Lecture Notes on the Empirical and Quantitative Methods in International Trade

Transcription

1 Lecture Notes on the Empirical and Quantitative Methods in International Trade Xin Tang 1 Department of World Economics Economics and Management School at Wuhan University November 2, The author is responsible for all possible mistakes. Please address all correspondence to: zjutangxin@gmail.com

2 To those who helped me in my Ph.D career

3 PREFACE THIS SET OF NOTES is designed for the one-semester graduate course, Empirical and Quantitative Methods in International Trade, that I taught in Fall 2015 at Wuhan University. We will take a stand from the perspective of individual firms to examine the empirical and quantitative aspects of various trade theories. The primary purpose of these notes is to help students develop the ability of bringing theories usually presented in the form of mathematical models to real world data. I also hope that the materials covered in these notes can improve students ability to build, test, and use economic theories in other branches of economics, for example quantitative macroeconomics, labor economics, and industrial organization, etc. Since our focus will be on the connection between theories and evidence, these notes will be different from those that aim at introducing the theories of trade, and those that attempt to explain the actual empirical and quantitative tools. For instance, we will not devote efforts to the Heckscher-Ohlin model, the Armington model, and models with explicit treatment of the internal organizational structure of multinational firms, etc. These materials are usually covered with great length in theory-oriented courses or books like those by Pol Antràs from Harvard, and Costas Arkolakis from Yale. We also will not dive deep into the algebraic and technical details in deriving a particular estimation equation, or constructing a particular estimator that has some desirable statistical properties. These topics are usually covered in details in courses or books on advanced econometrics, or computational methods. Instead, the topics that we will spend most of our efforts on, are why a particular research tool is selected, and how the results from different methods should be interpreted. Put differently, how to think in a scientific way, if you are to answer a research question that is empirical or quantitative in its nature. This choice of topics comes from both the successful and unsuccessful experience by myself and those from my classmates and colleagues in preparing ourselves to be researchers. Many students implement their own research through mimicking existing literature, but either fail to demonstrate their understanding of the correct way to interpret their empirical or quantitative results, or fail to explain how their results support the hypothesis they proposed. Given the technical sophistication of modern empirical and quantitative methods and the efforts that are required to master the use of them, many students find themselves lost in the technical details and generate numbers that they forget why these computations are needed in the first place. This class attempts to build in your mind the habit of always being aware of what your research question is, why a particular research tool is chosen, and how the results should be interpreted. Traditionally, empirical studies in international trade are dominated by reduced-form econometric analysis, which has the virtues of both the ease to implement, and the transparency in the identification of causal relationships. It, however, suffers from the famous Lucas critique, which in the wording of Chetty (2009), criticizes the reduced-form approach for estimating statistics that are not policy-invariant parameters of e- conomic models and therefore have limited relevance for policy and welfare analysis. Similar to the trend in other branches of economics, in the field of international trade, structural models are more frequently brought to confront data directly nowadays. Structural approach addresses the Lucas critique by fully speciiii

4 fy the complete models, and therefore an estimation of the full set of policy-invariant parameters (also called the deep parameters) enables researchers to use these models for simulating various counterfactual experiments. The costs to do this, are higher degree of sophistication of model specifications and estimations techniques, and vague identification of some primitive parameters. In fact, the hostility between structural and reduced-form advocates is among the fiercest of all those in economics. We will take an open view on these two methodologies in this class. Even the most ardent advocates of either method will not claim that one method is superior to the other in every aspect. My personal view is that there is no good or bad about a research method, what matters is whether the method chosen is appropriate for the research question that one attempts to answer. As a result, we will study together just as many academic articles which employ the reduced-form approach as those that adopt the structural approach. Research papers that use the sufficient statistics approach, which is a hybrid of the previous two approaches, will also be surveyed. We will discuss why a certain research method is chosen, and what are the benefits and costs when comparing against the others. In most cases, the answer is open, and it is up to you as a researcher, to make your own decision on which approach to pursue in your own studies. The purpose of the discussion in this course, is to deepen your understandings of different methods such that when the time of decision comes, you are choosing your own research strategies and not merely following the others blindly. The materials are organized according to their research topics. A more detailed introduction of these topics will be provided in Chapter 1, and therefore is omitted here. Such an organization is chosen because I think it best serves the purpose of this course that papers addressing related questions using different approaches be discussed together and compared directly. Another benefit from such an organization is that the course can also be viewed as a second year topic class in international trade. Students with a good understanding of the research methodologies can take the discussions as introductions to the current research frontier of international trade, and therefore formulate his or her own research questions. iv

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7 CONTENTS Preface iii Preface (Chinese) v Contents vii List of Figures ix List of Tables xi 1 Introduction 1 2 Trade and Heterogeneous Firms 8 1 Monopolistic Competition and Heterogeneous Firms Comparative Advantage and Heterogeneous Firms Market Penetration Costs Entry, Exit and Firm Dynamics Trade and Productivity 53 1 An Anatomy of International Trade Trade Liberalization and Firm-level Productivity Trade and Geography 78 1 Technology, Geography and Trade The Impact of Transportation Infrastructure The Topography of a Spatial Economy vii

8 5 Market Frictions 95 1 Misallocation and TFP Growing like China Credit Constraints and International Trade Bibliography 112 viii

9 LIST OF FIGURES 2.1 The Equilibrium Exit Threshold ϕ The Impact of Trade Trade Liberalization and Comparative Advantage Entry and Market Size Sales Distribution of Exporting Firms Pareto Distribution Sales of Different Percentiles across Markets: Data and Endogenous Model Sales of Different Percentiles across Markets: Data and Exogenous Model Average Sales Distribution Changes of Trading Volumes after Trade Liberalization Trade Liberalization and the Margins of Trade The Exit Decision Entry and Sales by Market Size Domestic Activity of Exporting Firms Domestic Activity of Exporting Firms Model Fit The Fréchet Distribution The Evolution of India s Railroad Network: Distribution of TFPQ Distribution of TFPR Distribution of Plant Size ix

10 5.4 Foreign Reserves and Domestic Deposits/Loans Difference Domestic Private Firms versus State-owned Firms SOE Employment Share across Industries Employment Share of DPE x

11 LIST OF TABLES 3.1 Unconditional Probability of Exporting to Top Seven Most Popular Destinations Number of Firms that Follow the Exporting Hierarchy Estimation Results xi

12 CHAPTER 1 INTRODUCTION THE ACCESSIBILITY OF MICRO-LEVEL DATA and the advancement of computational capability have dramatically reshaped the paradigm of modern economic research. That decreasing is the number of pure theoretical researches that are based on some anecdotal evidence or several stylized facts at the macro level, while on the contrary, that increasing is the quantity of empirical and quantitative researches that are based on solid statistical evidence. This trend has, of course, affected the methods that are adopted in the frontier researches. In the area of international trade, the traditional reduced-form econometric analysis is equipped with more and more sophisticated identification strategy in order to provide clear identification of key causal relationships under situations of various data availability condition. Simulation-based estimation techniques which are able to recover the structural parameters of a model as whole are taking an increasing share of modern empirical studies. General equilibrium analysis which usually adopts the calibration strategy is also demonstrating its importance gradually in the trade community. The toolbox portfolio of college mathematics plus classical micro-econometric techniques is neither enough for researchers to understand the cutting-edge studies, nor to answer research questions using the most appropriate approach. This set of notes is designed to introduce the empirical and quantitative methods used in modern international trade studies to students and researchers who are planning to launch their own researches in this area. Confronting Theory with Data. Early in your undergraduate introductory class, you should have learned that research in economics is associated with two different styles of reasoning: inductive reasoning which moves from specific observations to theories that could be generalized, and deductive reasoning which builds theories on top of several assumptions and applies the prediction of the theory to observations that the theory is deemed to explain. In most modern economic research, these two types of reasonings are used together. The following equation shows a stylized flowchart of a typical economic research: 1. Empirical or Stylized Facts 2. Assumptions 3. Theory 4. Predictions 5. Application You probably recall also from your introductory class that studies that involve step 1 through step 4 are called positive economics, and those that include also step 5 are named normative economics. In modern economic research, data are heavily involved in almost every single step of the above procedure. For instance, the facts that motivate the assumptions have to be empirically credible, the main mechanism of the theory has to be supported by evidence, and the facts that the theory is designed to explain must be empirically refined such that the definitions of the variables in the data are in accord with those in the model, etc. Except in some 1

13 CHAPTER 1. INTRODUCTION particular fields, theoretical papers like those in the 70s or 80s which built a model that delivers predictions that are qualitatively consistent with some stylized facts are increasingly rare in leading academic journals of mainstream economics nowadays. Researchers have realized that everything can be explained by models. From a scientific point of view, simply write a model that matches some observations qualitatively contributes no more than telling a bold story using Greek letters as opposed to using natural language. Therefore, the contribution of a modern research, is largely determined by to what extent, the components of a theory can be credibly supported by empirical evidence. The Lucas Critique. So how exactly are these empirical tests working, and how are different methods differing from each other? Let us illustrate the basic intuitions with a toy example. Suppose you want to examine the effect of a 10% increase in the import of a particular product on the domestic price of that product in a paper. The research question is rather straight forward here, how will the domestic price respond to a 10% increase in import, if we just rephrase it. The question is a positive one, but obviously normative questions like whether a tariff reduction of 10% should be implemented in the current economic downturn, or how much should the government set the tariff to be can follow up naturally. 1 Now, how do we answer this problem? First, this is a research question of quantitative nature. Therefore, a theoretical model with qualitative prediction obviously is not going to work here. If you have an empirical mind, you will probably dive into the history to pull out a series of data containing the time series of domestic product price p it and import volumes x it, where i is the country/region subscript and t is that for time. Assume that the estimation equation is: log p it = β 0 + β 1 log x it + X t γ 1 + X i γ 2 + ε it, where X t is set of time dummies, and X i is that of country dummies. Let us say that you have estimated an elasticity of β 1 = 0.2, meaning that a 1% increase in import volumes is associated with a 0.2% increase in domestic product price. Now, does this mean that if we lift the import quota restriction of some product by 10%, the domestic price will increase by 2% and therefore hurt consumers welfare? Not necessarily. It could very well the case that the domestic demand for that product is so large, that the domestic production can only meet a small proportion of the demand, and as a result, excess demand drives up the domestic market price, which in turn invites more import from foreign producers. The pure effect of the increase in import could on the contrary be the decrease in the market price. Put differently, in absence of the increase in import, the price may have gone up by 0.5%, instead of 0.2%. When asked about whether a lift of 10% import quota will increase consumers welfare of the domestic population, a researcher is likely to give misleading policy suggestions based on the estimates of β 1. This is a classical example of the Lucas (1976) critique. The reason that policy suggestions based on the estimates of β 1 is misleading is because it summarizes the total effect, as opposed to showing separately the individual effect of the contributing channels. The information that is used in the estimation only reflects the change in the equilibrium points, hence it should not be surprising that the results are not informative in the circumstances of evaluating the effect of a quota lift, where the demand side channel has to be isolated. To address this problem, additional information about how the supply responds to price changes has to be introduced into the solutions in some particular ways. 1 See for example, Eaton et al. (2015). 2

14 CHAPTER 1. INTRODUCTION If you are an applied microeconometrician, you probably have already realized that the above example is the classical example used to introduce the simultaneous equation models. The solution provided by microeconometricians, is of course as you expected, the instrumental variables. If we can find a subset of the variations of x it, where we can be assured that those variations of x it is not caused by those of p it, then the estimation is informative about the effect of policies that exogenously change the supply. The variables that we use to select this subset is then called the instrumental variable. In the extreme case where the instrumental variable is discrete, we get the special case of treatment effect estimation. Notice that in the instrumental variable estimation, we are introducing some new information (when the supply is exogenous), but we are also discarding some existing information (variations that are caused by price variations). As a benefit, a particular channel is isolated which validates the policy suggestions given, but the precision of the estimates deteriorates as a cost, since less numbers of samples are used to provide the inference. This, however, is not the only way that information about the demand curve can be introduced. Recall that by introducing instrumental variables, only those variations that are not associated with price variations are used. If the researcher is comfortable in specifying a complete structure for how the foreign exporters react to price changes, then all the information could be used in the estimation. The estimates from such a structural estimation is also an eligible candidate for the inference of the possible policy effects. For example, the researcher may have confidence that the supply function is characterized by log x it = log p it, then by plugging the above equation into the original estimation equation, we get log p it = (β β 1 ) β 1 log p it + X t γ 1 + X i γ 2 + ε it, which gives us the estimates of β 1. Mathematically, with no information on the price elasticity of supply, if we assume that the supply function takes the form of log x it = A + B log p it, then it can be seen from the resulting estimation equation that β 1 cannot be estimated. log p it = (β 0 + β 1 A) + β 1 B log p it + X t γ 1 + X i γ 2 + ε it, From the above discussion, we see that when responding to the Lucas critique, both the reduced-form and the structural estimation require the inclusion of additional information in the estimation. The disagreements are about what assumptions one is comfortable with, and how these assumptions should be incorporated into the estimation procedure. It is important to bear in mind that essentially, an estimation is a systematically way to extract useful information from the data. No new information will be created out of nowhere, simply because of a different estimation strategy is employed. Different models impose different structures to the way the data are presented, and therefore permit different views of how the data could be interpreted in 3

15 CHAPTER 1. INTRODUCTION an economically meaningful way. Using the previous example, IV estimation that selects the variations of imports that are not caused by those of the prices, implicit imposes the structure that B = 0, into the estimation. Going back to the flowchart in the last section, in order that the estimation results are credible, in this toy example, evidence on either the exogeneity of the supply variation (in the reducedform case), or the specific functional form supplied (the structural case) has to be provided. Later in this course we will see that for more sophisticated studies, tests on the mechanisms of the theory sometimes are also employed to support the hypothesis. In this toy example, the connection between the two estimation methods are easy to see. However, as we progress later in our study, the analysis will become more and more complicated. This of course, is why we will devote a large amount of efforts to this topic. To summarize, it is important that one realizes that reduced-form and structural methods are not mutually exclusive. They all make additional assumptions, whether explicitly or implicitly, and they are presenting essentially the same piece of information, with the difference being how the information is organized. The Organization of the Materials. Ever since the age of Adam Smith, one of the central issue in economics is whether international trade is beneficial to the welfare of domestic consumers, and if yes through which channel. Adam Smith (1776) proposed the theory of absolute advantage, and David Ricardo (1821) developed it further in the Principles of Political Economy and Taxation, which leads to the theory of comparative advantage. Both the absolute and comparative advantage theories still serve as the major driving force of intra- and international trade. Their modern descendants are equipped with more complicated treatment of micro structure of the economy, which enables the models to be used in explaining issues like how trade affects the behavior of individual firms, how resources are reallocated across firms, sectors, geographic regions, and what are the welfare implications of these factors, etc. We will start with the discussion of how trade affects the individual firms, and how responses from individual firms affect the evaluation of the gains from trade. We then introduce the analytical frameworks that are suitable to study resource allocations across geographic areas, and show how they can be confronted with data. Having exposed to a number of empirical studies, we will divert slightly to an equally important issue in connecting model directly with data, that is what are the correct data counterpart of model variables. The discussion emphasizes the importance that one be aware of the what the data actually measure, and how this will change the conclusions of one s research. We then move on to discuss several extensions of the basic models, namely those with multi-product firms, market frictions of various types, and realistic geographic structure. More specifically, in Chapter 2, we introduce the workhorse model that will be used throughout the whole semester, namely the Melitz (2003) model. The Melitz model is a heterogenous firms model with monopolistic competition market, where trade improves welfare by inducing resource re-allocation from less productive firms to firms with dominating productivity. The model builds upon the industry dynamics model by Hopenhayn (1992), and provides a general analytical framework that could be used to investigate various issues where resource allocation across different firms assumes an important role. This permits us to either quantify the structural model itself, or to deliver testable predictions that could be examined with reducedform econometric analysis. To show the potential of the Melitz model as a modifiable analytical tool, we 4

16 CHAPTER 1. INTRODUCTION will study the paper by Bernard, Redding and Schott (2007) which shows how comparative advantage and inter-sectoral resource allocation could be investigated with the Melitz model, and the paper by Arkolakis (2010) which demonstrates another example of how small changes in the modeling assumption lead to large improvement in the explanatory power of the model. The paper by Arkolakis (2010) adopts a calibration strategy, which is the dominating quantitative tool in modern macroeconomics. This probably is the first time that many readers encounter a calibration study. If time permits, we will also briefly summarize some basic stylized facts of the exporting firms in China [Manova and Zhang (2009)] and in the U.S [Bernard, Redding and Schott (2009)]. In the Melitz style model, the productivity of a firm is key to understand the predictions of the models. In Chapter 3, we will go over three papers where firm level productivity assumes a leading role. Two of the papers, Bernard and Jensen (1999) and Pavcnik (2002) are implemented with reduced-form technique, while the other one, Eaton, Kortum and Kramarz (2011) estimates the structural model. Using reduced-form analysis, Bernard and Jensen (1999) examines the causal relationship between firm s exporting decision and its productivity, that is, whether the statistical relationship between more productive firms and larger exporting volumes is because more productive firms choose to export or because exporting helps improving the productivity of a firm. Empirical evidence shows supports the hypothesis that productive firms choose to export, therefore validates the core mechanism of Melitz (2003). Using a different econometric specification, however, Pavcnik (2002) shows that trade liberalization also significantly improves the productivity of exporting firms. We will then investigate together possible reasons that these two papers draw different conclusions. Eaton, Kortum and Kramarz (2011) estimate a variation of the Melitz model to examine the relative contribution of several factors to some observed stylized facts. The estimation method the authors used is the Methods of Simulated Moments [Pakes and Pollard (1989)], which is also widely used in macroeconomics and labor economics. In Chapter 4, we will examine one particular channel that trade affects the productivity of the firm the technology upgrading and investigates subsequently its welfare implications. In particular, two papers will be covered in this chapter: Bustos (2011) and Atkeson and Burstein (2010). Using reduced-form analysis, Bustos (2011) studies the impact of the regional trade agreement the MERCOSUR (from the Spanish: Mercado Común del Sur, which means Southern Common Market) on technology upgrading by Argentinean firms. She found evidence which shows that trade liberalization boosts the pace of technology upgrading by exporting firms. It may seem natural to infer that technology upgrading facilitates as a channel through which trade liberalization could improve welfare. However, as is shown by Atkeson and Burstein (2010), such an inference must be done with caution. This paper demonstrates the importance of general equilibrium feedback in delivering the welfare implication of a model. It also is the second time that we encounter a calibrated general equilibrium quantitative model. Thus we will take this chance to explain in details the connection between calibration and estimation, which I find a significant number of students find confusing. The discussion on the welfare implications of trade liberalization continues in Chapter 5. We start with two empirical studies, Broda and Weinstein (2006) and Khandelwal, Schott and Wei (2013). We then will cover in great details a very influential theoretical work by Arkolakis, Costinot and Rodríguez-Clare (2012), which sets up a framework where the sufficient statistics approach could be applied. 5

17 CHAPTER 1. INTRODUCTION Research articles covered in previous chapters rely heavily on data at the micro-level, in Chapter 6, we will turn to those studies that are implemented with data at the aggregate level. This chapter also introduces the workhorse model of economic geography. We begin with a detailed coverage of the Eaton and Kortum (2002) model. The Eaton-Kortum model provides a tractable way that explicit geographic factors could be incorporated into traditional general equilibrium analysis, and is demonstrating increasing influence in recent cutting-edge studies. Later in this chapter, we first go over two applied studies where highly stylized geographic structures are modeled, Michaels, Rauch and Redding (2012) and Holmes and Stevens (2014) to demonstrate the empirical and quantitative potential of the Eaton-Kortum model. We will continue the discussion of recent studies where complicated geographic structures are explicitly modeled in Chapter 11. We then briefly go over Helpman, Melitz and Rubinstein (2008), which demonstrates how a Melitz model could be used in estimation in situations where only aggregate data are available. Having seen a number of quantitative and empirical studies in international trade, in Chapter 7 we will take our discussion to an important, yet often ignored issue, that is what are the trade data measuring, and what are their model counterparts? Recent studies in development and macroeconomics showed that different interpretation of the model counterparts of data variables can result in very different theoretical explanations for some observed phenomena [Herrendorf, Rogerson and Valentinyi (2013), Herrendorf and Schoellman (2014 forthcoming)]. We show in this chapter that the similar issue also exists in international trade. In particular, we will go over two papers that works on the measurement of trade data, those by Koopman, Wang and Wei (2014) and Johnson and Noguera (2012). In Chapters 8 through 11, we will introduce several extensions to the two workhorse models the Melitz model and the Eaton-Kortum model in recent literature. We begin with the discussion of the interaction of various market frictions with trade in Chapter 8. We first present an in-depth discussion of an influential paper on how market frictions could be measured by Hsieh and Klenow (2009). Hsieh and Klenow (2009) proposed an indirect measure of market frictions by contrasting the distribution of average factor productivity with the theoretical implications from a distortion-free neoclassical model with a Melitz structure. This method works essentially as measuring the residuals in econometrics or growth accounting. It has the advantage that it is less demanding on the data, however, it also has the disadvantage that explicit policy suggestions are very hard to come by. Although difficult, there are some rare cases where the effect of one particular source of market distortions could be measured. We will go over three studies in this chapter, one quantitative by Song, Storesletten and Zilibotti (2011), and two empirical by Manova (2013) and Manova and Yu (2014). We will also study the channel that goes the other way round, that is how the exposure to trade will affect the domestic market frictions. We have already seen one empirical paper in Chapter 5 [Khandelwal, Schott and Wei (2013)], and in this chapter we will study another quantitative one by Edmond, Midrigan and Xu (2015 forthcoming). In Chapter 9, we will turn to another issue on how firms that produce multiple products react to trade [Bernard, Redding and Schott (2010), Mayer, Melitz and Ottaviano (2014)], and what are the subsequent environmental implications [Barrows and Ollivier (2014), Shapiro and Walker (2015)]. Chapter 10 will survey the recent development of modeling sophisticated geographic structure in trade models. More specifically, we will refer heavily on the extensions based on the Eaton-Kortum model. We will start with the classical contribution by Krugman (1991), which illustrates the basic economic intuition behind the theory of 6

18 CHAPTER 1. INTRODUCTION economic geography. We then study Allen and Arkolakis (2014) together, which sets up a general analytical framework on how geography can be incorporated into traditional economic research. In particular, the tools they draw heavily upon are the path integral, and the resulting system of integral equations. Finally, we will see how such a framework can be used empirically, with the aid of Donaldson (2015 forthcoming), which studies the effects of Indian railroad network on various development issues in India. If time permits, we will survey several papers addressing the problem of foreign direct investment in Chapter 11 [Brainard (1997), Alfaro and Charlton (2009)]. Chapter 12 concludes. 7

19 CHAPTER 2 TRADE AND HETEROGENEOUS FIRMS In this chapter, we introduce the workhorse model of trade with heterogeneous firms. We start with Melitz (2003) in Section 1. We then go over two studies that extend the basic Melitz model Bernard, Redding and Schott (2007) and Arkolakis (2010) in Section 2 and 3. For readers who would like to understand the Melitz model from a more general perspective, an introduction to the industry dynamics model by Hopenhayn (1992) is provided as appendix in Section 4. 1 MONOPOLISTIC COMPETITION AND HETEROGENEOUS FIRMS For centuries, economists and policy makers have associated international trade with economic growth. Theoretical investigation could go back as far as Adam Smith (1776) and David Ricardo (1821). Channels like comparative advantage, resource complementarity (the Heckscher-Ohlin model), Foreign Direct Investment (FDI), technology spill-over have all been extensively studied. Empirical studies at the micro-level showed that exporting firms usually have higher productivity and exposure to trade often affects different firms d- ifferently. So, to answer the question of which firm exports and why, or how will trade affect the resource allocation across heterogeneous firms, the Hopenhayn (1992) model is a natural model to build upon. More specifically, we will study the most cited paper regarding these two questions so far, the Melitz (2003). 1 Melitz (2003) studies the effect of international trade on resource allocation using a general equilibrium heterogeneous firms model. First, it shows that the fixed cost associated with the access to foreign market selects the most productive firms into the export market. Second, it shows that general equilibrium feedback of increased wage in domestic factor market raises the cost of production and forces the least productive firms to quit. At an aggregate level, these two channels lead to allocation of resource favoring more productive firms, which increases the aggregate productivity and consequently contributes to welfare gain of consumers. The general idea of the paper goes as follows. The main structure resembles mostly that of the Hopenhayn s model. Firms differ in their productivity levels. Access to foreign markets requires a fixed cost f x. As a result, more productive firms can reap the increasing returns to scale and choose to export to foreign markets. Exports increase the total output and thus the total factor demand by these productive firms, which in a general equilibrium framework combined with immobile production factor leads to an increase in factor 1 As of January 2015, Melitz s paper has been cited by 7452 published papers, more than 500 times per year! 8

20 1. MONOPOLISTIC COMPETITION AND HETEROGENEOUS FIRMS prices. The rising costs thus drive the least productive firms out of the market. Similar to Jovanovic (1982) and Hopenhayn (1992), selection increases aggregate productivity. A major difference between the Melitz model and the Hopenhayn is the market structure adopted. While a perfect competitive market is assumed in the Hopenhayn model, in the Melitz model, a monopolistic competition structure is adopted. Monopolistic competition is a feature pertains to observed pattern of international trade and it has little to do with the above mechanisms. In fact, one can show that with a Lucas (1978) model, all the analysis relates to the selection of firms carry through. Regarding the selection mechanism, monopolistic competition merely is one way to introduce decreasing returns to scale to firms profit function, which is assumed directly in the Lucas model. The reason monopolistic competition is adopted is that large volume of trade within the same industry between countries with similar factor endowment and technology cannot be accounted for using a model with perfect competitive market. Models with monopolistic competition, however, as is shown in Krugman (1979, 1980) provides a useful analytical framework. For students who find the abstract analysis in Jovanovic (1982) and Hopenhayn (1992) challenging, Melitz (2003) should be much easier to follow. All assumptions in the model are made with specific functional form, and hence the analytical tractability keeps the model mechanism transparent. The rest of this section is organized as follows. We start with the setup of the model and the decision problem of individual firms in the first two sections. We then discuss the closed economy equilibrium in section III. The open economy equilibrium, and hence the effect of international trade, is analyzed in section IV. 1.1 The Model In this section, we introduce the building blocks of the model. We start with the assumptions on consumer s preference and firm s technology. We then show that the average productivity level is a sufficient statistic to track the evolution of various variables at the aggregate level. Preference. One difference between the current model and those we have seen so far, is that the consumer s problem is explicitly modeled. In particular, assume that the preference of a representative consumer is given by a CES utility function over a continuum of goods indexed by ω: [ U = q(ω) ρ dω ω Ω where the measure of the set Ω represents the mass of available goods. Assume that these goods are substitutes, which means the elasticity of substitution between any two goods equals σ = 1/(1 ρ) > 1. The problem of the consumer is essentially how to spread the expenditure across different ω Ω in the most efficient way, which requires the marginal utility generated from a dollar spent on each ω is equalized. If we let the price of the individual good be p(ω), and that of the aggregate bundle Q = U be ] 1/ρ [ ] 1 P = p(ω) 1 σ 1 σ dω, (2.1) ω Ω 9

21 CHAPTER 2. THE WORKHORSE MODELS then the optimal consumption and expenditure decision for an individual variety are [ ] p(ω) σ q(ω) = Q P [ ] p(ω) 1 σ r(ω) = R, (2.2) P where R = P Q = ω Ω r(ω)dω represents the aggregate expenditure. We see clearly from Equations (2.2) that the quantity purchased and expenditure spent on each ω is proportional to the total quantity and total expenditure. Notice that the elasticity of revenue r(ω) to normalized price (p(ω)/p ) is 1 σ, which is smaller in absolute value than the quantity-price elasticity σ. The reason is that price influences quantity only through demand, but itself can affects the expenditure directly in an opposite direction, which reduces the total effect. The problem we see here involves only the differentiated goods. Sometimes in the literature, you will see economies with a numeraire good ω 0 and a set of differentiated goods Ω (for example Grossman and Helpman (1994)). Under such scenario, the way to solve the consumer s problem is similar. Now a two-step problem is solved. In the first step, the consumer allocate budget between the numeraire good and an aggregate good Q. In the second step, the consumer allocate budget across the differentiated goods optimally. In the current framework, step one is trivial and only step two is included. See Dixit and Stiglitz (1977) for a general discussion. Production. There is a continuum of firms, with each firm produces a different variety ω. Further, each variety is only produced by one firm. Production requires only one factor, which we assume to be labor here. Labor is inelastically supplied and the total endowment across the country is L. The production technology is characterized by a cost function c(q) = w l = w(f + q ϕ ), where w is the prevailing wage, f is the overhead cost, ϕ is the firm-specific technology efficiency, and l = f + q/ϕ is the labor demand function. In the context of this model, higher ϕ leads to lower production cost, and hence is interpreted as higher productivity. Each firm faces a residual demand curve with constant elasticity σ regardless of their productivity, therefore monopoly yields the standard markup pricing rule with σ/(σ 1) = 1/ρ being the cost markup ratio: p(ϕ) = w ρϕ (2.3) where w is the prevailing domestic wage and are normalized to one hereafter. Notice that this assumption is equivalent to the existence of a numeraire good in other settings. With some algebra, firm profit can be shown to be π(ϕ) = r(ϕ) l(ϕ) = r(ϕ) σ f where r(ϕ) is the revenue and r(ϕ)/σ is the variable profit. Substitute in (2.2), we have r(ϕ) = R(P ρϕ) σ 1 (2.4) π(ϕ) = R σ (P ρϕ)σ 1 f (2.5) 10

22 1. MONOPOLISTIC COMPETITION AND HETEROGENEOUS FIRMS which further leads to q(ϕ 1 ) q(ϕ 2 ) = ( ϕ1 ϕ 2 ) σ, r(ϕ 1 ) r(ϕ 2 ) = ( ϕ1 ϕ 2 ) σ 1. (2.6) So in summary, a more productive firm (higher ϕ) produces more (higher q), charges lower price, and earn higher profits. Aggregation. Suppose that the distribution of firms over productivity levels in equilibrium is given by µ(ϕ), and the measure of µ(ϕ) is M. Given the one on one relationship between firms and goods, the total number of goods available to consumers is also M. Apply the Law of Large Number, the realized aggregate price P in (2.1) is [ P = 0 ] 1 p(ϕ) 1 σ 1 σ Mµ(dϕ) Substitute in (2.3), this can be written as P = M 1/(1 σ) p( ϕ), where [ ϕ = 0 ] 1 ϕ σ 1 1 σ µ(dϕ) (2.7) So ϕ is the weighted average productivity level, which also serves as a measure of aggregate productivity. Similarly, we can show that P = M 1 1 σ p( ϕ), Q = M 1/ρ q( ϕ), R = P Q = Mr( ϕ) Π = Mπ( ϕ) where R = 0 r(ϕ)mµ(dϕ) and Π = 0 π(ϕ)mµ(dϕ) are aggregate revenue and profit. A couple of discussions follow. First, under the current structure, an industry comprised of a distribution µ(ϕ) of firms with measure M, at the aggregate level looks just the same as an alternative industry with M representative firms, each shares a same productivity ϕ. As we shall see shortly, this is the key to the tractability of the model. Second, P and Q are not linear with respect to M, which follows from their definitions of aggregation. Third, the aggregate revenue, profit and production are all increasing in M, while when the goods are substitutes to each other, aggregate price P is decreasing in M. The reason is that more variety allows consumers to better optimize the composition of their consumption bundle. 1.2 Entry and Exit Given the pricing and producing decisions of firms with productivity ϕ, let us now turn to the discussion of entry and exit decision of firms. This will pin down the distribution of µ(ϕ) in equilibrium. Entry is the same as in the Hopenhayn model. That is, a continuum of ex ante identical prospective entrants pays a fixed entry cost f e > 0 to make a draw of initial productivity from a common distribution g(ϕ), with corresponding cumulative distribution G(ϕ). The productivity stays constant once is drawn. We assume that there is a positive probability δ which forces a firm to exit exogenously. Comparing to Hopenhayn (1992), the dynamics of productivity is highly simplified. Notice that the assumption of recurrence is met here. The benefit that such simplification buys is that the equilibrium is fully characterized by g(ϕ) and δ, which 11

23 CHAPTER 2. THE WORKHORSE MODELS combines with the results of aggregation in the last section leads to tractability. It is also in this sense that the Melitz model is usually considered as a static model. Though the shape of µ(ϕ) is determined exogenously, the selection threshold and hence the aggregate productivity is nevertheless determined endogenously. We consider also the stationary equilibrium. Since the productivity level of a firm is constant once drawn, its optimal per period profit π(ϕ) also remains still. Assume without loss of generality that there is no discounting, the expected value of firm is given by v(ϕ) = max { 0, } (1 δ) t π(ϕ) t=0 = max {0, 1δ } π(ϕ). Therefore as before, we can define the threshold productivity of production ϕ = inf{ϕ : v(ϕ) > 0}. Now, since π(0) = f < 0, continuity of π( ) means π(ϕ ) = 0. This condition, which says that the marginal entrant makes zero per period profit, is later referred to as the zero cutoff profit (ZCP) condition. Firms with productivity ϕ < ϕ will leave the market immediately. Since δ is exogenously determined and is independent of ϕ, so long as the Law of Large Number holds, the exiting process will not affect the shape of µ(ϕ). Therefore, the equilibrium distribution µ(ϕ) is simply the initial distribution g(ϕ) conditional on survival ϕ > ϕ. Formally, µ(ϕ) = { g(ϕ) 1 G(ϕ ) if ϕ ϕ 0 otherwise We will refer to p in = 1 G(ϕ ) as the ex ante probability of successful entry later in this section. The aggregate productivity ϕ (as a function of ϕ ) is thus given by (2.8) [ ϕ(ϕ 1 ) = 1 G(ϕ ) ϕ ] 1 ϕ σ 1 σ 1 g(dϕ) (2.9) Zero Cutoff Profit Condition. Now let us examine the ZCP in greater details. Notice the ZCP condition characterizes the exit decision of firms. Since the ratio of revenues of two firms depends only on the ratio of their productivity, we have Substitute in the definition of π(ϕ), this yields r( ϕ) r(ϕ ) = [ ϕ(ϕ ) ϕ ] σ 1 π( ϕ) = r( ϕ) σ f = [ ϕ(ϕ ] ) σ 1 r(ϕ ) ϕ f σ Recall that we have just shown π(ϕ ) = 0, therefore r(ϕ ) = σf, and the average profit per firm π is given by where k(ϕ ) = [ ϕ(ϕ )/ϕ ] σ 1 1. π = fk(ϕ ) (2.10) 12

24 1. MONOPOLISTIC COMPETITION AND HETEROGENEOUS FIRMS Free Entry Condition. The entry decision of firms is characterized by the usual free entry (FE) condition. In particular, the expected value of entry must equal to the entry cost f e in equilibrium. Let v = t=0 (1 δ) t π be the expected value of entry, then free entry condition implies p in v = 1 G(ϕ ) π = f e. (2.11) δ 1.3 The Closed Economy Equilibrium With the entry and exit decisions of firms properly defined, we can characterize analytically the stationary competitive equilibrium. For such purpose, we need to solve for: the threshold ϕ, the mass of entrants M e, and the mass of incumbent firms M. Solving for ϕ. The ZCP and free entry condition represent two different relationships between the average industry profits π (which is equivalent to ex ante expected profit) and the exit threshold ϕ : π = fk(ϕ ) π = δf e 1 G(ϕ ) (ZCP), (FE) (2.12) This is a two equations two variables nonlinear system in the (π, ϕ ) space. The intersection of the two curves gives the equilibrium level of ϕ. To see this, notice that ZCP reflects a decreasing relationship between π and ϕ, while FE captures an increasing one. The economic intuition goes as follows. The average industry profit π s along the ZCP are such that the productivity levels of the corresponding marginal firms are ϕ s. A higher ϕ means that even relatively productive firms can barely break-even, which suggests that the profitability of the industry cannot be too optimistic. Thus a negative relationship follows. On the other hand, the π s along the FE are those such that when the exit thresholds are ϕ, the ex ante expected value of entry equals f e. In this circumstance, a higher ϕ refers to a higher failure rate. In order that conditional on a high failure rate, prospective entrants still expect to earn positive profits, the profitability of the industry must be quite optimistic. Therefore, the relationship between π and ϕ is positive. Figure 2.1 demonstrates this graphically. Solving for M e and M. First notice that by the definition of stationary equilibrium, the mass of entry equals that of the exit: p in M e = δm (2.13) With ϕ given, we only need to solve for one of the two measures, and the other follows immediately from the equation above. Let us work on the total mass of active firms M. The total labor force L is divided into production (L p ) and entry (L e ), which writes L = L p + L e. The accounting identity of revenue, cost and profit gives L p = R Π. It is straightforward that L e = M e f e. Substitute in (2.13), we have L e = M e f e = δm p in f e. 13

25 CHAPTER 2. THE WORKHORSE MODELS FIGURE 2.1. THE EQUILIBRIUM EXIT THRESHOLD ϕ Substitute in (2.11), the above equation becomes L e = Mπ = Π. Therefore, R = L p + Π = L p + L e = L. Since in the previous section we have shown that π = r/σ f and R = Mr, where r is the average revenue of across the industry, the mass of active firms M can be solved from M = R r = L σ(π + f). (2.14) All other prices and allocations can be constructed from ϕ and M, hence the characterization of the stationary equilibrium is complete. 1.4 The Open Economy Equilibrium Now let us extend the model to include international trade. We start with the basic setup, i.e., how do we model the foreign market. We then discuss the entry, exit and export decision of firms when there exists a foreign market. After all these, we solve the equilibrium prices and allocations under the new setup. The general logic is pretty much the same as in Section II and III. As a result, we will only highlight where it is different from the closed economy case. Setup. Assume that there are n countries. The domestic economy is the same as in the closed economy case. To access the export market, firms need to pay another fixed cost f ex > 0, and the usual unit-level iceberg cost τ 1 > 0 is also required. To keep the tractability of the model, we further assume that all countries are identical, and as a result the equilibrium will be a symmetric one. This assumption is not innocuous for the implication of the model at the aggregate level, i.e., the trading pattern between different 14