INTERNATIONAL ECONOMIC ASSOCIATION SERIES

# **Macroeconomic Modelling of R&D and Innovation Policies**

*Edited by*  Ufuk Akcigit · Cristiana Benedetti Fasil Giammario Impullitti · Omar Licandro Miguel Sanchez-Martinez

# International Economic Association Series

The IEA was founded in 1950 as a Non Governmental Organization, at the instigation of the Social Sciences Department of UNESCO. This series of Conference Volumes publishes selected papers from past IEA World Congresses and other Roundtable events, representing issues discussed at sessions by both established and younger scholars from around the world. The IEA is committed to disseminating the discipline of economics at an international scale. Its aim from the beginning has been to promote personal contacts and mutual understanding among economists in different parts of the world through the organization of scientific meetings, through common research programs and by means of publications of an international character on problems of current importance.

More information about this series at http://www.palgrave.com/gp/series/13991 Ufuk Akcigit · Cristiana Benedetti Fasil · Giammario Impullitti · Omar Licandro · Miguel Sanchez-Martinez Editors

# Macroeconomic Modelling of R&D and Innovation Policies

*Editors* Ufuk Akcigit University of Chicago Chicago, IL, USA

Giammario Impullitti University of Nottingham Nottingham, UK

Miguel Sanchez-Martinez European Commission DG JRC Brussels, Belgium

Cristiana Benedetti Fasil European Commission DG JRC Brussels, Belgium

Omar Licandro University of Nottingham Nottingham, UK

Sadly, Cristiana passed away recently and any correspondence should be directed to Miguel

ISSN 2662-6330 ISSN 2662-6349 (electronic) International Economic Association Series ISBN 978-3-030-71456-7 ISBN 978-3-030-71457-4 (eBook) https://doi.org/10.1007/978-3-030-71457-4

© The Editor(s) (if applicable) and The Author(s) 2022. This book is an open access publication.

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*In memory of Cristiana, a unique person who will be deeply missed.*

# Acknowledgements

Being this publication to a large extent a result of the JRC-IEA Macroeconomic Workshop 2017, the authors acknowledge the contribution of the Roundtable speakers Philippe Aghion, Ufuk Akcigit, Francisco Buera, Guido Cozzi, Felipe Saffie and Petr Sedlacek, and the panelists Javier Barbero Jimenez, Henrik Barslund Fosse, Francesco di Comite, Martin Christensen, Jacques Mairesse, Dimitrios Pontikakis, Werner Roeger, Jan in't Veld and Paul Zagamé. We would also like to especially thank Peder Christensen, Janos Varga and Martin Christensen for their contributions and feedback, which were key for the completion of the book, as well as the support of Xabier Goenaga without whom the Roundtable would have not been possible.

# Contents

x CONTENTS

# Editors and Contributors

# **About the Editors**

**Ufuk Akcigit** is an Associate Professor of economics with tenure at the University of Chicago and a faculty research fellow at the NBER and CEPR. Akcigit's research focuses on economic growth, productivity, firm dynamics, and the economics of innovation. His work aims to uncover the sources of technological progress and innovation that serve as engines of long-run economic growth. He is also interested in the role of public policy in growth, with a focus on environmental regulations, public research and funding for universities, and industrial policies such as R&D tax credits and corporate taxation. His research has been published in leading economics journals and has been supported by the Kauffman Foundation, the Sloan Foundation, and the National Science Foundation. He is the recipient of a CAREER award from the National Science Foundation. Akcigit holds a B.A. in economics from Koç University in Istanbul and a Ph.D. in economics from MIT.

**Cristiana Benedetti Fasil** was an economist and policy analyst at the Directorate General Joint Research Centre of the European Commission. Previously, she was lecturer at University College London and research associate at the Instituto de Análisis Económico (CSIC) in Barcelona. Her research background includes applied macroeconomics, heterogeneous firms models, international trade and economic growth. At the European Commission-DG JRC, she worked on impact assessment of European innovation policies with the use of different modelling platforms. Cristiana was also member of the Adansonia project which runs Randomized Controlled Trials to study the impact of virtual social networks and peer-to-peer learning on start-ups success in developing countries. Driven by her passion for women empowerment, she also founded the 'Social Venture Africa' NGO offering vocational and business trainings in Ghana. She held a Ph.D. and M.S. in economics from the European University Institute.

**Giammario Impullitti** is an Associate Professor in the School of Economics at the University of Nottingham. Previously, he was an assistant professor at Cambridge University and IMT Lucca, and a Max Weber postdoctoral fellow at the European University Institute in Florence. He is currently a research Fellow of CEP at the London School of Economics, CESifo Munich and Centro D'Agliano Milano. His fields of specialization include International Trade, Macroeconomics and Economic Growth. His research focuses on two major lines: the first studies the effects of globalization on growth and on labour market outcomes, such as unemployment and income inequality. The second line of research zeros in on the role of innovation policy in promoting growth and development. He has published in top field international journals such as the Journal of International Economics, International Economic Review, Economic Journal, Journal of the European Economic Association, and the Review of Economics and Statistics.

**Omar Licandro** is a Professor at the University of Nottingham and Research Professor at the Instituto de Análisis Económico, Barcelona (on leave). He is an Affiliated Professor at the Barcelona GSE and fellow at CESIfo. He was previously associate professor at Universidad Carlos III de Madrid, senior researcher at FEDEA, and professor at the European University Institute. He was the Research Director of the Barcelona GSE and, since 2013, he has been Secretary General of the International Economic Association and the Executive Secretary of the Research Institute for Development, Growth and Economics (RIDGE). Licandro's main research interests is in growth theory, technical progress and innovation, with important contributions to the literature on vintage capital and the transition to modern growth. He is currently contributing to the literature on: trade and growth, skill obsolescence, growth and productivity measurement, and the pass-through of large devaluations. His research has been published in leading economic journals.

**Miguel Sanchez-Martinez** is an economist and policy analyst at the Directorate General Joint Research Centre of the European Commission. Previously, he held an appointment as Research Fellow at the National Institute of Economic and Social Research in London. In this post, he conducted both academic and policy-oriented research in various areas, such as model-based evaluations of the macroeconomic impact of immigration to the UK. He also conducted short and medium-term macroeconomic forecasting for a number of economies. In his current position, he is responsible for analysing the productivity growth slowdown and undertaking macroeconomic impact assessment of EU policies using different modelling platforms. He holds a Ph.D. in Economics from the European University Institute and a Master of Science from Universitat Pompeu Fabra. His research interests include international macroeconomics, economic modelling, economic growth, productivity and environmental economics.

# **Contributors**

**Ufuk Akcigit** University of Chicago, CEPR, and NBER, Chicago, Illinois, USA

**Cristiana Benedetti Fasil** European Commission, DG JRC, Brussels, Belgium

**Baptiste Boitier** SEURECO/ERASME, Paris, France

**Martin Aarøe Christensen** European Commission, DG JRC, Seville, Spain

**Giammario Impullitti** University of Nottingham, Nottingham, UK

**Jan in't Veld** European Commission-DG ECFIN, Brussels, Belgium

**Pierre Le Mouël** SEURECO/ERASME, Paris, France

**Omar Licandro** University of Nottingham, IAE-CSIC and Barcelona GSE, University Park, Nottingham, UK

**Julien Ravet** European Commission, DG RTD, Brussels, Belgium

**Werner Roeger** DIW Berlin and VIVES KU Leuven, Leuven, Belgium

**Miguel Sanchez-Martinez** European Commission, DG JRC, Brussels, Belgium

**Janos Varga** European Commission, DG ECFIN, Brussels, Belgium

**Paul Zagamé** SEURECO/ERASME, Paris, France

# List of Figures

#### xviii LIST OF FIGURES

# List of Tables

# Introduction

# *Cristiana Benedetti Fasil and Miguel Sanchez-Martinez*

The key role of innovation as a driver of economic development was first described by Schumpeter (1934). In current times, there is widespread consensus among academic economists and policymakers, that research and development (R&D) activities play a decisive role in fostering growth in productivity and, hence, in the standards of living, as innovation intensive industries create highly skilled jobs, exhibit higher wages, are more productive, are often export-led and enhance competitiveness during the thick and thin of business cycles.1

The productivity growth slowdown in Europe and other advanced economic blocs experienced since the 2008–2009 economic and financial crisis has further reinforced the interest of policymakers in promoting innovation. Improving innovation performance is complex, not least because of the numerous actors and pieces of the innovation system

1 See, among others, Kumar and Sundarraj (2018).

C. Benedetti Fasil (Deceased) · M. Sanchez-Martinez (B) European Commission, DG JRC, Brussels, Belgium e-mail<EMAIL_ADDRESS>
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U. Akcigit et al. (eds.), *Macroeconomic Modelling of R&D and Innovation Policies*, International Economic Association Series, https://doi.org/10.1007/978-3-030-71457-4\_1

involved. Thus, the intricacies surrounding the promotion of innovation, and especially the best approaches to fostering it, play a central role in the European Union's policy landscape. Testament to this are profound debates such as the ones reflected in the European Union's Europe 2020 strategy, which emphasise R&D and innovation as essential means to achieving the overarching goals of jobs, growth and sustainable development. The emphasis of the Europe 2020 strategy is notably on 'improving the conditions for innovation, research and development' (European Council, 2010), with the specific objective of 'increasing combined public and private investment in R&D to 3% of GDP by 2020' (European Commission, 2014).

In the innovation policy debate, the following topics usually take centre stage: (i) best policy practices to spur innovation by the private sector with as large society-wide impacts as possible, (ii) technology diffusion (both across countries and firms), (iii) the apprehension of disruptive innovations, and ways to promote them, iv) the role of non-R&D innovation, v) the role of public versus private R&D.2 For the specific case of the EU, the most pressing innovation challenges identified include: increasing knowledge-intensive industrial activities, improving access to finance in high growth, highly innovative activities, strengthening the role of higher education institutions in local innovation ecosystems and improving the governance of research and innovation systems.3

The increased interest of policymakers in innovation has naturally been accompanied by an increasing need to evaluate the impact of policy measures. EU funding instruments such as the Framework Programme (FP) for research and the regional Europe Structural Investment Funds (ESIF) include legal requirements to collect data on implementation and to undertake evaluations at certain stages of the implementation (midterm/ex-post, for example). Measuring the impact of innovation is an intricate question compounded by the often relatively long lag between policy initiatives and observed actual impacts. In addition to indicatorbased approaches, such as the European Innovation Scoreboard, there has also been mounting interest in undertaking macroeconomic assessments of the impacts on GDP, imports, exports, employment at the EU,

2 On this last point, see, among others, Mazzucato (2018).

3 For the most salient documentation on these issues, see European Commission (2015), European Commission (2016), European Commission (2017) and European Commission (2018).

national and regional levels (European Council, 2010). This has provided further momentum to conducting research on the modelling of R&D and innovation policies as an additional way to quantifying the economic impact of innovation policies.

This reader is aimed at bringing to the forefront the latest empirical and theoretical insights stemming from the most recent literature related to the modelling of the macroeconomic impact of R&D policies. The content of this book is thus relevant both to academic and policy-related audiences working in the fields of R&D and innovation. As such, it is a wide-encompassing manuscript containing clear messages and results in the area of R&D and innovation policy and their macroeconomic impact and modelling.

Specifically, the purpose of this volume is threefold. First, to dissect the most relevant empirical facts to date on innovation and growth, and their consequences for policy. Second, to provide an overview of the state-of-the-art of macroeconomic models featuring innovation channels, the new elements of this narrative and their drawbacks. Third, to briefly discuss the models that have been implemented to analyse some of the most relevant innovation policies managed by the European Commission, including succinct examples. Fourth, to bridge the technical discussions offered with precise suggestions on fruitful ways forward, with a view to tackling the most pressing policy demands.

These and other similar questions were the subject of a workshop jointly organised by the International Economic Association and the European Commission's Joint Research Centre in March 2017. Distinguished academics and Commission officials participated and discussed different modelling approaches and issues for modelling R&D and innovation. This book is an off-spring of the discussions in this workshop, and it includes its proceedings.

The book is divided into three parts. In line with the aforementioned objectives, the first part is devoted to overviewing the latest theoretical and empirical contributions in the field of the macroeconomic modelling of R&D and innovation policies.4 In particular, Chapter 2 overviews the most recent empirical literature and its implications for innovation policy.

4 Please note that since January 2020, the UK is no longer a member of the EU. However, the contents of this book were written at a time when this status was still not officially recognized. The authors and editors have thus decided to include the UK as part of the EU in all the discussions contained in this reader

Chapter 3 delineates where the literature on DSGE models with innovation dynamics currently stands, the main ingredients of these models, and the paths that the academic literature in this area is set to follow. Chapter 4 presents a succinct summary of the Proceedings of the joint IEA-JRC workshop on 'Macroeconomic Modelling for R&D and Innovation', held in Brussels in March 2017. Part II of the book presents concise overviews of the different macroeconomic models that have been used for innovation policy evaluations by the European Commission in the past. Some examples of such evaluations are also provided, together with brief discussions on them. Finally, Part III presents the main conclusions on the macroeconomic modelling of R&D and innovation policies, and the potential ways forward.

### References

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/ by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# Macroeconomic Modelling of Innovation Policy: State-Of-The-Art

# Innovation, Public Policy and Growth: What the Data Say

# *Ufuk Akcigit*

# 2.1 Introduction

Innovation and technological progress are the key determinants of longrun economic growth and welfare. For instance, in recent work, Akcigit et al. (2017) (henceforth AGN) show that those states in the US that have innovated more over the twentieth century grew much more rapidly than those that innovated less (see Fig. 2.1). Relatedly, more research effort has been devoted to understanding the social implications of innovation. Does higher GDP per capita or GDP growth increase happiness? The existing empirical literature on happiness and income looks at how various measures of subjective well-being relate to income or income growth. For instance, Aghion et al. (2016) analyze the relationship between creative destruction and subjective well-being. They show that: (i) the effect of creative destruction on expected individual welfare is unambiguously positive if the unemployment rate is controlled for, less so if it is not; (ii) job

U. Akcigit (B)

University of Chicago, CEPR, and NBER, Chicago, Illinois, USA e-mail<EMAIL_ADDRESS>
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creation has a positive and job destruction a negative impact on wellbeing; (iii) job destruction has a less negative impact in US Metropolitan Statistical Areas (MSA) within states with more generous unemployment insurance policies; (iv) job creation has a more positive effect on individuals that are more forward-looking.

Given the tight link between innovation, economic growth, and well-being, designing the right public policies to achieve inclusive and sustainable growth requires a good understanding of what lies behind the innovation process. The mapping between innovation and economic growth can be described broadly as

$$\text{FIRMs} \rightarrow \text{ INværrons} \rightarrow \text{ Ipeas} \rightarrow \text{ Aegærædæt} \text{ GraWTH}$$

where firms hire inventors to produce new ideas/technologies which lead to economic growth. In line with this mapping, I will center my discussion in this chapter around three categories: (*i*) firm studies, (*ii*) inventor studies, and (*iii*) idea (patent) studies.

**Fig. 2.1** 100 Years of Innovation and Economic Growth (US States, 1900– 2000) (*Source* Akcigit, Grigsby, and Nicholas [2017])

# 2.2 Firm Studies

**Tax Credit and R&D Incentives of Firms** Debates on public policy and economic growth cannot ignore the fact that innovations do not fall from the sky. They are created by firms and inventors who respond to economic incentives and, importantly, incentives are shaped by public policy. A large literature documents the important effects of tax incentives for R&D, thus justifying the detailed study of their optimal design.1 A recent paper by Akcigit et al. (2017) (henceforth AAI) studies the role of R&D Tax Credit for innovation. In the US, the 1970s was a period of productivity slowdown that raised concerns about the declining international competitiveness of the US. At the time, John McTague of the Reagan White House said, "Foreign competition in the technology intensive industries poses a serious threat to our country's position in the international marketplace than ever before in our history." There are possible policies to deal with this "problem," the most discussed being import tariffs. The result of these debates was the introduction of the Federal R&D Tax Credit for the first time in 1981 (which has been in effect ever since).

Figure 2.2 shows the evolution of average firm-level R&D spending (normalized by firm sales) and the total share of patents at the US Patent Office filed by US firms. There are two facts worth mentioning. First, there had been a massive loss of technology leadership as documented by the rapid decline in the US patent share from 1975 to 1985. Second, US firms showed a large response to policy change. Starting from 1981, firms in the US increased their R&D spending which then translated into more patented innovations and brought international technology catch-up to a halt. How effective were these policies of the 1980s and how do tariffs affect innovation incentives?

Akcigit et al. (2017) assess the effects of import tariffs and R&D subsidies as possible policy responses to foreign technological competition in a dynamic general equilibrium growth model. Their quantitative investigation illustrates that, statically, globalization (defined as reduced trade barriers) has an ambiguous effect on welfare, while, dynamically, intensified globalization boosts domestic innovation through induced international competition. Accounting for transitional dynamics, they compute

1 Among many others, see Goolsbee (1998), Bloom et al. (2002), Bloom and Griffith (2001), Bloom et al. (2002), and Serrano-Velarde (2009).

**Fig. 2.2** Introduction of R&D Tax Credit, Firm R&D Spending and Innovation in the US (*Source* Akcigit, Ates, and Impullitti [2017])

optimal policies over different time horizons. Their model suggests that the introduction of the Research and Experimentation Tax Credit in 1981 was an effective policy response to foreign competition, generating substantial welfare gains in the long run. A counterfactual exercise shows that increasing trade barriers as an alternative policy response produces gains only in the very short run, and only when introduced unilaterally, while leading to large losses in the medium and long run. Protectionist measures generate large dynamic losses from trade, distorting the impact of openness on innovation incentives and productivity growth. Finally, they show that less government intervention is needed in a globalized world, thanks to intensified international competition as a result of lower trade barriers.

*Key takeaway:* An important policy message from this example is that tax policy, or specifically the R&D Tax Credit, could contribute to the attractiveness of a country for R&D and be a powerful tool for making firms more innovative and competitive.

**Firm Selection and Public Policy** The goal of R&D policies is to incentivize firms to undertake greater R&D investment, produce more innovations, increase productivity, and create more jobs. However, these policies do not affect every firm in the economy in the same way. For instance, Criscuolo et al. (2012) have shown that large incumbents are better at obtaining government subsidies. One can argue that R&D subsidies to incumbents might be inefficiently preventing the entry of new firms and therefore slowing down the replacement of inefficient incumbents by more productive new entrants. The turnover and factor reallocation between incumbents and entrants is an important source of productivity growth. Foster et al. (2001, 2006) have shown empirically that the reallocation of factors across firms accounts for more than 50% of productivity growth in the US. Given the empirical importance of this reallocation margin, it is necessary that R&D policy takes into account the interaction between innovation and factor reallocation. This is the focus in Acemoglu et al. (2018) (henceforth AAABK).

AAABK build a model with heterogeneous firm types where type is determined by innovative productivity. For instance, high-type firms produce more innovation for any given level of R&D input than lowtype firms. The authors estimate the model by matching various empirical moments capturing key features of firm-level R&D behavior, shipment growth, employment growth, and exit, and the variation of these moments with size and age. They then use the estimated model as a lab to run counterfactual experiments and test the impacts of various observed R&D policies on economic growth and welfare. The policies that the authors consider include a subsidy to new entrants, a subsidy to R&D by incumbents, and a subsidy for the continued operation of incumbents.

The main findings are summarized as follows. Using 1% of the GDP to subsidize new entrants, R&D or continued operations of incumbents have small effects, and some of them even reduce welfare in the economy. This result might suggest that the decentralized equilibrium is already efficient and any subsidy in this environment is making the economy move away from its efficient level. To the contrary, the decentralized equilibrium may be highly inefficient due to the usual intertemporal R&D spillovers and competition (Schumpeterian) effects. However, in this model there is another important margin: *firm selection*.

In order to understand the role of selection, AAABK first solve for the economy's allocation from the viewpoint of a social planner who internalizes all the externalities of R&D spending. What they find is that the social planner forces low-type firms to exit the economy much more frequently, so that all their production resources are reallocated to the high-type firms. Then they turn to the *optimal* public policy experiments in which they assume that the policymaker cannot observe firm types but has access to the usual policy tools such as an R&D subsidy, an entry subsidy, and a subsidy/tax to firm operations. What they find is that the optimal policy requires a substantial tax on the operation of incumbents combined with an R&D subsidy to incumbents. The reason for this result is that taxing operations makes it harder for low-type firms to survive and forces them to exit. The freed-up factors of production are then reallocated to high-type firms, which make use of them much more effectively.

Their general equilibrium analysis, which incorporates both reallocation and selection effects, highlights the fact that the economy in equilibrium might contain too many low-type firms and policies that ignore the selection effect might help low-type firms survive. Another point that is highlighted is the fact that intertemporal spillovers are sizable and overall R&D investment is below the efficient level. Therefore a combination of R&D subsidies and taxes on firm operations could be an effective way of providing innovation incentives to firms, while also leveraging the selection margin in the economy.

*Key takeaway:* The authors conclude that (i) governments often subsidize industries, (ii) these subsidies typically go to all firms, regardless of performance, (iii) focused subsidies could be more effective since they could redistribute key resources by letting low-type firms exit, and hence exploit the selection of firms in the economy.

# 2.3 Inventor Studies

**Who Becomes an Inventor?** Inequality of opportunities to get proper education could prevent the citizens as well as society from realizing their full innovative potential. The strong complementarity between innovation and education is documented by AGN for the US and by Aghion et al. (2017) for Finland.

In Figure 2.3, AGN show that increased education makes it more likely for someone to become an inventor. Figure 2.4, on the other hand, shows that kids with rich parents are also more likely to become inventors. If parental income is the only resource to accessing education, Figures 2.3 and 2.4 suggest that financial constraints could be important impediments to inclusive growth whereby a broader fraction of the society participates in the innovation and growth process.

*Key takeaway:* An important takeaway from these findings is that public policy needs to ensure access to education for potential future inventors who could generate economic growth through their creative ideas.

**Fig. 2.3** Becoming Inventor and Education (*Source* Akcigit, Grigsby, and Nicholas [2017])

**Fig. 2.4** Becoming Inventor and Parental Income (*Source* Akcigit, Grigsby, and Nicholas [2017])

**Taxation and Inventor Mobility** When it comes to policy debates, it is important to also take into account the disincentive effect of taxes on individuals and inventors in particular. Many of the prolific inventors around the world are international migrants and their location choice is affected by country-specific policies. In their work, Akcigit et al. (2016) (henceforth ABS) analyze the impact of top marginal income tax rates on the international mobility of inventors. Among many other things, they study the changes in tax codes in various countries, as illustrated in Figures 2.5 and 2.6.

Figure 2.5 shows the 1986 Policy Reform that reduced the top marginal tax rate in the US. The effect has been a rise in the number of foreign superstar (highest-quality) inventors who migrate to the US. Similarly, Figure 2.6 shows the policy change in Denmark in 1992 which lowered the top tax rate for high-income foreign researchers. The result of this change is again a significant rise in the number of foreign inventors in the country.

*Key takeaway:* The analysis by ABS shows the (dis)incentive effects of tax policies. Their findings suggest that some policies (top marginal tax rates, in this case) could impose significant costs on the society through their adverse effects on innovation incentives and economic growth.

**Innovation, Inequality and Social Mobility** Rising top-income share has been at the center stage of the current policy debates and many of the proposals to combat this trend focus on imposing heavy taxes on top-income groups. These discussions should also take into account

Elasticity= 0.71 (0.242)

.4

4

**Fig. 2.6** Becoming

the link between top-income inequality and innovation, which has been studied by Aghion et al. (2018) (henceforth AABBH).

Innovation has important and nuanced implications for inequality and social mobility. On the positive side, AABBH show that those US regions (commuting zones) that produced more innovations have also experienced greater social mobility (see Figure 2.7).2

Innovation, however, comes with an important trade-off. In Figure 2.8, AABBH also show that the states which had an increase in

2 Social mobility here is the expected percentile or "rank" (from 0 to 100) for someone aged 30 in 2011–2012 whose parents belonged to some percentile of the income in 1996 when the person was aged 16.

patented innovations also experienced, on average, a rise in top-income share between 1980 and 2005. These findings highlight the fact that while innovation is associated with faster growth and social mobility, it also comes with an increase in top-income inequality.

*Key takeaway:* Policies should take into account this tradeoff before leaping to conclusions for or against heavy taxation. In particular, high taxation may induce more equality but less social mobility.

# 2.4 Ideas

**Market for Ideas** New ideas are the seeds for economic growth. The rise in living standards depends not only on the production of new ideas (as it was discussed in Section 2.2), but also on the effectiveness of transforming new ideas into consumer products or production processes. Incarnating an idea into a product or a production process is by no means immediate. What happens to ideas and patents once they are produced? While a lot of the policy discussions center around increasing the number of ideas/patents/technologies produced, very little attempt is made at understanding how these new ideas are utilized after their invention. Akcigit et al. (2016) fill this gap by studying the secondary market for patents.

Ideas are not necessarily born to their best users and firms often develop patents that are not close to their primary business activity. This initial "mismatch" could potentially be mitigated in a secondary market where firms can buy and sell patents through patent agents (intermediaries). In Akcigit et al. (2016), the authors study the secondary market for ideas (patents) in the US. They build an endogenous growth model where firms invest in R&D to produce new ideas. An idea increases a firm's productivity. By how much depends on the technological distance between an idea and the firm's line of business. Ideas can be bought and sold on a market for patents. A firm can sell an idea that is not relevant to its business or buy one if it fails to innovate. The model is matched to stylized facts from the market for patents in the US. The analysis gauges how efficiency in the patent market affects growth. They find that the contribution of the secondary market for patents to the overall productivity is quantitatively significant.

*Key takeaway:* The immediate policy implication of this study is that strengthening the market for technologies could make economies use their scarce innovations and ideas better by allocating them to better users.

**Patent Trolls** The secondary market for patents suffers from various frictions and so-called "patent trolls" or non-practicing entities (NPEs) have emerged due to these frictions. Despite the attention on NPEs in the media and in policy circles, there is almost no systematic evidence on their business activities. How do NPEs impact innovation and technological progress? The question has enormous importance for industrial policy, with virtually no direct empirical evidence to start answering it.

A recent paper (Abrams et al., 2017) takes a major step in this direction by making use of some NPE-derived patent and financial data to answer this question. In doing so the authors inform the debate that has portrayed NPEs alternatively as benign middlemen that help to reallocate IP to where it is most productive or stick-up artists that exploit the patent system to extract rents, thereby hurting innovation. They find that NPEs target patents coming from small firms that are more litigation-prone, and patents from large firms that are not core to a company's business. When NPEs license patents, those that generate higher fees are closer to the licensee's business and more likely to be litigated. These findings suggest that NPEs could serve as middlemen in the market for technologies when frictions like high search costs or informational asymmetry between potential licensors and licensees are present.

*Key takeaway:* Taken together, the evidence in this paper is mixed and does not solely support the benign middleman or the stick-up artist theory. Rather it suggests that there are some aspects of NPEs that may increase innovation and some that may not. Therefore a more nuanced perspective on NPEs as well as additional empirical work is necessary before informed policy decisions can be made.

# 2.5 Conclusion

To sum up, innovation is good for society for at least three reasons: it leads to economic growth, social mobility, and happiness.

On the firm side, industrial policies could encourage more innovation, if we guide our innovation policy in an informed way, especially thinking about how the competition will have differential effects in different industries. The analysis on trade and innovation also shows that protectionist policies are detrimental for competition and growth, suggesting that the single-market policies of EU that remove trade barriers would stimulate more innovation and productivity growth. When it comes to individuals, a strong education policy could be a very influential innovation policy. High taxation could have significant disincentive effects for innovators, which could also harm social mobility in the society. Finally, having a well-functioning market for technologies could make economies utilize their scarce innovative ideas much more effectively.

Even though some of these studies use data from the US, their findings are much more general and relevant for all frontier countries that aim to grow through innovations. These findings show that public policy, innovation, market for ideas, and economic growth are tightly interlinked. Therefore any discussion on public policy and growth cannot be pursued in isolation from innovations and their effective use in practice, which are the main sources of long-run economic growth and prosperity.

The main lessons from these studies for Europe can be summarized as follows: First, international competition is healthy for innovation incentives. Second, innovation policies, such as R&D subsidies, require patience on the policymaker side, as these subsidies impact the economy in the medium-to-long run. Third, industrial policy needs to take into account the firm composition and factor reallocation in the economy. Bailing-out unproductive firms could slowdown factor reallocation from unproductive incumbents to more productive entrants. Fourth, education policy could be a very effective innovation policy in Europe. Providing as much equal opportunity for education as possible could improve the quality of the inventor pool and the overall innovation capacity. Fifth, the design of income tax policy has to take into account the fact that inventors do respond to incentives. Therefore one policy direction could be to couple income tax with tax breaks or research grants to inventors in order to undo the potential disincentive effects of taxes. Finally, the use of new technologies is at least as important as their inventions. Hence, Europe might also have to focus on its secondary market for technologies, in particular on technology sale and licensing, in order to improve its overall productivity.

# References

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# Innovation and Growth: Theory

# *Omar Licandro*

# 3.1 Introduction

This chapter surveys the literature on innovation, endogenous growth and firm dynamics, aiming to better understand the mechanisms through which innovation policies affect the progress of technology, productivity growth, output growth and welfare. When modeling the macroeconomy with the objective of evaluating the effect of innovation policies, the modeler has to fundamentally understand the different mechanisms through which a policy is expected to affect the dynamics of the economy through innovation. Since innovations fundamentally diffuse through a complex process of firm, plant and product creation and destruction, it is critical to understand the relation between innovation and the dynamics of market selection.

In writing this survey, an effort has been made to keep notation consistent across different models, imposing assumptions and interpreting results under a common framework, making models as comparable as possible. Section 3.2 gives a preliminary picture of these similarities by

23

O. Licandro (B)

University of Nottingham, Nottingham, UK e-mail<EMAIL_ADDRESS>
© The Author(s) 2022

U. Akcigit et al. (eds.), *Macroeconomic Modelling of R&D and Innovation Policies*, International Economic Association Series, https://doi.org/10.1007/978-3-030-71457-4\_3

pointing out some fundamental issues that arise when modeling innovation in a context of heterogeneous firms. It stresses the dynamic nature of the innovation process, describes the usual assumptions about firm heterogeneity in the context of innovation models of perfect, monopolistic and oligopolistic competition, uses aggregation theory to relate models of heterogeneous firms with the one-final-good Neoclassical and endogenous growth models, draws attention to the equivalence between (sunk) entry costs and R&D irreversible investments, as well as the embodied nature of technical progress.

Section 3.3 describes and analyzes firm heterogeneity in models of exogenous growth, starting with the perfectly competitive model of heterogeneous firms first developed by Hopenhayn, to then study a close economy version of the monopolistic competitive model first suggested by Melitz (2003) to finally refer to the close economy version of the oligopolistic model developed by Impullitti and Licandro (2018).

Finally, Sect. 3.4 studies firm heterogeneity in models of endogenous growth in order to understand the role of selection in shaping innovation and productivity growth. This section relates the traditional Romer (1990) and Schumpeterian (Aghion & Howitt, 1992) models to the recent literature on endogenous growth with firm heterogeneity, discussing the selection and imitation mechanism suggested by Luttmer (2007) and Klette and Kortum (2004).

# 3.2 Preliminaries

Before surveying the literature on firm dynamics and innovation, this section revises some critical concepts.

**Time.** Since the Industrial Revolution, modern economies live in a permanent state of innovation and progress. In this sense, innovation has to be understood as a dynamic process fueling technological developments. For this reason, the literature on economic growth belongs to the family of dynamic stochastic general equilibrium (DSGE) models where time is a fundamental dimension of the economic system. Static models are some times used as a shortcut, however, by construction they miss a fundamental dimension of the innovation process: It takes a long time to implement, adopt and diffuse new technologies.1

1 Different authors have measured the time it takes for innovation to diffuse. Comin and Hobijn (2010) estimate that new technologies take around fifty years to be adopted

Aggregate macroeconomic and microeconomic data are generally collected monthly, quarterly or annually. As a consequence, models designed to simulate and evaluate innovation policies often assume that time is discrete. In this chapter instead, we choose to follow the main tradition of economic growth theory and assume that time is continuous. Moreover, we abstract from aggregate shocks, even when some of the models reviewed originally embody aggregate stochastic processes.

**Firm Heterogeneity.** New technologies are fundamentally developed and implemented by the private sector. In a decentralized world, technical progress operates through the creation and destruction of products, plants and firms. In this sense, understanding the innovation process requires a minimum degree of firm heterogeneity and a good understanding of the dynamics of firms and markets.

The recent literature on firm dynamics usually models firm heterogeneity by assuming that the productivity of a firm can be characterized by some variable *z*. A firm entering the economy at time *t* draws its initial *z*, let us denote it by *zt*, from some known entry (density) distribution ψ*t*(*z*). The entry distribution may be evolving over time. The support of the entry distribution is usually assumed to be in the real line, with some lower bound ζ*t* ≥ 0. 2 As time passes, the productivities of these firms evolve following independent Markov processes. Equilibrium at time *t* will be then characterized as an equilibrium productivity (density) distribution that we denote by φ*t*(*z*), for *z* ≥ *z*∗ *t* , where *z*∗ *t* is the productivity of the least productive firm still surviving on the market. This is commonly called the cut-off productivity level.

In this review, we mainly concentrate on the study of economies where the productivity of a firm is time invariant, meaning that at entry firms draw a productivity *z* from ψ*t*(*z*), for *z* ≥ ζ*t*, ζ*t* ≥ 0, and keep this productivity constant all along their active life. In stationary economies, the entry distribution ψ(*z*) and the lower bound of its support ζ are assumed to be time invariant. Instead, in growing economies the entry distribution ψ*t*(*z*)

worldwide after their invention. When compared to the US, Comin et al. (2008) estimate that the lag in the use of new technologies by most countries is measured in decades. Adams (1990) measures in roughly 20 years the time it takes academic knowledge to contribute to productivity growth. Mansfield (1989) quantifies in 8 years the mean adoption delay of twelve mayor 20th-century innovations. Jovanovic and Lach (1997) estimate at 8.1% the annual diffusion rate of new products.

2 Some papers, like Melitz and Redding (2015), assume also that the support of entry distribution has a finite upper bound.

will move overtime guided by some form of spillover, as well as the lower bound of its support ζ*t*. Hence, a stationary entry distribution will not result in growing average productivity.

A standard assumption in this literature is that a firm with productivity *z* employs some flexible production factor *x* to produce output *y*. In the following, inputs and output of a firm with productivity *z* will be denoted *x*(*z*) and *y*(*z*), respectively. In perfectly competitive economies, this technology is assumed to have decreasing returns on *x*. However, under monopolistic competition, including also other forms of imperfect competition, technology is frequently assumed to be linear on *x*. Hopenhayn and Melitz, respectively assume *y*(*z*) = *z x*(*z*)α, α ∈ (0, 1) and *y*(*z*) = *z x*(*z*). Following Collard and Licandro (2018), we will in some sections assume that *y*(*z*) = *F* - *z*, *x*(*z*) , with *F* (.) being a Neoclassical technology.

This literature often abstracts from capital accumulation by assuming that labor is the sole production factor, i.e. *x* = . Hopenhayn (2014) generalizes it to a two production factor economy, with *x* = *F*(*k*, ); *k* represents capital and *F*(*k*, ) is assumed to be a Neoclassical technology.

In line with the Neoclassical growth framework, we survey first models where the productivity of firms is stationary or evolves exogenously, to then study models where firm heterogeneity is guided by innovation and technological developments.

**Entry Cost, Innovation and Capital Reversibility.** It is generally assumed that firms have to pay some entry cost before they draw productivity *z* from ψ*t*(*z*). On top of that, the entry cost is frequently assumed to be sunk, i.e. the investment realized to create the firm is irreversible: When a firm closes down, nothing is recovered from this investment. Moreover, it is usual that net revenues of operative firms are strictly positive, implying that fixed production costs need to be assumed for the least productive firms to exit.3

Interestingly, the sunk entry cost, even when fully irreversible, can be interpreted as a form of intangible investment. Since operative firms make positive profits at equilibrium, the value of the firm, namely the expected

3 In Hopenhayn (1992), net revenues are strictly concave due to decreasing returns to labor; at equilibrium, low productivity firms optimally hire few workers making net revenues strictly positive. In Melitz (2003), monopoly profits are strictly concave implying that low productive firms also optimally hire few workers making net revenues strictly positive.

discounted flow of profits, is the value of the associated intangible investment. Firms close down and exit when the value of their intangible capital is zero.

As an alternative and consistent with national accounts, Collard and Licandro (2018) assume that the entry cost is a form of capital investment (tangible and intangible), with capital being partially reversible, i.e. it has a market value smaller than the replacement cost. Under very general assumptions about firms'technology, they show that aggregate technology is Neoclassical on aggregate capital and labor. Moreover, since capital is partially reversible, no fixed production cost is needed for the least productive firms to exit: Firms exit when the value of being operative is smaller than the market value of capital.

In the endogenous growth literature, when innovation is assumed to be undertaken by new entrant firms, R&D investment is a form of entry cost.4 Firms have to invest in R&D in order to innovate and then enter the economy. In Romer (1990), new firms innovate by creating a new variety. Since firms are never displaced in the Romer's model, the R&D investment can be seen as fully reversible. Patents can be transferred at no cost. In the Schumpeterian model of Aghion and Howitt (1992) or in the Grossman and Helpman (1991)'s quality-ladder model the entering firm displaces an incumbent firm, which is known as business-stealing effect. R&D investments are then fully irreversible in these two models.

**One-Final-Good Economy.** Macroeconomic models are designed to understand the behavior of GDP as measured by national accounts. Consistently, in the tradition of the Neoclassical growth theory, an economy is modeled as producing a sole final good, directly associated to GDP in the data. The final good is then allocated to different uses, such as consumption or investment. Macroeconomic models of heterogeneous firms belong to this tradition.

For example, in Hopenahyn (1992) the production side of an economy is modeled as a mass of heterogeneous firms that produce the sole final good under perfect competition. Hence, in these economies, firm's technology has decreasing returns on labor, in line with the Lucas (1978)'s span of control model.

4 In Sect. 4.4, some models of innovation by incumbents, where R&D does not play the role of an entry cost, are also surveyed. Another example of such models can be found in Akcigit and Kerr (2017).

Alternatively, in the monopolistic competitive approach inspired in Dixit and Stiglitz (1977), a perfectly competitive, representative firm produces the sole final good by the mean of a constant elasticity of substitution technology defined on a continuum of heterogeneous intermediary inputs, which are assumed to be gross substitutes. The market for intermediary inputs is assumed to be monopolistically competitive. Each heterogeneous intermediary firm has monopoly power on the production of a particular intermediary input and owns a constant return to scale technology defined on a vector of production factors (usually labor only).5 An alternative and isomorphic way of representing the same economy is to assume that the monopolistically competitive firms produce a continuum of final consumption goods that households order by the mean of a constant elasticity of substitution utility function. Consumption in national accounts is then interpreted as the aggregate of all the different consumption goods, aggregated by the mean of household preferences.

**Aggregation.** As shown by Hopenhayn and Collard and Licandro (2018), most of these economies share some simple aggregation properties that cause aggregate technology to be Neoclassical in exogenously driven growth models. These aggregation properties are shared with most endogenous growth models, where aggregate technology indeed belongs to the family of AK production functions. The main implication is that aggregate conditions are quite standard despite the complexity added by firm heterogeneity.

**Spillovers.** In the Neoclassical growth model, technical progress is a gift offered by Nature that instantaneously diffuses over the whole economy without limit: All firms and countries may access the frontier technology. In particular, the representative firm benefits from it without paying any cost. In this sense, technical progress in the Neoclassical growth model is nothing else than spillovers! Of course, since technical progress is part of the environment, and Nature gives rise to it without facing any trade-off, a perfectly competitive economy reacts to it optimally. Hence, in the Neoclassical growth mode technical progress diffuses through inconsequential spillovers.

However, when innovations are added to the picture endogenizing the rate of technical progress, spillovers become consequential. For example,

5 This framework has been extended to alternative forms of imperfect competition, see Atkeson and Burstein (2008) and Impullitti and Licandro (2018).

endogenous growth in the learning-by-doing model is based on a particular form of spillover: the state of technology depends on past capital production. Consequently, investors (i) do not internalize the effect of their actions on technical progress, (ii) invest less than optimal and (iii) the economy grows at a rate smaller than the optimal growth rate. Similarly, in Romer (1990) expanding-product-variety model, innovators increase the mass of intermediary inputs, which affect the productivity of final producers through another form of spillover externality.

In the Schumpeterian and quality-ladder models, the technology of innovators builds upon the pre-existing state of knowledge. This operates as a form of spillover, since the knowledge that innovators create, indeed, flows back to the economy improving the innovation technology of the following innovators. Hence, when innovators substitute Nature, investing resources to make the technology progress, since they do not internalize knowledge spillovers, spillovers become consequential.

Technological spillovers result from a fundamental property of knowledge, the so-called *non-rivalry*: The use of some knowledge by an individual or firm does not prevent another by using it simultaneously. The fact that an engineer is using the Pythagoras theorem to calculate some structures does not impede others from using it too. For this fundamental reason, a model designed to evaluate innovation policies has to include knowledge spillovers, as well as the potential distortions generated by the policies, in particular those addressed to protect intellectual property rights. In this sense, it is important to understand that innovation policies have to be analyzed in a second-best framework.

**Embodied Technical Change.** The evidence of a permanent decline in the price of durable goods (including equipment investment, structures, durable consumption and some forms of intangible capital), relative to the price of non-durable consumption and services, gave raise to a large literature stressing the importance of modeling the economy as a twosector model with durable and non-durable goods. The standard way of modeling is in line with Greenwood et al. (1997).6

As aforementioned, in the Neoclassical growth model technical progress is disembodied: new technologies diffuse all over the economy at no cost. Instead, when technical progress is embodied in capital, it requires investments to diffuse. The frontier of technology moves in

6 See also Felbermayr and Licandro (2005).

the investment sector, but investments are needed to allow for technical progress to diffuse to the consumption sector.

Finally, in the vintage capital literature the gift of Nature is only addressed to new capital units.7 Investment is key for an economy that wants to benefit from the progress of technology, since technical progress does not spillover previously produced capital. For this reason, technical progress in the vintage capital model, is said to be embodied in new capital. Moreover, a perfectly competitive vintage capital economy optimally reacts to technical change. The fact that the gift of Nature only flows over the capital producing industry is also inconsequential (Solow, 1962).

# 3.3 Firm Dynamics and the Neoclassical Model

The seminal work of Jovanovic (1982) and Hopenhayn (1992), and the subsequent application to international trade by Melitz (2003), gave raise to an extensive literature on the macroeconomic implications of firm dynamics pointing to the fundamental role of market selection on economic performance and welfare. Even if Jovanovic (1982) stresses the role of incomplete information and learning, in the Hopenhayn (1992) framework heterogeneous firms operate in perfectly competitive markets, making selection to be optimal by construction. In the Melitz (2003) model of monopolistic competition, instead, selection interacts with different types of market frictions, making welfare gains from selection less obvious.

As mentioned, time is assumed to be continuous and denoted by *t*, with *t* = 0 being the initial time. Population is assumed to be constant and normalized to one, implying that aggregate variables are measured in per capita terms. There is a sole final good which is adopted as numeraire, even if in some sections of the chapter the implications of multiple final goods (consumption and investment, for example) are discussed.

A representative household, with additively separable constant intertemporal elasticity of substitution (CIES) preferences, inelastically offers one unit flow of labor. Households face perfect financial markets with riskless interest rate *rt*. The saving behavior of the representative

7 See Solow (1962) and Solow et al. (1966), and more recently, Boucekkine et al. (1997, 2005) and Gilchrist and Williams (2000). Bambi et al. (2014) develop an endogenous growth model of vintage technologies.

household then reduces to the well-known Euler equation

$$\frac{\dot{c}\_l}{c\_l} = \sigma(r\_l - \rho),\tag{\text{EE}}$$

where *ct* is per capita consumption, σ > 0 is the intertemporal elasticity of substitution and ρ > 0 is the subjective discount rate. When financial markets value time more than individuals subjectively do, i.e. when *rt* > ρ, individuals optimally save and postpone consumption, making *c*˙*t ct* > 0. The intensity of consumption postponement depends on the intertemporal elasticity of substitution. In the extreme case of intertemporal perfect substitutability, when σ goes to ∞, for a given difference *rt* − ρ > 0, individuals postpone any consumption, making *c*˙*t ct* = ∞. In the other extreme of perfect complementarity, when σ goes to 0, any given difference *rt* − ρ > 0 has no effect on the consumption path that will be in any case constant.

#### *3.3.1 Ramsey-Hopenhayn Model*

**The Economy.** A continuum of perfectly competitive heterogeneous firms produces at time *t* a sole final good using capital as a fixed production factor and labor as a flexible factor.8

A firm, when associated to a particular unit of capital, has a time invariant productivity *z*, with φ*t*(*z*) representing the equilibrium (density) productivity distribution, for *z* in the support (*z*∗ *t* ,∞); the so-called cutoff productivity *z*∗ *t* is endogenous. To fix ideas, let us see each unit of capital as a plant. Firms may own different plants with different productivity. φ*t*(*z*) is the distribution of productivity across plants. Buying one unit of capital costs η units of the final good, η > 1. However, when transformed back into the final good, the capital unit is worth just one unit. In line with the misallocation literature, investment distortions are measured by η − 1 > 0. 9 In this sense, investment is partially sunk, since firms cannot recover their investment fully when a plant closes down.

8 This section is inspired in Collard and Licandro (2018).

9 See Hsieh and Klenow (2009). For a survey on this literature, see Restuccia and Rogerson (2017). Hopenhayn (2014) shows the intrinsic relation between the literature on firm dynamics and the literature on misallocation.

A plant with productivity *z* has access to technology10

$$
\partial\_I \mathbf{y}\_l(\mathbf{z}) = A\_l \, \mathbf{z}^\alpha \ell\_l(\mathbf{z})^{1-\alpha}, \tag{3.1}
$$

with α ∈ (0, 1). 11 The state of technology *At* exogenously grows at the rate (1−α)γ , γ > 0. Variables *yt*(*z*) and *t*(*z*) denote output and employment, respectively. It is easy to see that, for a given wage rate w*t*, the optimal labor demand is

$$\ell\_l(z) = \left(\frac{(1-\alpha)A\_l}{w\_l}\right)^{\frac{1}{\alpha}} z.$$

Since operative plants produce all the same final good, per capita production (remind that population has been normalized to one) is

$$\mathbf{y}\_{l} = k\_{l} \int\_{z\_{l}^{\*}}^{\infty} \mathbf{y}\_{l}(z) \phi\_{l}(z) \mathbf{d}z,$$

where *kt* represents the mass of operative plants, which by assumption is equal to the stock of capital per capita.

Labor market clearing implies that the equilibrium wage rate, plant profits and per capita output are, respectively,

$$w\_l = (1 - \alpha)A\_l \left(\overline{z}\_l k\_l\right)^{\alpha}, \quad \pi\_l(z) = \alpha A\_l \left(\overline{z}\_l k\_l\right)^{\alpha - 1} z \quad \text{and} \quad y\_l = A\_l \left(\overline{z}\_l k\_l\right)^{\alpha}, \tag{3.2}$$

where the average productivity of firms is

$$\bar{z}\_l = \int\_{z\_l^\*}^{\infty} z \phi\_l(z) dz.$$

At equilibrium aggregate technology is Cobb-Douglas with total factor productivity given by *At z*¯ α *t* . 12 Wages and profits are equal to the marginal product of labor and capital, respectively. Selection raises the average

10 The argument below applies to any Neoclassical technology *F*(*z*, ).

11 This technology is in line with the span of control assumption in Lucas (1978).

12 Alternatively, Collard and Licandro (2018) interpret productivity *z* as being embodied in capital, meaning that *z*¯ represents the average quality of the physical capital *k* and *zk*¯ measures capital in quality adjusted units.

productivity of firms, increasing output *yt*, wages w*t*, and average profits π*t*(*z*¯*t*).

**Selection.** New plants buy one unit of capital and then draw productivity *z* from the entry distribution ψ(*z*), for *z* in the positive real line. Let us assume ψ(*z*) is Pareto, with tail parameter κ > 1 and expected productivity at entry equal to one (which implies that the lower bound of the support of *z* is κ−1 κ . As shown in Collard and Licandro (2018) under some general conditions, for all time *t* ≥ 0, the equilibrium cut-off productivity is *z*∗ *t* = *z*∗ (time invariant) and the equilibrium productivity distribution is the entry distribution truncated at *z*∗; i.e.,

$$\phi\_l(z) = \frac{\psi(z)}{1 - \Psi(z^\*)},$$

for all *t* ≥ 0 and for *z* ∈ (*z*∗,∞), where (*z*) is the cumulative of ψ(*z*). 13

Since profits are linear on *z* and the equilibrium *z*∗ is time invariant, the value of any operative plant v*t*(*z*) is linear on *z* too. Notice that operative plants at *t* = 0 will optimally like to be operative forever. At equilibrium, then

$$v\_l(z) = v\_l z,\qquad \text{and}\qquad v\_l = \int\_I^\infty \pi\_s(1) \, \mathbf{c}^{\int\_s^\infty (r\_h + \delta) \, \mathrm{d}h} \, \mathrm{d}s$$

where v*t* is the expected discounted flow of profits of a firm with productivity *z* = 1 and δ > 0 is an exogenous exit rate, equivalent to the physical depreciation rate in the Neoclassical model. The path of v*t* depends on the path of the aggregates.

The equilibrium cut-off productivity *z*∗ results then from combining the exit and free entry conditions

$$v\_t z^\* = 1 \tag{\text{EC}\_{\text{RH}}}$$

$$
\Psi(z^\*) + \left(1 - \Psi(z^\*)\right)\upsilon\_l \tilde{z} = \eta \qquad\qquad\qquad\left(\text{FE}\_{\text{RH}}\right).
$$

13 There are two critical assumptions behind this result. Firstly, the economy is assumed to be at steady state at the initial time *t* = 0. Secondly, a permanent and unanticipated shock makes the economy become more selective. The first is a very usual assumption in macro dynamics. The second restrict the analysis to policies that promote selection, which in this framework, are welfare improving.

From the exit condition (ECRH), the value of the marginal plant, v*tz*∗, is equal to the value of capital (which is equal to one, since capital can be transformed back into one unit of the final good). From the free entry condition (FERH), the investment cost η has to be equal to the expected value of entry. Notice that a new plant expects to get a productivity smaller than *z*∗ with probability (*z*∗), in which case immediately exits and recovers one. Otherwise, with probability 1 − (*z*∗), the plant will produce and get an expected value v*tz*¯.

The equilibrium cut-off productivity results from combining (ECRH) and (FERH) to get rid of v*t*. Collard and Licandro (2018) show that under very general conditions the solution exists and is unique, with *z*∗ > 1/η depending only on the entry distribution ψ(*z*) and the investment distortion η − 1. 14 Both, a reduction in investment distortions and an increase in the variance of the entry distribution raise *z*∗ by reducing the cost of entry and increasing the likelihood of reaping the benefits of a high productivity draw, respectively.

**Aggregate Economy.** Since the capital of exiting plants (those with productivity smaller than *z*∗) is fully recycled, the efficiency condition reads

$$y\_l + \Psi(z^\*)e\_l = c\_l + \eta e\_l,$$

where *et* represents entry, i.e., the mass of new plants created at time *t*. Each new plant needs a unit of capital, which costs η. Moreover, a fraction (*z*∗) of them close down and their capital reverts to the economy, being consumed or invested.15

Capital per capita evolves following

$$\dot{k}\_l = (1 - \Psi(z^\*))e\_l - \delta k\_l, \dot{z}$$

14 This result generalizes the separation result in Melitz (2003), making selection to be independent of the path of the aggregates.

15 It is implicitly assumed that the selection process at any time *t* repeats infinitely until all firms get a productivity larger than *z*∗. Collard and Licandro (2018) use the alternative assumption that the capital of plants closing down at *t* cannot be recycled until *t* + d*t*, in which case the dynamics of Ramsey-Hopenhayn economy is slightly different even if it still shows standard (saddle-path) monotonic convergence properties.

where δ > 0 is the physical rate of capital depreciation. The feasibility condition results from combining the previous two equations

$$\dot{k}\_{l} = \frac{1 - \Psi(z^\*)}{\eta - \Psi(z^\*)} \Big( A\_l \left( \tilde{z} k\_l \right)^\alpha - c\_l \Big) - \delta k\_l. \tag{\text{FC}}$$

Notice that the rate at which the final good transforms into physical capital is smaller than one, since investment distortions make the selection process costly.

Combining the exit condition (EC\_RH) and the Euler equation (EE), the last one reads

$$\frac{\dot{c}\_l}{c\_l} = \sigma \left( z^\* \alpha A\_l \left( \overline{z} k\_l \right)^{\alpha - 1} - \rho - \delta \right), \tag{\text{EE'}}, \tag{\text{EE'}}$$

where the marginal product of capital corresponds to the profits of the marginal plant *z*∗.

Given an initial *k*0 > 0, the equilibrium cut-off *z*∗ and the associated average productivity *z*¯, an aggregate equilibrium of the Ramsey-Hopenhyan model is a path {*ct*, *kt*}, for *t* ≥ 0, such that both (EE') and (FC) conditions hold (together with a transversality condition) It is important to notice that (FC) and (EE') are the same as in the Neoclassical growth model, with a few constant terms depending on the equilibrium value of *z*∗. At the balanced growth path the economy then grows at the constant rate γ .

Collard and Licandro (2018) show that a policy that decreases investment distortions, by making the economy more efficient, increases capital, output and consumption (measured in efficiency units) at the balanced growth path, generating steady state welfare gains. Moreover, the dynamic system has standard stability properties, meaning that the economy monotonically converges to a unique balanced growth path.

**Transitional Dynamics.** Let us assume the economy was initially at steady state with past investment distortions given by η*p* > 1. 16 For simplicity, the rate of technical progress is γ = 0. Consistently, the economy at the initial time *t* = 0 has a distribution of firms φ*p*(*z*) = ψ(*z*)/- 1 − (*z*∗ *p*) in the support *z* ∈ (*z*∗ *p*,∞), as well as an initial stock of physical capital *k p*; both *z*∗ *p* and *k p* solve the steady state equilibrium conditions.

16 These distortions may represent different forms of barriers to entry.

**Fig. 3.1** Transition dynamics: permanent decline in investment distortions (η) (*Note* This figure was obtained setting σ = 1, ρ = 0.05, δ = 0.06, α = 0.3, κ = 3 and the initial investment distortion is η = 1.2. We then consider a 5% once for all reduction in η)

Let us also assume that from *t* = 0 a new policy permanently reduces investment distortions. For simplicity, let us refer to the new policy as η, with η<η*p*. The cut-off productivity jumps then to a new steady state *z*∗ > *z*∗ *p* and the equilibrium distribution jumps to φ(*z*) = ψ(*z*)/- 1 − (*z*∗) .

Interestingly, the initial stock of capital is partially destroyed because of selection, implying that

$$k\_0 = \left(1 - \frac{\eta - 1}{\eta - \Psi(z^\*)} \Phi\_p(z^\*)\right) k\_p < k\_p.$$

Of course, the average productivity *z*¯ jumps up at *t* = 0 making output to increase at the initial time. Similarly to the Neoclassical growth model, consumption at the initial time jumps down to the new saddle path solution converging monotonically with capital to the new higher steady state.

#### *3.3.2 Monopolistic Competition*

This section builds on a close economy of the Melitz (2003) type.

**The Economy.** Heterogeneous intermediary firms produce a continuum of intermediary inputs used in the manufacture of a sole final good. The final good is produced by a representative competitive firm under perfect competition; the final good is used as numeraire. Intermediary inputs, indeed, in line with Dixit and Stiglitz (1977), are produced under monopolistic competition. For comparability, we adopt a similar notation as in the previous section (Fig. 3.1).

There is a mass *mt* of heterogeneous intermediary firms. Firms differ in their productivity *z*, producing each a differentiated intermediary input by the mean of the following linear technology

$$\mathbf{y}\_{\mathbf{l}}(z) = Az \,\ell\_{\mathbf{l}}(z),$$

where *yt*(*z*) and *t*(*z*) represent output and labor of a firm with productivity *z*, respectively; the state of technology *A* > 0 is assumed to be constant.17 As in the previous sections, operative firms have productivity *z* ≥ *z*∗ *t* .

The sole final good is allocated to consumption *ct* only and it is produced by a mass unit of identical perfectly competitive final firms by the mean of the constant elasticity of substitution (CES) technology

$$c\_{l} = \left(m\_{l} \int\_{z\_{l}^{\*}}^{\infty} \mathbf{y}\_{l}(z)^{\frac{\varrho - 1}{\varrho}} \phi\_{l}(z) \mathrm{d}z\right)^{\frac{\varrho}{\varrho - 1}}$$

defined on a mass *mt* of intermediary inputs, with constant elasticity of substitution > 1.

Final firms are price takers in both the final and the intermediary markets, optimally demanding of each intermediary input the quantity

$$\chi\_{\mathbf{l}}(z) = p\_{\mathbf{l}}(z)^{-\varrho} c\_{\mathbf{l}}.$$

The demand elasticity of any intermediary input is equal to the elasticity of substitution across varieties. More substitutable intermediary inputs are, more the final firm reacts to changes in input prices. Love-for-variety, in the sense of Dixit-Stiglitz, means that firms would like to use all available intermediary inputs, with relative quantities depending on relative prices.

Intermediary firms have monopoly power in the production of intermediary inputs. They maximize profits subject to the demand function

17 It is easy to extend the Melitz model to an environment where the aggregate state of technology *At* grows at a constant exogenous rate, as assumed in the previous section.

above, optimally setting price

$$p\_l(z) = \frac{\varrho}{\varrho - 1} \frac{w\_l}{Az}.$$

Intermediary firms charge a markup −1 > 1 over marginal costs w*t Az* . More productive firms set a lower price, producing and selling more. The markup is inversely related to the demand elasticity.

An important property of the monopolistic competitive model is that all monopolistically competitive firms charge the same markup, implying that their relative prices are equal to their relative marginal productivities. The direct implication is that the allocation of production factors within a monopolistically competitive sector is efficient, since relative prices are equal to relative marginal productivities.18 Of course, the allocation of production factors between the monopolistic competitive sector and other sectors of the economy may be distorted because of the markup. A recent literature stresses the role played by the dispersion of markups on the allocation of resources within an industry.19

Aggregating over intermediary firms, it can be shown that consumption per capita, the wage rate and total net revenues are

$$c\_{l} = A m\_{l}^{\frac{1}{\varrho - 1}} \bar{z}\_{l} L\_{l}, \quad w\_{l} = \frac{\varrho - 1}{\varrho} \; A m\_{l}^{\frac{1}{\varrho - 1}} \bar{z}\_{l} \quad \text{and} \quad \pi\_{l} = \frac{1}{\varrho} c\_{l},$$

with average productivity defined as

$$\bar{z}\_l = \left(\int\_{z\_l^\*}^{\infty} z^{\varrho - 1} \phi\_l(z) dz\right)^{\frac{1}{\varrho - 1}},$$

where *Lt* represents the share of total labor allocated to production (excluded any fixed production costs). The mass of intermediary inputs *mt* shows up in the aggregate technology as an externality, usually referred in this literature as love-for-variety. The more intermediary inputs are available for final production, the more efficient final production is. Moreover, selection positively affects output since it raises the average productivity of firms *z*¯*t*.

18 See Koeninger and Licandro (2006) and Epifani and Gancia (2011).

19 See Impullitti and Licandro (2018).

The production of the consumption good requires both labor and intermediary inputs, which mass is represented by *mt*. Wages are the return to labor and profits the return to the investment required to create an intermediary input (the entry cost). It is possible to interpret the mass of intermediary inputs as the stock of intangible capital. In this framework, the distribution of income between intangible capital and labor critically depends on the elasticity of substitution across intermediary inputs. An increase in substitutability reallocates income from intangible capital to labor.

Net revenues of firm *z* are, indeed,

$$
\pi\_l(z) = \frac{1}{\varrho} \left. \frac{c\_l}{m\_l} \left( \frac{z}{\bar{z}\_l} \right)^{\varrho - 1} \right|
$$

Notice that net revenues of the average firm *z*¯*t* are equal to average net revenues π*t* /*mt*. Firms with productivity larger than the mean make profits larger than the average profit.

**Selection.** Following Melitz (2003), let us assume intermediary firms have to pay a sunk entry cost w*t fe* to enter, *fe* > 0 being the amount of labor required to create a new intermediary input. After entry, firms draw a productivity*z* from an entry distribution (*z*) with support in the real line. Since new firms produce new varieties, the sunk entry cost may be interpreted as an R&D investment; i.e. the investment required to be able to produce the new input variety. Of course, if the technology producing the new intermediary input is not productive enough, the firm will close down making the value of this R&D investment to be zero ex-post.

At any time *t*, intermediary firms require a fixed amount of labor *f* , *f* > 0, to be operative, facing then a fixed production cost *f* w*t*. At the steady state of the Melitz model, the marginal firm *z*∗ is defined by the (zero-profit) exit condition

$$
\pi(z^\*) = \frac{1}{\varrho} \left. \frac{c}{m} \left( \frac{z^\*}{\overline{z}} \right)^{\varrho - 1} \right| = fw. \tag{\text{ECM}}
$$

Any firm with productivity *z* < *z*∗ exits since net revenues are not large enough to cover the fixed production costs. Notice that, for any operative firm with *z* ≥ *z*∗, profits can then be expressed in relation to the marginal

#### 40 O. LICANDRO

firm as

$$
\pi(z) - f\,w = \left( \left( \frac{z}{z^\*} \right)^{\varrho - 1} - 1 \right) f\,w.
$$

Any operative firm with productivity larger than *z*∗ makes positive profits, equilibrium profits being proportional to the fixed production cost.

The value *v*(*z*) of a firm with productivity *z* at steady state is the expected discounted flow of profits, which collapses to

$$v(z) = \frac{\left(\left(\frac{z}{z^{\ast}}\right)^{\varrho - 1} - 1\right)fw}{r + \delta},$$

since expected profits are discounted at *r* + δ, where δ > 0 is the Poisson destruction rate of any operative firm.20

The free entry condition makes expected profits equal to the entry cost

$$\int\_{z^\*}^{\infty} v(z) \mathrm{d}\Psi(z) = f\_{\mathfrak{E}} w. \tag{F \to\_{\mathfrak{M}}}$$

Remember that a firm is assumed to invest in intangibles before knowing its productivity. Under the assumption that the entry distribution is Pareto, i.e., (*z*) = 1 − ζ *z* κ , with κ > 1 and ζ > 0, by combining the exit (ECM) and free entry (FEM) conditions, the steady state equilibrium cut-off becomes21

$$z^\* = \left(\frac{f}{f\_\varepsilon} \frac{1}{r+\delta} \left(\left(\frac{\kappa}{\kappa-1}\right)^{\varrho-1} - 1\right)\right)^\frac{1}{\kappa} \chi. \tag{z\_{\mathrm{MJ}}^\*}$$

Any policy addressed to reduce the entry cost *fe* or the equilibrium interest rate *r* makes the economy more selective.

21 At steady state, the interest rate *r* = ρ is constant.

20 Notice that the R&D entry cost, even if sunk, it has a value. We will interpret it as a form of intangible capital, which has different value depending on the productivity of the firm.

There is a stationary allocation of labor to production and R&D investments, such that at the stationary equilibrium the mass of new intermediary firms is equal to the mass of exiting firms and the labor market clears. At the stationary equilibrium of the Melitz model, the mass of intermediary firms is given by

$$m = \left(\varrho - 1 + \frac{\delta + r\left(\frac{\kappa - 1}{\kappa}\right)^{\varrho - 1}}{r + \delta}\right)^{-1} \frac{1}{f} \left(\frac{\kappa - 1}{\kappa}\right)^{\varrho - 1}.$$

In the Melitz model, the entry (R&D) cost depends on wages, which raises with selection. A more selective economy faces then a larger entry cost, which reduces the incentives to enter. This mechanism causes the economy to converge to a stationary mass of varieties and a stationary cutoff productivity. In Sect. 3.4, we analyze economies where both the cutoff productivity and the mass of intermediary inputs permanently increase, making the economy to be more innovative with growing productivity and output.

#### *3.3.3 Oligopolistic Competition*

In the monopolistic competitive framework, since intermediary firms share the same elasticity of substitution with each other, they all set the same time-invariant markup. As discussed by Koeninger and Licandro (2006), equal markups cause the monopolistic competitive allocation to be optimal. In this section, we discuss a close economy version of Impullitti and Licandro (2018), who develop an oligopolistic competitive framework allowing to understand the fundamental role of competition in shaping the relation between competition, selection and growth.22

**The Economy.** As in the monopolistic competitive model of Sect. 3.3.2, let us assume a sole consumption good is produced by a representative, perfectly competitive final firm by the mean of the constant elasticity of substitution (CES) technology with constant elasticity of substitution > 1. Final firms are price takers in both the final and the intermediary markets, and optimally demand

$$\mathbf{y}\_{\mathbf{l}}(z) = p\_{\mathbf{l}}(z)^{-\varrho} c\_{\mathbf{l}},$$

22 See also Peretto (1996, 2003) and Navas and Licandro (2011).

where *ct* is total production, *yt*(*z*) and *pt*(*z*) are, respectively, the demand and price of intermediary input *z*. As before, the final consumption good is used as numeraire.

Following Impullitti and Licandro (2018), intermediary inputs, indeed, instead of being produced under monopolistic competition as in Melitz (2003), are produced under Cournot competition. There are *n* firms, *n* > 1, producing variety *z* by the mean of technology

$$
\chi\_{i,I}(z) = Az \,\ell\_{i,I}(z),
$$

where *yi*,*t*(*z*) and *i*,*t*(*z*) represent output and labor, respectively, of firm *i* sharing productivity *z* with other *n* − 1 firms, its direct competitors; the state of technology *A* > 0 is assumed to be constant. Of course, *yt*(*z*) =  *i yi*,*t*(*z*). As in the previous sections, operative intermediary inputs have productivity *z* ≥ *z*∗ *t* .

The equilibrium price of the Cournot game, the same for all firms producing *z*, is

$$p\_l(z) = \frac{1}{\theta} \, \frac{w\_l}{Az},$$

where the inverse of the markup rate is θ = *n*−1/ *n* , with the markup going from −1 to one, as the economy moves from monopolistic competition (*n* = 1) to perfect competition, when the number of firms goes to infinity.

Aggregating over intermediary firms, it can be shown that consumption per capita, the wage rate and total net revenues are

$$c\_l = Am\_l^{\frac{1}{\varrho - 1}} \overline{z}\_l L\_l, \qquad w\_l = \theta A m\_l^{\frac{1}{\varrho - 1}} \overline{z}\_l \quad \text{and} \quad \pi\_l = (1 - \theta) c\_l, \overline{z}\_l$$

with average productivity defined as

$$\bar{z}\_l = \left(\int\_{z\_l^\*}^{\infty} z^{\varrho - 1} \phi\_l(z) dz\right)^{\frac{1}{\varrho - 1}},$$

where *Lt* represents the share of total labor allocated to production (excluded the fixed production costs). For a given cut-off productivity *z*∗, the Cournot and the monopolistic competitive economies produce the same output. However, the share of labor is larger in the Cournot equilibrium, increasing with competition. An increase in competition reallocates income from (intangible) capital to labor.

**Selection.** Interestingly, if the number of competitors *n* is given, and the entry decision were jointly taken by the *n* potential entrants, since profits of the marginal firm and expected profits of the potential entrant are both affected proportionally by θ, the equilibrium cut-off at steady state is equal to *z*∗ in the equilibrium condition (*z*∗ M ) of the Melitz model. An exogenous change in the number of competitors *n* does not affect selection. However, the fraction of labor allocated to production and the mass of intermediary inputs do. At steady state,

$$L = \left( 1 + \frac{1 - \theta}{\theta} \left( \frac{\delta + r \left( \frac{\kappa - 1}{\kappa} \right)^{\varrho - 1}}{r + \delta} \right) \right)^{-1}$$

and

$$m = \left(\frac{\theta}{1-\theta} + \frac{\delta + r\left(\frac{\kappa - 1}{\kappa}\right)^{\theta - 1}}{r + \delta}\right)^{-1} \frac{1}{f} \left(\frac{\kappa - 1}{\kappa}\right)^{\theta - 1}.$$

An increase in competition, measured by a raise in θ, renders the static allocation more efficient, which moves labor toward production, increasing *L*. As an implication, less labor has to be allocated to create new varieties and to cover the fixed production costs, which implies a reduction in the mass of varieties.

In fact, Impullitti and Licandro (2018) analyze the problem under a very different perspective. They assume that the entry condition determines endogenously *n*. They also assume that there is a mass one of potential varieties, *mt* being the equilibrium mass. Potential entrants face a zero entry cost, but can only enter by producing a particular variety. At equilibrium, then, 1 − *mt* varieties are introduced at any time *t*; from them a fraction 1 − (*z*∗) is produced, the others exit instantaneously. As a consequence, the equilibrium mass of varieties is determined by the stationary condition

$$(1 - m)(1 - \Psi(z^\*)) = \delta m...$$

The free entry condition, instead, determines the number of competitor *n* that produce any intermediary input. Since *n* is determined before the productivity *z* is known, all varieties have the same number of competitors at a balanced growth path equilibrium.

#### *3.3.4 Physical and Intangible Capital*

An alternative way to Collard and Licandro (2018) of adding capital to the Hopenhayn model is in Hopenhayn (2014), which assumes that technology in (3.1) is defined in a composite production factor, such as,

$$\chi\_l(z) = z^{\alpha} F(k\_l(z), \ell\_l(z))^{1-\alpha}$$

where *F* (.) is a Neoclassical production function, *kt*(*z*) and *t*(*z*) are physical capital and labor employed for firm *z*. Hopenhayn (2014) shows that at equilibrium the aggregate technology is

$$\gamma\_l = \left(\bar{z}\_l m\_l\right)^{\alpha} F(k\_l, 1)^{1-\alpha}$$

where *mt* is the mass of firms, *kt* is physical capital per capita and, as before, total labor is assumed to be equal to one. Notice that in this economy there are two forms of capital: physical capital *kt* and intangible capital *z*¯*tmt*. The aggregate technology shows constant returns on labor, physical and intangible capital like in Corrado et al. (2009).

# 3.4 Firm Heterogeneity in Models with Innovation

#### *3.4.1 Romer Model*

Romer (1990) is based on the monopolistic competitive model developed by Dixit and Stiglitz (1977). Simplifying the model in Sect. 3.3.2, let us assume that identical intermediary firms (all with productivity *z* = 1) monopolistically compete in the intermediary goods market. In this framework, a typical intermediary firm sets price and produces quantity

$$p(z) = \frac{\varrho}{\varrho - 1} w\_l \quad \text{and} \quad \mathbf{y}(z) = \left(\frac{\varrho}{\varrho - 1} w\_l\right)^{-\varrho} c\_l, z$$

respectively, where w*t* is the equilibrium wage rate and *ct* is aggregate consumption; > 1 is the elasticity of substitution between intermediary inputs in the production of the final (consumption) good. Since intermediary firms are symmetric, they all set the same price and produce the same quantity. At equilibrium, the technology producing the final consumption good is

$$c\_l = A m\_l^\upsilon L\_l,$$

where the mass of intermediary inputs *mt* shows up as an externality and *Lt* is the fraction of total labor allocated to the production of intermediary goods (there are no fixed productions costs in Romer). Aggregate technology shows the well-known love-for-variety externality: labor productivity in the final good sector raises with the mass of intermediary inputs *mt* at the rate ν = 1 −1 . 23

Concerning innovation, let us assume that new intermediary varieties are produced by the mean of the R&D technology

$$
\dot{m}\_l = B m\_l (1 - L\_l),
$$

where R&D productivity *B* is normalized to *B* = ( − 1)*A* > 0 in order to simplify notation. Since the total labor supply is normalized to one, 1− *Lt* is the fraction of it allocated to research activities. The productivity of labor in the R&D sector is critically assumed to depend on the mass of varieties *mt*.

Let us define the state of knowledge as *kt* = *m*ν *t* . This allows us to interpret the Romer model in line with the Arrow (1962) learning-bydoing model. The economy learns by producing new varieties of the intermediary input. By doing so, technology becomes more productive in both the final good sector and the R&D sector. Combining the two last equations, the feasibility condition becomes

$$
\dot{k}\_l = Ak\_l - c\_l.
$$

With the state of knowledge *kt*, the economy produces *Akt*, which can be consumed or allocated to the production of new knowledge—a form of intangible investment in the sense of Corrado et al. (2009). Notice that the normalization used to define *kt* as a function of *mt*, including that of *B*, is inconsequential since knowledge has no natural unit. Technology in the Romer model then collapses to a one-good AK technology like in Rebelo (1991), sharing with Rebelo (1991) the conditions for a constant endogenous growth rate.

The optimal allocation of output *Akt* to consumption and savings (adopting the form of intangible investment) is as usual governed by the

23 Benassy (1996) adopts a more general framework, arguing that the love-for-variety externality ν may be any number between zero and infinity, independent of the elasticity of substitution .

Euler equation (EE). At equilibrium, the return on R&D is

$$r\_l = \frac{1}{\varrho - 1} A L\_l,$$

decreasing in the elasticity of substitution, but increasing in the final good productivity parameter *A* and the fraction of employment allocated to production. Notice that an increase in the degree of substitutability between intermediary goods raises the demand elasticity, reducing markups and profits and then decreasing the return on R&D.

Substituting the equilibrium interest rate *rt* in the Euler equation (EE), it can be shown that the equilibrium growth rate is

$$\mathbf{g} = \sigma \left( \frac{A - \rho}{\varrho} - \rho \right),$$

which negatively depends on the elasticity substitution , since it negatively affects the return on R&D.

**Firm Heterogeneity.** The Romer model can be combined with the Melitz model to generate endogenous growth with firm heterogeneity, where selection by affecting the productivity of the final good sector will have a direct effect on the growth rate. Aggregating over intermediary firms,

$$c\_l = A\bar{z}\_l k\_l L\_l \dots$$

By assuming that productivity *B* in the R&D technology also depends on the average productivity *z*¯*t*, knowledge evolves following

$$k\_l = A\bar{z}\_l k\_l (1 - L\_l),$$

implying that the feasibility condition becomes

$$
\dot{k}\_l = A\bar{z}\_l k\_l - c\_I.
$$

Technology is AK with the marginal product of capital depending on selection through *z*¯*t*. The productivity gains through selection spillover to the consumption and R&D sectors.

Since new firms draw their productivity from a time-invariant distribution, the productivity cut-off is constant at a balanced growth path, as well as the average productivity *z*¯. As in the Romer model, the return on R&D and the growth rate depend on the average productivity *z*¯. At steady state

$$\mathbf{g} = \sigma \left( \frac{A\bar{z} - \rho}{\varrho} - \rho \right).$$

Selection makes the average productivity of capital larger, positively affecting the stationary growth rate. Those parameters positively affecting selection in the Melitz model, have here also a positive effect on the growth rate.

#### *3.4.2 Selection and Imitation*

Following Luttmer (2007), selection by itself can generate endogenous growth through imitation, since selection raises the productivity of incumbents.24 How do new firms react to this raise in productivity? Instead of drawing their productivity from a time-invariant distribution, the initial productivity of new firms is randomly drawn from an entry distribution that follows key moments of the incumbents equilibrium distribution. In this sense, new entrants learn from incumbents, imitating them, which causes productivity gains from selection to spillover innovators. A similar mechanism is used by Sampson (2016) to study the dynamic gains from trade.25

Let us follow the argument as developed by Sampson (2016), adapting his notation to be consistent with the notation in the previous sections. Sampson's model belongs to the family of monopolistic competitive models with labor as the sole production factor as developed by Melitz (2003) and reviewed in Sect. 3.3.2. Monopolistically competitive intermediary firms draw at entry a time-invariant labor productivity *z* from an entry productivity distribution ˜ *t*(*z*), which differently from Melitz is assumed to be time dependent. Firms productivity is time invariant. However, due to selection, learning spillovers cause the distribution from which they draw their productivity follow these improvements in technology. Intermediary firms require a variable and fixed (*f* ) amount of labor to produce with w*t* being the equilibrium wage rate.

24 See also Luttmer (2011, 2012).

25 Gabler and Licandro (1979) develop the same idea in a framework similar to the one in the Ramsey-Hopenhayn model.

As in Sect. 3.3.2, an intermediary firm with productivity *z* sets price

$$p\_I(z) = \frac{\varrho}{\varrho - 1} \frac{w\_I}{z}.$$

The zero profit condition implicitly defines the equilibrium cut-off productivity *z*∗ *t* ,

$$\frac{1}{\varrho} \left. \frac{c\_l}{m\_l} \left( \frac{z\_l^\*}{\bar{z}\_l} \right)^{\varrho - 1} \right| = f \, w\_l \dots$$

where *ct* /*mt* is total consumption per firm (reminds that final output is fully allocated to consumption in the Melitz model).

**Entry and Spillovers.** Like in Romer (1990), potential entrants undertake R&D activities to discover new varieties of intermediary inputs. They pay the R&D (entry) cost *fe*w*t*, where *fe* > 0 is the amount of labor required to create a new intermediary input. R&D plays exactly the same role as an entry cost in the Melitz model. As usual in this literature, innovators are assumed to be protected by infinitely lived patents.

The critical assumption is the following: at any time *t* innovators (the entrants), after paying the R&D (entry) cost, draw a time-invariant productivity *z* = ω*z*¯*t*, where *z*¯*t* is the average productivity of incumbents and ω is a random variable distributed (ω) ˜ . To fix ideas, let us assume that (ω) ˜ is Pareto with tail parameter κ > 1. The only difference with the Melitz model is that the entry distribution *t*(*z*) = ( ˜ *z*/*z*¯*t*), from which innovators draw *z*, is time dependent. That is, it depends on the time varying average productivity of incumbents. Innovators learn from incumbents through this particular type of spillover.26

Let us define the firm-specific relative productivity *z*˜*t*, *z*˜*t* = *z z*∗ *t* , relative to the cut-off productivity *z*∗ *t* . Since the domain of *z* is (*z*∗ *t* ,∞), the domain of *z*˜ is (1,∞). In a growing economy, *z*∗ *t* will be permanently moving to the right. Since firm's productivity is time invariant, the relative productivity of any firm will eventually converge to one on a finite time. When the lower bound is reached, the firm exits. Firms born at different moments in time belong to different technological cohorts, and since new cohorts are in average more productive, firms face a finite life,

26 Imitation, since in this framework comes at a zero cost, can also be interpreted as diffusion.

i.e. firms are losing value over time since their technology becomes slowly obsolete.27

Despite the fact that firms have a finite life, under the assumption that the entry distribution (ω) ˜ is Pareto, with tail parameter κ > 1, the stationary productivity distribution is *t*(*z*) = 1 − *z*∗ *t z* κ , for *z* ≥ *z*∗ *t* . It is Pareto distributed, inheriting the tail parameter κ from the entry distribution. Since at a balanced growth path selection moves cut-off productivity *z*∗ *t* to the right, the equilibrium distribution *t*(*z*) is said to be a traveling wave.28

Combining the exit and entry conditions with the Euler equation, Sampson (2016) shows that the steady state endogenous rate of technical progress is

$$g = \frac{\sigma}{1 + \sigma(\kappa - 1)} \left( \frac{\varrho - 1}{\kappa + 1 - \varrho} \left. \frac{f}{f\_e} - \rho \right) .\right)$$

Output per capita and consumption also grow at the rate *g*. An increase in the variance of the entry distribution (a reduction in κ), which is equivalent to an increase in the probability of exceptionally good innovations, or a reduction in the R&D (entry) cost *fe* both make the economy more selective. They increase the productivity of incumbent firms and through learning affects the productivity of innovators, inducing faster growth at steady state. An increase in the elasticity of substitution between intermediary inputs makes the economy more competitive, also inducing more selection and growth.

27 In this sense, Sampson's model belongs to the vintage capital tradition. In Gilchrist and Williams (2000), for example, the productivity of new firms is drawn from a lognormal distribution, which mean has an exogenous trend. As shown by Boucekkine et al. (1997) and Boucekkine et al. (2005), vintage models involve (periodic) medium term movements which may be of relevance for the propagation of innovation. Bambi et al. (2014) extend this idea to vintage models with R&D, showing that the long delay that innovation takes to diffuse generates medium term cycles, which has to be considered when evaluating the performance of innovation policies.

28 Even if not proved by Sampson (2016), under similar conditions as in Collard and Licandro (2018), the equilibrium distribution will likely be a truncated Pareto with cutoff productivity moving systematically to the right following the endogenous progress in technology.

#### *3.4.3 Schumpeterian Model*

In the Schumpeterian framework, the dynamics of firms is modeled through the fundamental concept of creative destruction, which takes two forms: business stealing and crowding-out (or obsolescence).

In Aghion and Howitt (1992), the final consumption good *ct* is produced by the mean of technology

$$c\_l = \left(\int\_{z\_l^\*}^{\infty} \left(z \ge\_l(z)\right)^{\frac{\rho-1}{\varrho}} \phi\_l(z) \,\mathrm{d}z\right)^{\frac{\rho}{\varrho-1}},$$

where *xt*(*z*) represents the quantity of intermediary input *z* used in the production of the final consumption good *ct*, > 1 is the elasticity of substitution between them, and φ*t*(*z*) is the equilibrium (density) distribution. Differently from Romer (1990), the mass of varieties *mt* is assumed to be time invariant and normalized to one, but the quality of goods *z* is assumed to be heterogeneous. In the Schumpeterian model, R&D activities are addressed to improve the quality of exiting varieties, which makes φ*t*(*z*) time dependent, reflecting changes in technology induced by innovation.

Technology in the intermediary sector is assumed to be

$$
\alpha\_l(z) = A\ell\_l(z),
$$

where parameter *A* > 0 and *t*(*z*) is labor allocated to the production of the intermediary input *z*. Notice that the variable change *yt*(*z*) = *zxt*(*z*) brings us back to the Melitz model, where *yt*(*z*) is measured in quality adjusted units.29 Even if the Schumpeterian model is usually interpreted as a model of product innovation (addressed to improve the quality of intermediary goods), it can also be interpreted as a model of process innovation (addressed to reduce production costs).

Following the analysis in the previous sections, at equilibrium

$$c\_t = A\bar{z}\_t L\_t,$$

29 Price indexes are built with the objective of keeping quality constant, meaning that real quantities in national accounts are measured in quality adjusted units.

where

$$\bar{z}\_l = \left(\int\_{z\_l^\*}^{\infty} z^{\varrho - 1} \phi\_l(z) dz\right)^{\frac{1}{\varrho - 1}},$$

is the average product quality interpreted as in Aghion and Howitt (or average productivity as in Melitz) and *Lt* is labor allocated to the production of intermediary goods.

In the Schumpeterian model, an innovation is a new technology able to produce a better quality input, which is a perfect substitute of an already existing intermediary input (or a cheaper version of an intermediary input of the same quality). The particular input is randomly selected among the unit mass of existing intermediary inputs. When a new technology is discovered, the previous one becomes obsolete, making the previous producer to exit.30 Creation of new technologies is then associated with the destruction of old ones. The probability that a new version arrives follows a Poisson process with arrival rate *b* for unit of labor allocated to R&D, *b* > 0. 31

The productivity of the new version is assumed to be equal to the frontier, *leading-edge* technology ω*t*, which given some ω0 > 0 at time *t* = 0, is assumed to follow

$$\frac{\dot{\phi}\_l}{\omega\_l} = b(1 - L\_l),\tag{3.3}$$

where 1 − *Lt* represents the share of total labor allocated to R&D activities.32 Individual research effort spills over into the whole economy by

30 The distance in productivity between two consecutive innovations of a particular variety depends on the time interval between them. It may be that this distance is small enough to make the incumbent compete with the innovator. In this case, the innovation is said to be non-drastic. To avoid the associated complications of studying market structures with non-drastic innovations, it is assumed that the incumbent's technology is destroyed with the discovery of a new way of producing the same intermediary input.

31 In this literature, an innovation is a random event. Let us denote by *F*(*T* ) the probability that this event occurs before a period of length *T*. A Poisson process assumes *F*(*T* ) = 1− e−μ*T* , μ > 0. The associated density function is *f* (*T* ) = μ e−μ*T* , implying that the probability that the event occurs around *T* = 0 is μ. The probability that the event does not occur before *T* is e−μ*T* .

32 Note that technology has a vintage structure. Innovations introduced at time *t* have the frontier productivity ω*t*, which will be growing at equilibrium.

moving the frontier of knowledge, mimicking a form of learning-bydoing.

Finally, let us assume that new varieties receive infinitely lived patents, giving to the innovator the exclusivity on the use of this technology to produce the corresponding intermediary input.

Let us denote by *at* = *z* ω*t* the productivity of *z* relative to the frontier technology and by *H*(*a*), *a* ∈ [0, 1], the cumulative distribution of firms across technologies. It can be easily shown that at steady state the distribution *H* (*a*) is uniform in (0, 1) .33 It can also be shown that

$$
\bar{z}\_l = \mu \,\,\omega \quad \text{with} \quad \mu = \varrho^{\frac{1}{1-\varrho}} < 1.
$$

The average technology *z*¯*t* is at a distance μ from the frontier technology ω*t*. The *distance to the frontier technology* is increasing in , for > 1, with lim→∞ μ = 1, meaning that the average distance to the technological frontier approaches unity when intermediary inputs are close to perfect substitutes. On the other extreme, it goes to *e*−1 when goes to one.

At the balanced growth path, at any time *t*, the return to a patent aged *s* is

$$r = \underbrace{\frac{\pi\_{s1}}{v\_{s1}}}\_{\text{dividend-to-value}} + \underbrace{\frac{\dot{v}\_{s1}}{v\_{s1}}}\_{\text{capital gains}} - \underbrace{b(1 - L)}\_{\text{business-scalling}} - \underbrace{(\rho - 1)b(1 - L)}\_{\text{obsolsscance}}.$$

As usual, the return to assets adds capital gains to the dividend-to-value ratio. The remaining terms represent the negative effect that technical progress has on the patent protecting existing intermediary inputs. The third term is the so-called *business-stealing effect*. It measures the Poisson rate at which the patent will eventually die, when the associated intermediary input is substituted by a subsequent innovation. The last term represents the *obsolescence cost* produced by the emergency of cheaper (better quality) versions of other intermediary inputs, reducing the demand for the input produced by the patent.34 As time passes, other varieties become more and more productive, reducing the demand and

33 The distribution of firms across relative productivities is uniform because, by assumption, the rate at which innovations arrive is the same as the rate at which the frontier technology grows. Otherwise, the distribution is Pareto, as it is usually assumed in this literature.

34 Aghion and Howitt (1992) refer to it as the *crowding-out effect*.

profits of the variety we are evaluating. The obsolescence cost depends on the velocity at which the frontier technology moves, *b*(1− *L*), and the elasticity of substitution across varieties. When varieties become perfect substitutes, old varieties are substituted out by new varieties fully, making the crowding-out effect infinity.

#### *3.4.4 Innovation and the Life Cycle of Firms*

Klette and Kortum (2004) extends the Melitz model in line with the literature on endogenous growth with the aim of describing better the life cycle of firms. In this framework, the productivity of firms does not depend on their own intrinsic characteristics, but it is randomly assigned.35

**Firms and Products.** In the Klette and Kortum (2004) framework, a continuum of firms produces a continuum of measure of one of intermediary inputs, with each firm producing an integer number *nt* of them, *nt* ∈ {1, 2, 3, ...}. The integer number of intermediary inputs produced is heterogeneous and endogenously determined at equilibrium.

As in the Schumpeterian model, producing one unity of any intermediary input requires one unit of labor; labor productivity is normalized to one, and the same holds for all inputs. However, intermediary inputs are heterogeneous in their quality. The quality frontier of an intermediary input is denoted by *zt*.

**Innovation.** Technical progress in each intermediary input is represented by a quality ladder model as in Grossman and Helpman (1991). The dynamics of the frontier technology for the different intermediary inputs is governed by two types of innovations: innovation by incumbent firms and innovation by potential entrants. When a discovery takes place, it is randomly assigned to a single intermediary input moving its quality frontier one step in the quality ladder. The gain in quality is given by a firm-specific factor *q* > 1, which is specific to the firm that makes the discovery.

The firm-specific factor *q* maps one-to-one to a firm-specific profit per intermediary input π, π ∈ (0, 1), and it is the same for all intermediary inputs produced by the same firm; π and *q* are positively related and time invariant. In the following, it is assumed that firms draw π from

35 See also Acemoglu and Cao (2015).

the continuous distribution function (π ), which is equivalent to draw factor *q* from a known distribution. An operative intermediary firm is then characterized by a duple {*nt*, π}, with *nt* evolving over time.

**Innovation by Incumbents.** A firm that exercises intensity λ, when undertaking R&D activities, has a Poisson rate λ*n* of making a discovery, where *n* is the number of intermediary inputs being currently produced by the firm. This discovery allows the firm to move one step in the quality ladder of the frontier technology of a randomly selected intermediary input. The particular input is unknown to the firm at the time the firm undertakes the R&D activities. The R&D cost function of a firm {*n*, π} is

$$\frac{\pi}{\bar{\pi}} \, c(\lambda) \, n.$$

Function *c*(λ) is increasing and strictly convex (some other technical assumptions s.t. *c*(λ) > λ*c* (λ) are also required). The cost function depends on the firm-specific innovation factor *q* through π; more innovative firms, those with larger *q*, face larger innovation costs. The optimal (interior) innovation policy requires

$$\begin{aligned} c'(\lambda) &= v \\\\ (r + \mu - \lambda)v &= \bar{\pi} - c(\lambda), \end{aligned}$$

where *v* represents the expected value of a product produced by a firm of average type π¯ , *r* is the equilibrium interest rate and μ is the rate of creative destruction (measuring the rate at which the firm may lose a product line because another firm has just innovated in this particular product line). The first condition states that the marginal cost of innovation has to be equal to its marginal value. The second condition states that the expected return on innovation has to be equal to its opportunity cost. Irrespective of the firm-specific duple {*n*, π}, all firms optimally chose the same innovation intensity λ, the Poison rate of innovation of a firm with *n* products being λ*n*. Indeed, more profitable firms face larger R&D costs and have larger per product value.

**Innovation by Potential Entrants.** There is a mass of potential entrants investing at the rate *F*, *F* > 0, in return for a unit Poisson rate of entering the economy with a single product. Potential entrants, after entering, draw a firm-specific profit π from (π ). The firm-specific profit π and the associated innovation factor *q* apply to the first and any subsequent discovery of the firm. As in the case of incumbents, a new entrant randomly chose an intermediary input.

**Equilibrium Innovation.** Notice first that at equilibrium μ = λ + η, where η is the Poisson rate of innovation by potential entrants. Free entry requires *F* = v, since in expected terms the potential entrant covers the entry cost *F* with the value of the innovation. Combining the free entry condition and the optimal innovation policy of incumbents, the equilibrium innovation intensity of incumbents is determined by the condition

$$c'(\lambda) = F,$$

and the equilibrium innovation intensity of potential entrants by

$$
\eta = \frac{\bar{\pi} - c(\lambda)}{F} - \rho,
$$

where the interest rate *r* = ρ at the balanced growth path.

These two equations are fundamental to understand the incentives to innovation in the Klette-Kortum model, and hence the potential effects of innovation policies. An increase in the entry cost *F* reduces the R&D activity of potential entrants η, but raises the incumbents' innovation intensity λ. An increase in average profits π¯ raises the innovation intensity of new entrants but has no effect on incumbents. A raise in the innovation cost of incumbents (affecting both average and marginal costs) will have negative incentives for both incumbents and potential entrants. In this model, innovation policy affects innovation only through these channels, Of course, any policy that reduces the financial costs of firms, as reflected by ρ, also promotes innovation by potential entrants.

**Limit Pricing.** From the point of view of the final firm, the quality frontier version of any intermediary input is a perfect substitute of any previous version of the same input, with all versions weighted by their respective qualities. Under Bertrand competition, the firm producing the frontier quality optimally charges a markup *q* to its marginal cost in order to deter any competitor. Consequently, at equilibrium only the frontier quality is produced with the last innovator charging a markup equal to its specific factor *q*.

Intermediary inputs aggregate into the final output through the Cobb-Douglas technology

$$\mathbf{y}\_t = \int\_{\mathcal{z}} \log \left( \mathbf{z} \,\mathbf{x}(\mathbf{z}) \right) \phi(\mathbf{z}) \,\mathrm{d}z,$$

where φ(*z*) is the equilibrium distribution of the frontier quality across intermediary inputs. It is easy to see that the optimal demand implies *p*(*z*)*x*(*z*) = 1. A firm with quality improvement factor *q* yields then the same constant profit flow π = 1 − *q*−1, π ∈ (0, 1), for each intermediary input it produces irrespective of its quality *z*, since *p*(*z*)*x*(*z*) = 1 for all *z*.

# 3.5 Further Contributions

This survey does not review other important dimensions of the innovation process that may also be relevant for policy analysis, which should be considered when designing macro models for the evaluation of innovation policies.

First, there is a large literature analyzing the role of financial frictions shaping market selection and innovation.36 The evaluation of policies addressed to reduce financial frictions in order to promote innovation and productivity growth requires a rigorous modeling of the financial sector and the associated frictions.

Second, innovation policy needs also to be evaluated by its redistributive effects on the labor market, with regard to the correction of the negative effects that technical progress has in the evolution of employment and wages across industries and occupations. The recent literature on job polarization, automatization and skill obsolescence is addressed to study the labor market effects of innovation and technical progress. Technical progress develops differently in different sectors, affecting unevenly the dynamics of jobs and occupations. One of the most striking implications of these diverse sectorial evolutions of technology is stressed by the literature on structural transformation.37 This literature looks at replicating the evidence of non-balanced patterns of the three main

36 See Cooley and Quadrini (2001), Buera et al. (2011), and Midrigan and Yi Xu (2014), among others.

37 See Duarte and Restuccia (2010), as well as Herrendorf et al. (2014) for a survey on this literature.

sectors of modern economies: agriculture, manufacturing and services. An appropriate modeling of this dimension, likely in line with Kongsamut et al. (2001), Ngai and Pissarides (2007), and Acemoglu and Guerrieri (2008), will be of great importance in order to evaluate industrial policies addressed to promote innovation.

The unbalanced evolution of industries is mimicked by an unbalanced evolution of occupations (see Duernecker & Herrendorf, 2017), which is reflected in the polarization of wages and employment observed in the data (see Autor & Dorn, 2013). Modeling the joint evolution of technology and occupations, in line with the skill obsolescence hypothesis in Licandro and Poschke (2017), is of fundamental importance to evaluate the labor market effect of innovation policies.

Third, trade, although omitted in this chapter, is fundamental to understand the impact of innovation policies. This is particularly important for the evaluation of innovation policies in the European Union, where policies are expected and frequently addressed to affect regions and countries differently (See Atkeson and Burstein [2010], Aw et al. [2011], Baldwin and Robert-Nicoud [2008], and Broda and Weinstein [2006], among others). A model of this nature is needed to evaluate the trade-off between promoting excellence, by addressing resources toward the most efficient regions, and regional convergence or catching-up.

# References

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# The Frontier of Macroeconomic Modelling: Proceedings of the JRC-IEA Workshop 2017

# *Omar Licandro*

# 4.1 Introduction

The JRC-IEA Roundtable on Macroeconomic Modelling for R&D and Innovation was jointly organized by the DG Joint Research Centre (JRC) of the European Commission and the International Economic Association (IEA). The design and development of macroeconomic models addressed to study the impact of innovation policies is critical for the European Union, for which innovation policies are one of the highest priorities. The Roundtable aimed to discuss, in the framework of the recent development of the literature on economic growth and innovation, alternative modelling strategies for innovation and medium/long-term productivity and economic growth. The debate was organized having in mind the need for new ideas that may help the design of economic models addressed to evaluate the impact of innovation and related policies.

During the Roundtable, top researchers, including Philippe Aghion (Harvard), among others, presented some key new developments in the

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O. Licandro (B)

University of Nottingham, Nottingham, UK e-mail<EMAIL_ADDRESS>
© The Author(s) 2022

U. Akcigit et al. (eds.), *Macroeconomic Modelling of R&D and Innovation Policies*, International Economic Association Series, https://doi.org/10.1007/978-3-030-71457-4\_4

field of innovation and growth. The Roundtable aimed to better understand where the frontier of knowledge in the field of innovation and growth is to, and in a second stage, figure out the key elements a macro model designed to evaluate innovation policies should include. In particular, Ufuk Akcigit (Chicago) presented a survey on his views on the current academic research agenda on R&D and innovation. The session was closed by a panel composed mainly of practitioners, and a few academics, with the object of giving the perspective of those more directly involved in the evaluation of innovation policies or in the development of those models designed to evaluate these policies. A short summary of each contribution and my reading of the debate that followed are provided in Section 2. Section 3 discusses the proposed alternative lines of modelling that emerged from the Roundtable. It reflects the views of the author on a highly fruitful, sometimes controversial, debate that took place during the Roundtable.

# 4.2 Macroeconomic Modelling of Innovation

This section contains some of the lessons from the papers presented at the JRC-IEA Roundtable on Macroeconomic Modelling for R&D and Innovation.

• **Missing Growth from Creative Destruction** by Philippe Aghion, Antonin Bergeaud, Timo Boppart, Peter J. Klenow and Huiyu Li (Aghion et al., 2019).

Statistical agencies aim to compute price indexes for representative baskets of constant quality products. However, in practice, some products disappear being displaced by better quality ones. The authors point out that, in these cases, statistical agencies typically impute inflation for disappearing products from the inflation for surviving products, when likely its inflation may be lower because of quality improvements embodied in the substituting product. As a result, creative destruction may result in overstated inflation and understated growth. The authors use a simple model to relate this missing growth to the frequency and size of various kinds of innovations. Using US Census data, they assess the magnitude of missing growth for all private non-farm businesses from 1983 to 2013. They find: (i) missing growth from imputation is substantial, between 0.5 and 1 percentage points per year; and (ii) almost all of the missing growth is due to creative destruction (as opposed to new varieties).

The paper points to a key issue on evaluating the macroeconomic impact of innovation policies: the critical problem of measuring real output and productivity in a world where technical progress is embodied in new, better quality versions of existing products. The measurement strategy suggested by the authors is model-based. However, statistical agencies are reluctant to explicitly use models to measure price changes and strongly prefer well-designed methods based on data collection, which depend much less on highly specific modelling assumptions. Of course, there is no measurement without theory. Hence, data collection and statistical methods used to aggregate individual data are both based on theory. However, the theory behind these methods is usually quite general and does not depend on specific functional forms and parameter values.

In the same direction, Broda and Weinstein (2006) suggest a different strategy, based on love-for-variety theories, to measure gains associated with new products. Contrary to Aghion et al. (2019)' s findings reported above, Broda and Weinstein (2006) conclude that the US missing growth from increasing the product variety is of around 1.2 yearly percentage points for the period 1972–2001. Indeed, this estimation strongly depends on some strong assumptions about the extent of utility gains coming from love-for-variety.

The measurement of productivity at the firm level raises also some important measurement problems. It is generally accepted now that productivity at the firm level has at least two components: product value (or demand shock) and technical efficiency (generally referred as TFPQ) whose estimation faces some important issues. Indeed, the propagation of productivity gains in a network, by reducing the cost of inputs of upstream firms, calls for a third dimension of productivity: the quality and price of production inputs.

The main lesson to retain from the Aghion et al. (2019) paper is that a careful analysis of the way GDP growth is measured in the data is needed to make a correct evaluation of innovation policies. This problem has to be seriously taken into account when comparing model simulations used to evaluate innovation policy with the data. If gains from innovation are not in the statistics, we will never find them in the data and it will be difficult to find them in model's simulations.

• **The Dynamics of Development: Innovation and Reallocation** by Francisco Buera and Roberto Fattal-Jaef (Fattal Jaef & Buera, 2015).

Buera and Fattal-Jaef study the aggregate and firm-level properties of the dynamics of economic development, by investigating the macro and micro features of successful growth take-offs in the data and find that, while every episode exhibits sustained growth in TFP and investment rates, there are substantial differences in the evolution of the firm size distribution between the experiences of post-communist economies and the rest of the successful take-offs. The pattern is that firms tend to get larger on average during a typical acceleration, while the average size of a firm is declining along a post-communist transition. To understand this behaviour, the authors provide a quantitative theory of transitions featuring endogenous innovation, entry and exit, and the dismantling of idiosyncratic distortions. They evaluate hypothetical reforms in which the rate of progress in the reversal of distortions is calibrated to the experiences of China and Chile, to find that the mechanisms in the model are able to capture the salient features that they document in the data. The approach may be relevant for economies undergoing a similar transition or catching-up.

• **Fewer but Better: Sudden Stops, Firm Entry, and Financial Selection** by Sina Ates and Felipe Saffie (Ates & Saffie, 2021).