Y f. the labor market clears, i.e., L = LA + LY ; 14. C. G;.

(22) g. the government budget constraint is balanced, i.e.,. K rK K + L wL + Z. = G.. The balanced growth path is characterized by a set of constant growth rates of all economic variables. Let. denote the growth rate of technology and a “~”. over the variable denote its steady-state value. Along the balanced growth path, we have K_ Y_ C_ w_ A_ = = = = = ~ , L_ = L_ Y = L_ A = V_ = 0. K Y C w A. 治政 Long-run growth e¤ects of capital 大 taxation 立學. ‧ 國. 2.3. (17a). We now turn to examine the long-run growth e¤ects of the capital tax rate. In this section to obtain analytical solutions, we assume that. = 0. As a result, to. ‧. maintain a constant proportion of government spending, raising the capital tax. sit. y. Nat. must be accompanied by a reduction in another tax. As revealed in eq. (14), this. io. er. can be either a reduction in the lump-sum tax (if it is available) or a reduction in the labor income tax (if the lump-sum tax is not available). In the analysis that. n. al. Ch. i n U. v. follows, we deal with each of the two scenarios in turn.. 2.3.1. engchi. Tax shifting from lump-sum tax to capital income tax. Equipped with the de…nition of the decentralized equilibrium in Section 2.2.7,and de…ning ! = w=A, c = C=A; and z = Z=A, we can express the steady-state. 15.

(23) equilibrium conditions as follows: ~ = (1 ! ~ = ~Y L. rK K )~. c~=(1. (18b). L );. = (1. ~Y ( x~ = L. (18a). ;. 2. ~ 1 =~ )~ x L !; Y. (18c). =~ rK )1=(1. (18d). ). ;. ~Y ; L. r~ = r~ = (1. (18e) (18f). rK ; K )~. 政~ =治L~ ; 大 ~ ~ ~. (18g). A. ~ = (1 ~K x~ Kr. +. (18h). L = LY + LA ;. ~ + z~ =. )~ x. ~1 ; x~ L Y. ~L L!. 1 ~1 LY. 學. c~=~ x;. (18i) (18j). ‧. ‧ 國. 立. ~Y , L ~ A , L, ~ ! in which ten equations are used to solve ten unknowns ~ , r~K , L ~ , c~,. Nat. sit. y. x~, r~ and z~. We brie‡y discuss how we obtain eqs (18). Equation (18a) is derived. er. io. from the usual Keynes-Ramsey rule (3b). Equation (18b) is derived from the opti-. al. iv n C the demand functions for …nal-goods labor andUintermediate goods, (6) and (7). hengchi _ n. mality condition for labor supply (4). Equations (18c) and (18d) are respectively. Equation (18e) is derived from inserting V = 0 into the no-arbitrage condition in the R&D sector (11), and by using eqs (6), (10) and (13). Equation (18f) is the no-arbitrage condition of asset. Equation (18g) is derived from the innovation function of varieties (12). Equation (18h) is the labor-market clearing condition. Equation (18i) is derived from dividing both sides of the resource constraint (17) by A and using the condition Ax = K. Equation (18j) is derived from dividing both sides of the government constraint (14) by A and using the condition G = Y . We …rst use eqs (18a), (18e) and (18f)-(18h) to eliminate f~ r; ~ ; r~K g and express 16.

(24) ~Y ; L ~ A g as functions of L ~ given by fL ~ ~Y = L + = , L 1+ ~ L = . 1+. ~A = L. ~ A; L ~ Y g and L. ~ These two equations indicate a positive relationship between fL ~ A , we can derive the condition ~ = Moreover, from the previous condition for L (. ~ L. )=(1 + ), which shows that the steady-state equilibrium growth rate of. ~ Thus, we have technology is increasing in L. ! ~A @~ @L sgn = sgn = sgn @ K @ K. 政治大. 立. ~ @L @ K. !. .. (19). ‧ 國. 學. Accordingly, to investigate the growth e¤ect of the capital tax rate, it is convenient ~ to draw an inference from examining the e¤ect of the capital tax rate on labor L.. ‧. ~ By using eqs (8) and (9), We now derive an equilibrium expression of labor L.. y. sit. 2. Y . Then, dividing both sides of eq. (17) by Y yields. io. n. al. By inserting C=Y = (1 and rK K =. 2. Ch L )(1. ~. C . Y. K =1 Y. er. rK K =. 1)~ rK K=A. This expression together with eq. (10) implies that. Nat. = (1. we have. i n U. v. i e n)=(gL~c ),hwhich is derived from eqs (4) and (6), Y. Y into the above equation and using eqs (18e), (18f), and (18g). ~ Y and L ~ A , we can obtain the following equation along with the conditions for L ~ with one unknown L: 1. (1 + ) ~+ = ) (L. 2. (1. K). (1. =1. L )(1. )(1 + ) . ~ (L + = ). Simplifying this equation yields ~= L. 1. 1 (. 1. L. K). 17. (1 (1. K). ). ,. (20).

(25) where. (. K). 2. (. 2. K )=(1. ) is a composite parameter and. L. is an ex-. ogenous policy parameter. Then, from eq. (20), we can obtain the following relationship: ~ @L = @ K. 2. (1. 1. 2 )[1. (. K. L. )]2. (1 (1. K). [1. ). (. K )]. 1+. ,. which can be further simpli…ed to8 ~ @L = @ K. ~A + 2 [(1 + )L 2 )[1 (1 (. = ] < 0. K )]. (21). 政治大. From eqs (19) and (21), we have established the following proposition:. 立. Proposition 1 In the case of tax shifting from lump-sum tax to capital income. ‧ 國. 學. tax, raising the capital income tax rate reduces the steady-state equilibrium growth rate.. ‧. Nat. sit. y. Equation (19) is the key to understanding Proposition 1. It essentially says. er. io. that the e¤ect of the capital tax rate on long-run growth hinges on its e¤ect on ~ When the capital tax rate is higher, households tend to reduce their labor L.. n. al. Ch. i n U. v. investment rate and increase their consumption rate. The increase in consumption. engchi. raises leisure and reduces labor supply (by shifting up the horizontal labor supply curve). Therefore, a higher capital tax rate reduces the equilibrium levels of labor input, R&D labor and economic growth. 8. 2 The following reasoning ensures that 1 ( K ) = [1 (1 K )]=(1 2 2 steady-state consumption-output ratio is C=Y = 1 (1 (1 K) + 2 Therefore, lim !0 C=Y = 1 (1 K ). In other words, one can restrict 1 appealing to the fact that C=Y > 0 for all values of .. 18. 2. ) > 0. The =(e + ). ( K ) > 0 by. K).

(26) 2.3.2. Tax shifting from labor income tax to capital income tax. A lump-sum tax is not a realistic description in most economies. In this subsection, we therefore set aside the possibility of a lump-sum tax and deal with the more realistic case in which a rise in the capital tax rate is coupled with a reduction in another distortionary tax. This kind of tax shifting has been extensively investigated in the literature on factor taxation; see e.g., Judd (1985), Chamley (1986),. 政治大 uation we drop z~ from 立the model in this subsection. Thus, eq. (18j) is rewritten ~K x~ Kr. 學. as:. ‧ 國. Niepelt (2004), Aghion et al. (2013) and Chen and Lu (2013). Under such a sit-. ~ = x~ L ~1 . + ~L ! ~L Y. (22). ‧. It is useful to note that in eq. (22) the labor income tax rate ~L becomes an endogenous variable because it needs to adjust in response to a change in the. y. Nat. al. er. io. sit. capital tax rate.. n. The macroeconomy is now described by eqs (18a)-(18i) and (22) from which. Ch. i n U. v. ~Y , L ~ A , L, ~ ! we solve for ten unknowns ~ , r~K , L ~ , c~, x~, r~ and ~L . By arranging eq.. engchi. (22) with eqs (6), (16), (18c), and the condition rK K =. 2. Y , we can obtain. ~Y L = 1+ ( K ), ~ ~ 1 L L ~ Y = (L ~ + = )=(1 + ). Using the above condition where the second equality uses L ~L =. 2. (. K). ~ ~L g and obtain the following and eq. (20), we can solve the two unknowns fL; quadratic equation: ~2 L. 1. (1. (1 )[1. K). (. K )]. ~+ L. ( [1. K). (. K )]. = 0.. ~ 1 and L ~ 2 , which are given This quadratic equation has two solutions, denoted as L by: 19.

(27) where B (. ~1 = L. B(. K). ~2 = L. B(. K). K). =( ). +. p. B(. p. B(. 1. 2 K). 4 ( 2 =. K). =f[1. (. K )]. g. ,. (23a). 2 K). 4 ( 2 =. K). =f[1. (. K )]. g. ,. (23b). )[1. (. (1. K )=f(1. K )]g. is a composite. parameter.9 ~ is positive, we assume that the set of parameters jointly To ensure that L p satis…es the condition B > 4 =[(1 ) ]. Moreover, we restrict our analysis. 政治大. 立. to the case where an increase in the capital tax rate is coupled with a decrease in. ‧ 國. 學. ~ 1 is the only possible solution the labor tax rate. By doing so, we can show that L. =. [1. +. 2. (1. K )=(1. 2. )] =f(1. y. K. )(1. sit. Nat. where @B =@. ‧. to this system.10 From eq. (23a), we can derive the relationship: ( ) 2 ~1 @B @L B @B =@ K + 2 2 =[(1 )(1 )2 ] p = + >0 @ K 2 @ K B 2 4 =[(1 ) ]. (24). )2 g > 0. The. er. io. result in eq. (24) leads us to establish the following proposition:. al. n. iv n C Proposition 2 In the case ofhtax shifting from U e n g c h i labor income tax to capital income. tax, raising the capital income tax rate increases the steady-state equilibrium growth rate.. It would not be di¢ cult to understand the intuition underlying the positive growth e¤ect given that we have already shown the importance of equilibrium 9. For notational simplicity, we suppress the argument of ( K ) and B ( K ) in the following equations. 10 Based on the de…nition of tax shifting, an increase in one tax rate should be coupled with a ~ 2 , to hold a constant fall in another tax rate. In Appendix 2.A, we will show that when L = L proportion of the government spending, the labor tax rate actually increases in response to an increase in the capital tax rate. In this paper, we rule out this unrealistic case and only focus on ~1. the solution L = L. 20.

(28) labor input on economic growth from previous discussion. In the present case, there are two con‡icting e¤ects on labor supply. The …rst is the consumption e¤ect that we discussed in Proposition 1; i.e., raising the capital tax rate induces the households to lower the investment rate and increase the consumption rate, which in turn reduces labor supply. The second e¤ect emerges from the channel of shifting taxes from labor income to capital income. A rise in the capital income tax rate leads to a reduction in the labor income tax rate, which tends to boost labor supply. In particular, this latter tax-shifting e¤ect has a more powerful direct. 政治大 net e¤ect is positive such that a higher capital income tax rate stimulates economic 立 growth in the long run. impact on the labor market so that it dominates the former one. As a result, the. ‧. ‧ 國. 學. 2.4. Quantitative analysis. sit. y. Nat. io. follows:. er. To examine the robustness of our results, we generalize the utility function as. n. aU l= Z ln C + (1 L) iev dt; Ch 1 Un e n chi 0 determines the Frischgelasticity of labor supply. 1. 1. t. (25). 0. where. nests eq. (1) as a special case when f ; ; ; ; ; ;. K g.. Equation (25). = 0. The model features 7 parameters:. We consider the following standard parameter values or em-. pirical moments in the literature. First, we set the discount rate to the capital share to. = 0:30. Second, we set. = 0:04 and. = 1:67, which implies a Frisch. elasticity of 1.2; see Chetty et al. (2011). Third, in line with Belo et al. (2013), the government spending ratio is set to of two-thirds (i.e., L = 1=3), we set. = 0:20. Fourth, to obtain a leisure time = 1:17. Fifth, to generate a steady-state. output growth rate of 1.92%, which is the per capita long-run growth rate of the 21.

(29) US economy, we set rate is set to. K. = 0:65. Finally, the benchmark value of the capital tax. = 0:36; see for example Lucas (1990). The parameter values are. summarized below.. Table 2.1: Calibrated parameter values K. 0:04. 1:17. 1:67. 0:30. 0:65. 0:20. 0:36. Figure 2.1 presents the growth e¤ects of varying the capital income tax rate. 政治大. from 0 to 0.6. We can clearly see that, as the capital tax rate increases, the steady-. 立. state equilibrium growth rate increases. From this illustrative numerical exercise,. ‧ 國. 學. we …nd that if the government raises the capital tax rate from the benchmark value of 36% to a hypothetical value of 50%, the steady-state equilibrium growth. ‧. rate increases from 1.92% to 2.02%. The intuition can be explained as follows.. y. Nat. Although an increase in the capital tax rate exerts a negative e¤ect on economic. io. sit. growth by depressing capital accumulation, it also causes a fall in the labor income. n. al. er. tax rate, which boosts labor supply and thus is bene…cial to R&D and economic. i n U. v. growth. In the long run, the latter e¤ect dominates. Consequently, the steady-. Ch. engchi. state equilibrium growth rate increases in response to a rise in the capital income tax rate.11 In the rest of this section, we simulate the transition dynamics of an increase in the capital income tax rate. The dynamic system is presented in Appendix 2.B. We consider the case of an increase in the capital income tax rate by one percentage point (i.e., from 36% to 37%).12 First of all, the higher rate of capital taxation 11. Our simulation result is robust if we introduce dividend income taxes into our model, see Appendix 2.C. 12 In the case of a larger increase in the capital income tax rate, the qualitative pattern of the transitional paths of variables remains the same. Results are available upon request.. 22.

(30) Figure 2.1: Long-run growth e¤ect of capital taxation. 政治大. leads to a decrease in the investment rate and an increase in the consumption rate. 立. _ as shown in Figures 2.2 and 2.3, where investment I = K.. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 2.2: Transition path of the investment rate. 23.

(31) 政治大 Figure 2.3: Transition path of the consumption rate 立. ‧ 國. 學. The lower capital-investment rate gives rise to an initial fall in the capital. ‧. growth rate as shown in Figure 2.4, which contributes to an initial fall in the output growth rate as we will show later. The rise in the consumption rate increases leisure. y. Nat. sit. and decreases labor supply as shown in Figure 2.5. This decrease in labor supply. n. al. er. io. reduces the amount of factor input available for R&D. As a result, the growth rate. i n U. v. of technology also decreases initially as shown in Figure 2.6.. Ch. engchi. 24.

(32) 立. 政治大. Figure 2.4: Transition path of the capital growth rate. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 2.5: Transition path of labor supply. 25.

(33) 政治大 Figure 2.6: Transition path of the technology growth rate 立. ‧ 國. 學. Although tax shifting resulting from a higher capital income tax rate gives rise. ‧. to a lower labor income tax rate, this e¤ect is weak in the short run. However, it becomes a stronger force in the long run as shown in Figure 2.7. As a result, labor. y. Nat. sit. supply eventually rises above the original level, which in turn leads to a higher. n. al. er. io. steady-state equilibrium growth rate of technology. Therefore, the initial drop in. i n U. v. the growth rates of output and capital is followed by a subsequent increase. In. Ch. engchi. the long run, the steady-state equilibrium growth rate of output is higher than the initial steady-state equilibrium growth rate as shown in Figure 2.8. To sum up, the reason for the contrasting short-run and long-run e¤ects of capital taxation on economic growth is that the consumption e¤ect is stronger (weaker) than the tax-shifting e¤ect in the short (long) run.. 26.

(34) 立. 政治大. Figure 2.7: Transition path of the labor tax rate. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 2.8: Transition path of the output growth rate. 27.

(35) 2.5. Conclusion. In this chapter, we have explored the short-run and long-run e¤ects of capital taxation on innovation and economic growth. Our results can be summarized as follows. An increase in the capital income tax rate has both a positive tax-shifting e¤ect and a negative consumption e¤ect on innovation and economic growth. In the long run, increasing the capital tax rate has an unambiguously positive e¤ect on the steady-state equilibrium growth rate because the positive tax-shifting e¤ect. 政治大 path, increasing the capital tax rate …rst decreases the equilibrium growth rates of 立 technology and output before these growth rates converge to a higher steady-state strictly dominates the negative consumption e¤ect. However, along the transitional. ‧ 國. 學. equilibrium level. These contrasting implications of capital taxation on economic growth suggest that a complete empirical analysis of capital taxation and economic. ‧. growth needs to take into consideration the possibility that the e¤ects of capital. n. al. er. io. sit. y. Nat. taxation change sign at di¤erent time horizons.. Ch. engchi. 28. i n U. v.

(36) Appendix 2.A The system has ten equations, (18a)-(18i), and (22). After some calculations, we ~ and ~L : can derive the following expressions for L. ~2 L. (1. 1. (1. K). )(1. ). ~+ L. (1. ). (A1). = 0;. ~ ; ~ = [1 + 治=( L)] 政大. (A2). L. n. (1. Ch. 4. =[(1. ) ]. 4. y. B2. =[(1. ) ]. sit. p. (A4). ;. 2. K )=[(1. (A3). ;. 2. ‧. al. 1. p B2. 學. B. ~2 = L. io =( ). ~ ). Equation (A1) gives the two solutions for L:. B+. ~1 = L. Nat where B. 2. K )=(1. )(1. engchi. er. 立. 2. (. ‧ 國. where. )].. i n U. v. As mentioned in the main text, our analysis focuses on the case where the notion of tax shifting is sustained. That is, we impose the condition @~L =@ We can then show that the condition @~L =@. K. where. p B2. = ~2 L=L. 4. =[(1. (1. + (1 2 1. ) ] and. 29. < 0.. ~ 2. < 0 does not hold if L = L. ~ 2 into eq. (A2) and di¤erentiating it with respect to By plugging L. @~L @ K. K. = )). +. @ @. =2: K. K. yields:. (A5).

(37) @ @ where @B =@. K. =. 1. B. K. @B + @ K (1. 2 2 2 )(1. > 0;. )2. > 0. It is clear from eq. (A5) that @~L =@. ~2 K jL=L. (A6) is positive,. which contradicts the assumption of tax shifting. Therefore, we should rule out ~=L ~ 2. the possibility L. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 30. i n U. v.

(38) Appendix 2.B This appendix solves the dynamic system of the model under tax shifting from labor income taxes to capital income taxes (Z = 0 ). The set of equations under the model is expressed by:. (B1). 1=C =. K. (B3). _ C=C = (1. K )rK. (B4). wLY. )Y. K. = (1 2. =. Y. (1. )Y. io. n. V_ + r = V V. Ch. G =. Y. i n U. eYn =g c hr i K + K K. Y. y. Nat. A. ). sit. 2. rK K =. =rK )1=(1. v. L wL. = K (ALY )1. K_ = Y. C. (B5). ‧. x = LY (. al. (B2). L. 學. ‧ 國. 立. L) = (1 )w 政治大 r = (1 )r. er. (1. G. (B6) (B7) (B8) (B9) (B10) (B11) (B12) (B13). A_ =. ALA. (B14). V. w A. (B15). =. L = LY + LA. (B16). in which 16 equations are used to solve 16 unknowns endogenous variables fC; L; A; K; LY ; x; rK ;. ; r; G;. L;. Y;. ; LA ; V; wg; where 31. denotes the Hamil-.

(39) tonian multiplier. Based on K = Ax, eqs (B1), (B2), (B5), and (B12) and let f = C=K be the ratio between consumption and capital, we can obtain:. L=1. [(1=( f ))(1. )(1=LY )(LY =x)1. L )(1. ]. 1=. :. (B17). Based on eqs (B5), (B7), and (B11), we have:. 2 L. =. K. 1. (. LY ): L. (B18). 政治大 We now turn to deal with the transitional dynamics of the model. By using 立 x = K=A; eqs (B16), (B17), and (B18), we can infer the following expression: ‧. ‧ 國. 學 K );. (B19). n. a l@L = @xC h. (1 x(. @L = @f f(. @L @LY. =. @L @ K. =. 1 L. +. (1. 1 L. +. (1. 2. K. 1. LY ( (. 2 LY L. (. ;i n U. ). e n g c1h i (. er. io. sit. y. Nat. where. L = L(x; f; LY ;. LY L 1. L L )L. ). )=(1 + (1 L. )=((1 1 L. v. (B20b). ; L). + ; ) L )L. L. )(1 +. (B20a). ) L )L. L. L. (1. L )L. L )). ). :. (B20c) (B20d). Based on eqs (B14) and (B15), we have:. V_ w_ = V w 32. A_ : A. (B20e).

(40) From K = Ax, eqs (B5), and (B12), we can further obtain:. _ w=w _ = A=A. L_ Y =LY + x=x: _. (B20f). Additionally, substituting eq. (B20f) into eq. (B20e) yields:. V_ = (x=x _ V. L_ Y =LY ):. (B20g). Combining eqs (B3), (B5), (B7), (B8), (B9), (B12), and (B15), we can obtain:. 政治大. 立. K). 2. (. LY 1 ) x. =. V_ : V. LY +. (B21a). 學. ‧ 國. (1. Substituting eq. (20g) into eqs (B21a), (B21a) can be rearranged as:. ‧. K). io. LY 1 ) x. :. (B21b). n. er. Based on x = K=A, we have the result:. al. (. y. (1. sit. Nat. L_ Y =LY = LY + x=x _. i _ _ n C hx=x _ = K=K A=A: engchi U. v. (B21c). Substituting f = C=K, eqs (B10), (B13), (B14), and (B16) into eq. (B21c), we have:. x=x _ = (1. )(. LY 1 ) x. f. (L. LY ):. (B21d). From eqs (B21b) and (B21d), we can obtain:. L_ Y =LY = (1. (1. K ))(. LY 1 ) x 33. f. (L. 2LY ):. (B21e).

(41) Moreover, from eqs (B3), (B4), (B5), (B7), and (B12) we can obtain:. _ C=C = (1. 2. K). (. LY 1 ) x. (22a). :. Based on f = C=K, we have the following expression:. _ f_=f = C=C. _ K=K:. (B22b). 政治大. Substituting eqs (B10), (B12), (B13), and (B22a) into eq. (B22b), we can derive:. 立. ‧ 國. K). 2. (1. ))(. LY 1 ) x. 學. f_=f = ((1. (B22c). + f:. Based on eqs (B19), (B21d), (B21e), and (B22c), the dynamic system can be. ‧. expressed as:. L_ Y =LY. f. (L(x; f; LY ;. K. Y. 2. Y. = (1. K). K. (B23a). LY );. a l ) (1 ))( L ) i vf; + n x Ch e n))(gL c) h i Uf (L(x; f; L (1. n. f_=f = ((1. LY 1 ) x. er. io. )(. sit. y. Nat x=x _ = (1. 1. 1. Y;. x. (B23b) K). 2LY ): (B23c). Linearizing eqs (B23a), (B23b), and (B23c) around the steady-state equilibrium yields:. 0. 1. 0. B x_ C B b11 b12 b13 B C B B f_ C = B b B C B 21 b22 b23 @ A @ L_ Y b31 b32 b33. 10 CB CB CB CB A@. x f LY 34. x~. 1. 0. C B b14 C B B f~ C C + B b24 A @ ~Y L b34. 1. C C Cd C A. K,. (B24).

(42) where. Y; ( @L(x;f;L @f. b12 = x( 1 b13 = (1. (1. ‧ 國 (1. )(1. LY. ))( LxY ). (1. ( fx );. @L(x;f;LY ; @x. LY 2 K ))( x ). (1. al. n. b34 =. LY ( LxY )1. ( fx );. f;. @L(x;f;LY ; @f. )(1. ))( LxY )1. (1. 政治大. 2. K). io b33 = (1. 1);. ‧. 2 LY 1 ( x ). K). 學. b32 =. 2. K). )((1. Nat. b31 =. )); Y; x( @L(x;f;L @LY. 立. b23 = (1. ;. );. )((1. b22 = f;. b24 =. K). K). K). LY ;. y. b21 =. K). )( LxY ). )(1 Y; ( @L(x;f;L @ K. b14 =. Y; x @L(x;f;L @x. )( LxY )1. )(1. K). (1. sit. (1. LY ;. Ch. LY 1 K ))( x ) K). v ni. Y; ( @L(x;f;L @LY. e n gLc):h i U. Y; ( @L(x;f;L @ K. er. b11 =. K). LY. 2LY );. Y. Let `1 , `2 , and `3 be the three characteristic roots of the dynamic system. We do not analytically prove the saddle-path stability of the dynamic system; instead, we show that the dynamic system features two positive and one negative characteristic roots numerically. For expository convenience, in what follows let `1 be the negative root and `2 , and `3 be the positive roots. From eq. (B24), the general solutions for xt ; ft , and LY;t , can be described by:. 35.

(43) xt = x~ + D1 e`1 t + D2 e`2 t + D3 e`3 t ;. (B25a). ft = f~ + h1 D1 e`1 t + h2 D2 e`2 t + h3 D3 e`3 t ;. (B25b). ~ Y + `1 LY;t = L. b11 b12 h1 `2 D1 e`1 t + b13. b11 b12 h2 `3 D2 e`2 t + b13. b11 b12 h3 D3 e`3 t : b13 (B25c). where h1 = [(`1 b11 ). b33 )(`1. b11 ). b31 b13 ]=[b32 b13 + b12 (`2. b12 (`3. b31 b13 ]=[b32 b13 + b12 (`1 b33 )]; h3 = [(`3. b33 )]; h2 = [(`2. b33 )(`3. b11 ). 政治大. b33 )(`2. b31 b13 ]=[b32 b13 +. b33 )], and D1 ; D2 and D3 are undetermined coe¢ cients.. 立. The government changes the capital tax rate. K. from. to. K0. K1. at t=0, based. ‧ 國. 學. on eqs (B25a)-(B25c), we employ the following equations to capture the dynamic adjustment of xt , ft , and LY;t :. ‧ y. al. K0 );. Ch. > : f~( K1 ) + h1 D1 e`1 t 8 > < L ~ Y ( K0 ); > ~Y ( : L. K1 ). +. 2. 2. i n U. 3. 3. (B26a). 0+. e hi + hnDge c + h D e `2 t. sit. + D1 e`1 t + D2 e`2 t + D3 e`3 t ; t. er. K1 ). t=0. n. LY;t =. > : x~( 8 > < f~(. K0 );. io. ft =. Nat. xt =. 8 > < x~(. `3 t. v. t=0. ; t. (B26b). 0+ t=0. `1 b11 b12 h1 D1 e`1 t b13. +. `2 b11 b12 h2 D2 e`2 t b13. +. `3 b11 b12 h3 D3 e`3 t ; b13. t. 0+ (B26c). where 0 and 0+ denote the instant before and after the policy implementation, respectively. The values for D1 ; D2 ; and D3 are determined by:. 36.

(44) = x0+ ;. (B27a). D2 = D3 = 0:. (B27b). x0. Equation eq. (B27a) indicates that the level of intermediate goods remains unchanged at the instant of the policy implementation. Equation (B27b) is the stability condition which ensures that all xt , ft , and LY;t converge to their new steady-state equilibrium. By using eqs (B27a) and (B27b), we can obtain:. 政治大. 立. D1 = x~(. K0 ). x~(. (B28). K1 ):. ‧ 國. 學. Inserting eqs (B27b) and (B28) into eqs (B26a)-(B26c) yields:. LY;t =. > : 8 > <. f~(. y. x( K1 ) + (~. al. n. ft =. t=0 K0 ). x~(. `1 t K1 ))e ; t. (B29a). sit. io. > : x~( 8 > < f~(. K0 );. 0+. er. xt =. ‧. Nat. 8 > < x~(. iv t = 0 n C ) +h he (~ xn ( g)c hx~(i U))e ; t 0. K0 ); K1. ~Y ( L. > ~Y ( : L. 1. K0. K1. `1 t. K0 ); K1 ) +. (B29b). +. t=0 `1 b11 b12 h1 (~ x( K0 ) b13. 37. x~(. `1 t K1 ))e ; t. 0+. (B29c).

(45) Appendix 2.C In the case of tax shifting from labor income taxes to capital income taxes, raising the capital income tax rate increases the steady-state equilibrium growth rate. This result relies on some assumptions. In this appendix, we will relax them and examine whether the Proposition 2 is still robust. Speci…cally, this appendix takes into account dividend income taxes. We will use a quantitative analysis to show that Proposition 2 still holds if we introduce dividend income taxes into our model.. 政治大 capital income taxes.立 In this appendix we relax this assumption by considering. In subsection 2.3.2 we deal with a tax-shifting from labor income taxes to. ‧ 國. 學. tax-shifting from labor income taxes to both capital income taxes and divident income taxes. with this consideration, household’s budget constraint reported in. ‧. eq. (2) and government’s budget constraint reported in eq.(22) can be respectively modi…ed as follows:. n. al. A )rA. + V_ =V )a + (1. Ch. K rK K. er. io. sit. y. Nat K_ + a_ = ((1. +v(1 i n. K )rK K. e+ n rgVcAh+i UwL = A A. L. Y:. L )wL. C,. (C1) (C2). where rA a is agent’s total dividend income. The rates of return on the two assets, physical capital and equity shares, must follow a no-arbitrage condition at any time:. r. (1. A )rA. + V_ =V = (1. K )rK. (C3). Given that the government imposes the same tax rate on both capital income and dividend income (. K. =. A). and the long-run market value of an invented variety 38.

(46) V is equal to constant since V = w=( A), and eq. (17a) hold in the long run, we then have: (C4). rK = rA :. From eqs (18a)-(18i), (C1), (C2), (C3), and (C4), after some tedious calculations ~ as the level of steady-state labor supply, we then have: as well as de…ning L (1. 2. 1. L). (1. 1 2. = 1. K. (1. ~A = ( L ~ From eq. (18g), L. 立. 1+. K). 2. (1 ) K (1 K ). ~+ (L. ) + (1. K). (1. ). (C3). ~+ L ~ L. 政治大 = )=(1 + ); eq. (C3), and the following standard 2). parameter values, we can obtain values of long-run growth rate with respect to. ‧ 國. 學. varying the capital income tax rate. The parameter values are summarized below in Table 2.C.1. ‧. io. y. sit. K. 0:04. n. al. 1:17. Ch. 1:67. 0:30. 0:65. engchi. er. Nat. Table 2.C.1: Calibrated parameter values. 0:20. i n U. v. 0:36. Figure 2.C.1 presents the growth e¤ects of varying the capital income tax rate from 0 to 0.6. We can clearly see that, as the capital tax rate increases, the steadystate labor supply increases (see Figure 2.C.2), and thus the growth rate increases (see eq.(19)). The intuition can be explained as follows. Although an increase in the capital tax rate and dividend income tax rate exerts a negative e¤ect on economic growth by depressing household’s saving, it causes dramatic fall in the labor income taxes, which boosts labor supply and thus is bene…cial to R&D and economic growth. In the long run, the latter e¤ect dominates. Compared with tax-shifting from labor income taxes to capital income taxes, tax-shifting from. 39.

(47) labor income taxes to both capital income taxes and dividend income taxes leads to lower wage income taxes and hence results in higher labor supply (see Figure 2.C.2). However, our result is robust if we consider tax-sh…ting from labor taxes to both capital income taxes and dividend income taxes.. 立. 政治大. ‧. ‧ 國. 學 Figure 2.C.1. n. er. io. sit. y. Nat. al. Ch. engchi. Figure 2.C.2. 40. i n U. v.

(48) CHAPTER 3 OPTIMAL CAPITAL TAXATION AND R&D EXTERNALITIES. 3.1. Introduction. Capital income is taxed worldwide. The estimated e¤ective average tax rates on capital income are around 40% in the United States and 30% in EU countries. In some countries, such as the United Kingdom and Japan, the capital income tax. 政治大. rates are even up to near 60%. From the perspective of welfare maximization,. 立. whether these capital tax rates are too high or too low is a question that will never. ‧ 國. 學. cease being debated by economists and policymakers.. Despite that capital taxes are commonly levied in the real world, a striking. ‧. theory put forth by Judd (1985) and Chamley (1986) suggests that the government. Nat. sit. y. should only tax labor income and leave capital income untaxed in the long run. A. io. al. er. number of subsequent studies, including Chari et al. (1994), Jones et al. (1997),. n. Atkinson et al. (1999), and Chari and Kehoe (1999), relax key assumptions in. Ch. i n U. v. Judd (1985) and Chamley (1986), and …nd their result to be quite robust. The. engchi. idea of a zero optimal capital tax has then been dubbed the Chamley-Judd result, which turns out to be one of the most well-established and important benchmarks in the optimal taxation literature. In this paper, we revisit the Chamley-Judd result in an innovation-based growth model. There are several reasons with regard to why we choose this environment to study optimal taxation. First, as stressed by Aghion et al. (2013), the consideration of growth seems not to play much of a role in the debate of the Chamley-Judd result. However, given that the recent empirical evidence suggests a signi…cant. 41.

(49) impact of the tax structure on economic growth (e.g., Arnold et al., 2011), it is more plausible to bring the role of growth into the picture. Second, along the line of the optimal taxation literature, production technology is treated as exogenously given. The role of endogenous technological change driven by R&D has thus been neglected in previous models. In view of the fact that innovation is a crucial factor in economic development as well as in the improvement of human well-being, overlooking this facet could lead to inadequate design of tax policies. Our study thus aims to …ll this gap. Third, as pointed out by Domeij (2005), a key premise in early. 政治大 distortions (externalities) in the economy. If incomplete markets are present, the 立 optimal capital income tax might be di¤erent from zero. Thus, we introduce an contributions supporting the Chamley-Judd result is that there exist no inherent. ‧ 國. 學. innovation market featured with various R&D externalities put forth by Jones and Williams (2000). Within this framework we can study how the optimal capital. ‧. taxation and R&D externalities interact in ways not heretofore understood.. y. Nat. sit. By calibrating the model to the US economy, our numerical analysis shows that. er. io. the optimal capital income tax is signi…cantly positive at a rate of 12 percent. The. al. n. iv n C h e nIngessential, can be brie‡y explained as follows. c h i Uthe Chamley-Judd result involves reason for a positive optimal capital income tax in our R&D-based growth model. a tax shift between capital income tax and labor income tax. The basic rationale behind a zero optimal capital tax is that taxing capital generates more distortion than taxing labor, because taxing capital creates a dynamic ine¢ ciency for capital accumulation. In our R&D-based growth model, by contrast, labor is considered as the main input of innovation, as typically speci…ed in standard R&D-based growth models (e.g., Romer, 1990; Jones, 1995; Acemoglu, 1998). Under such a framework, taxing labor has a detrimental e¤ect on the incentives to innovation and growth. This introduces a justi…cation for taxing capital income instead of. 42.

(50) labor income. On this ground, it might be optimal to have a non-zero capital income tax rate. The main contribution of this study is to link optimal capital taxation to the features of innovation process. We vary the parameters capturing important R&D externalities and see how the optimal capital income tax changes in response. Our main …ndings can be brie‡y summarized as follows. First, under most circumstances, the positive optimal capital income tax still holds. Second, when knowledge spillovers are strong and/or the duplication externalities are small (in. 政治大. which cases the underinvestment of R&D is more likely), it is more likely to have. 立. a positive optimal capital income tax rate. Third, when creative destruction is. ‧ 國. 學. more important during the R&D process, the optimal capital income tax should be higher (smaller) if the monopolistic markup is constrained (unconstrained) by. ‧. the parameter of creative destruction. Fourth, a higher government spending ratio pushes toward a positive optimal capital income tax.. sit. y. Nat. io. al. er. Finally, it is well-known in the existing studies (e.g., Aiyagari, 1995; Judd,. n. 1997, 2002; Coto-Martínez et al, 2007) that when the intermediate …rms are im-. Ch. i n U. v. perfectly competitive, the production level is too low compared to social optimum.. engchi. Accordingly, the government should subsidize capital to induce a higher level of production. This means that the optimal capital income tax tends to be negative, in particular when the monopolistic markup is higher. However, our results show that the optimal capital income tax and the markup display an inverse-U shaped relationship, meaning that there is another e¤ect of the markup on the optimal capital tax, which we call the R&D e¤ect. To be more precise, in an R&D-based growth model, the monopolistic rents go to the upstream R&D sector. A higher markup means that the R&D sector is more important. Under such a situation,. 43.

(51) subsidizing capital …nanced by taxing labor has a detrimental e¤ect on the incentives to innovation, which reduces growth and welfare. Considering this R&D e¤ect, an increase in the monopolistic markup does not necessarily result in a lower optimal capital income tax. Our study related to a vast literature attempting to overturn the ChamleyJudd result and obtaining a positive optimal capital income tax (e.g., among others, Chamley, 2001; Erosa and Gervais, 2002; Domeij, 2005; Golosov et al., 2006; Conesa et al., 2009; Aghion et al., 2013; Chen and Lu, 2013; Piketty and. 政治大. Saez, 2013). This paper contributes to the literature by introducing the role of. 立. endogenous technological change. Two papers studying optimal factor tax within. ‧ 國. 學. the framework of an endogenous growth model are closely related to the present paper. Chen and Lu (2013) consider a human capital-based endogenous growth. ‧. model developed by Lucas (1988). They …nd that a switch from labor income taxes to capital income taxes always enhances growth and welfare. Thus, the gov-. y. Nat. sit. ernment should tax capital income to a maximum level of 99%. Aghion et al.. er. io. (2013) also introduce R&D-based growth into the debate of the Chamley-Judd. al. n. iv n C h e na gSchumpeterian First, Aghion et al. (2013) consider c h i U quality-ladder growth model, result. However, our paper di¤ers from Aghion et al. (2013) in the following ways.. while we adopt an expanding-variety R&D model (Romer, 1990) incorporating the feature of creative destruction by following Jones and Williams (2000). Second, Aghion et al. (2013) consider a lab-equipment innovation process (i.e., R&D uses …nal goods as inputs), while we assume a knowledge-driven innovation process (i.e., R&D uses labor as inputs). Under our setting, therefore, the welfare costs of taxing labor would be larger than that in their model. Third, in Aghion et al. (2013), the positive optimal capital income tax sustains when the government spending of output ratio exceeds about 38%, which is much larger than the empirical value.. 44.

(52) In our analysis, by contrast, the optimal capital income tax is positive even if the government spending ratio is quite small (around 14%). Finally, Aghion et al. (2013) do not examine how the optimal capital income tax responds to various R&D externalities, which is the main focus of our analysis. The remaining of the paper proceeds as follows. In Section 3.2 we describe the R&D-based growth model featuring creative destruction and various types of R&D externalities elucidated by Jones and Williams (2000). In Section 3.3 we analyze in the long run how capital tax changes a¤ect the economy. In Section 3.4 we. 政治大. quantify the optimal capital income tax and examine how it interacts with R&D. 立. externalities. Section 3.5 provides concluding remarks.. ‧ 國. 學. The model. ‧. 3.2. y. Nat. sit. Our framework builds on the non-scale R&D-based growth model of the seminal. er. io. work developed by Jones and Williams (2000). The main novelty of the Jones and. al. iv n C tant dimensions of R&D intoh the original variety-expending R&D-based model of engchi U n. Williams model is that it removes the scale e¤ects and introduces several impor-. Romer (1990). In this paper, we extend their model by incorporating (i) elastic labor supply and (ii) factor taxes, namely the capital and labor income taxes. To conserve space, the familiar components of Romer’s variety-based model will be brie‡y described, while the new features will be described in more detail.. 45.

(53) 3.2.1. Households. We consider a continuous-time economy that is inhabited by a unit continuum of identical in…nitely-lived households. At time t, the population size of each household is Nt , which grows at an exogenous constant rate n. Each member of households is endowed with one unit of time that he/she can supply labor to a competitive market or enjoy leisure. The lifetime utility function of a representative household is given as:1 Z 1 e U=. t. 0. 立. 政 ln(1治 l )] dt;大. [ln ct +. t. > 0;. (1). 0;. where ct is per capita consumption and lt is the supply of labor per capita. The and. denote respectively the subjective rate of time preference. ‧ 國. 學. parameters. and leisure preference. The representative household maximizes (1) subject to the. ‧. following budget constraint: n. ]kt + (rt. n) et + (1. L;t )wt lt. (2). ct ;. sit. Nat. K )rK;t. y. k_ t + e_ t = [(1. al. er. io. where a dot hereafter denotes the rate of change with respect to time, kt is phys-. n. iv n C equity shares of R&D ownedhby eeach n gmember, c h i Ur is the capital rental rate, r ical capital per capita,. is physical capital depreciation rate, et is the value of K;t. is real interest rate, wt is the wage rate. The policy parameters. t. K;t. and. L;t. are. respectively the capital and labor income tax rate. Solving the dynamic optimization problem yields the following …rst-order conditions: 1 = qt ; ct (1. L;t )wt (1. 1. lt ) = ct ;. (3) (4). In line with Chu and Cozzi (2014) we assume that the utility function is based on per capital utility function.. 46.

(54) rt = (1. K )rK;t. (5). :. where qt is the Hamiltonian co-state variable on eq. (2). Equations (3) and (4) are respectively the optimality conditions for consumption and labor supply, and eq. (5) is a no-arbitrage condition which states that the net returns on physical capital and equity shares must be equalized. We denote the common net return on both assets as rt (i.e., rt = (1. ). The typical Keynes-Ramsey rules. K )rK;t. is: c_t = rt ct. (6). :. 政治大. 立. The …nal-goods sector. 學. ‧ 國. 3.2.2. n. A perfectly-competitive …nal-good sector produces a single …nal output Yt (treated. ‧. as the numéraire) by using labor and a continuum of intermediate capital goods,. n. al. xt (i). i=1. Ch. y. !1. ;1 >. sit. io. Yt = L1Y;t. At X. > 0; 1= >. er. Nat. according to the CES technology:. i n U. > 0;. (7). v. where LY;t is the labor input employed in …nal goods production, xt (i) (i 2 [0; A]). engchi. is the ith intermediate capital good, and At is the number of varieties of the intermediate goods. As will be introduced later, intermediate goods and capital has a one-to-one relation. Therefore, in eq. (7) we have followed Jones and Williams (2000) and Comin (2004) to separate the capital share ( ) and the elasticity of substitution across varieties (. ).. Pro…t maximization yields the following conditional demand functions for the labor input and intermediate goods: wt = (1. ). 47. Yt ; LY;t. (8).

(55) pt (i) =. At X. L1Y;t. xt (i). i=1. !1. 1 1. xt. (i);. (9). where pt (i) is the price of the ith intermediate good.. The intermediate-goods sector. 3.2.3. Each intermediate good is produced by a monopolistic producer who owns a perpetually protected patent for that good. The producer needs to use one unit of. 政治大 function is x (i) = v 立 (i), where v (i) denotes the capital input employed by mo-. physical capital to produce one unit of intermediate goods. Thus, the production t. t. t. is:. x;t (i). = pt (i)xt (i). rK;t vt (i):. ‧. ‧ 國. 學. nopolistic intermediate …rm i. Accordingly, the pro…t of intermediate goods …rm i. (10). y. Nat. Pro…t maximization subject to the production function xt (i) = vt (i) and eq. (9) 1. rK;t :. t. (11). iv n C (i) denote the grosshmarkup i Uith intermediate …rm can charge e n g that c h the n. Let. er. io. al. pt (i) =. sit. yields the pricing rule:. over its marginal cost. Then, we have:. pt (i) =. t (i)rK;t :. (12). By some manipulations, the pro…t of the ith intermediate …rm can be obtained as: x;t (i). =. t (i). 1 Yt : At t (i). (13). It follows from eqs (11) and (12) that if the monopolistic intermediate …rm freely sets the price, the markup would be equal to the elasticity of substitution between intermediate capital goods, i.e.,. t (i). 48. = 1=(. ). This is the case of an.