Dina I Waked

Antitrust Enforcement in Developing Countries: Reasons for Enforcement & Non-Enforcement using Resource-Based Evidence

Keywords: Information. These countries are: Barbados, Croatia, Czech Republic, Estonia, Hungary, Saudi

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I. INTRODUCTION

Developing countries have been increasingly adopting specialized laws aimed at regulating competitive behaviour and prohibiting anticompetitive activities. By 2007, out of the world's 151 developing countries 1 , 77 had an antirust law in force and an agency set up to enforce the adopted law, a surge from less than 10 before 1990. 2 The trend to adopt these laws in the developing world has been a phenomenon of the 1990s, where the number of countries adopting antitrust laws post 1990 is astonishing, compared to the decades before (Figure 1). (Table 1 lists all developing countries with a competition law). This raises the question of why did these countries adopt them in the 1990s and not before? And also, why did so many of them suddenly become interested in competition law adoption? There is no simple answer to either, except to say that many countries had provisions either in their penal codes, civil codes or commercial legislations dealing with competition law issues before formally adopting a legislation that is solely concerned with competition matters. 3 This made them less interested in adopting particular laws seeing that they had general provisions in other legislations dealing with the same issues. With regards to the second question, why so many became suddenly interested in these kind of laws in the 1990s, it is simplistic to argue, yet quite true, that many countries were entering trade agreements in the 1990s that made the adoption of competition law a prerequisite to the implementation of the trade deals. 4 These treaties were either trade agreements creating free trade zones, or were part of structural programs that intended to open up the developing world economies and facilitated the entry of foreign entities that considered a competition law a necessity and guarantee for their work abroad, in particularly in a developing country. 5 More generally, the 1990s are considered the era where developing world countries started to put an end to their former protectionist policies that were either inspired by communist or socialist regimes or simply by efforts to industrialize and strengthen national champions and local producers. The 1990s introduced the new era of international trade, encouraging foreign direct investment, and membership in regional trade agreements or the World Trade Organization (WTO). With the emergence of many of these structural changes, open door policies and participation in world trade relations, competition laws were suddenly prescribed as necessities to facilitate much of the impending changes. It is important to understand the role played by the WTO in encouraging and often requiring new members to adopt these laws in order to understand the surge in the developing world. 6 Similarly, the role played by the EU in encouraging new members and trade partners to adopt competition law is even more Arabia, and Slovak Republic (where the Czech Republic and the Slovak Republic were considered developing in the 2008 IMF Survey and are no longer so in the 2009 survey). 3 See for example Egypt's Penal Code of 1937 in Article 345 prohibits raising or lowering prices to achieve illegal benefits. Also, Law 241 for 1959 states that it is prohibited for any distributor to have a monopoly in distributing any domestically produced good that is subject to an import ban. 4 Francisco Marcos, Do Developing Countries Need Competition Law and Policy?, 3 (September 2006). Available at SSRN: http://papers.ssrn.com.ezp-prod1.hul.harvard.edu/sol3/papers.cfm?abstract_id=930562&download=yes ("[Competition Policy] mandates are also contained in most of the bilateral trade agreements and Free Trade Agreements in which young and developing countries take part. Parties to those treaties normally are required to have in place a domestic antitrust regime as one of the main conditions before entering into the agreement.") 5 See e.g The Euro-Mediterranean Association Agreement between the EU and Egypt in Article 72 states that, a "financial cooperation package shall be made available to Egypt" focused among others on "the accompanying measures for the establishment and implementation of competition legislation." Similar provisions are found in other Euro-Mediterranean Association Agreements which have been concluded between the EU and each of the following: Algeria, Egypt, Israel, Jordan, Lebanon, Morocco, Palestinian Authority, Tunisia, and Turkey. 6 World Trade Organization; Paper of 18 September 1998; WT/WGTCP/W/80. 3 straightforward. 7 Adopting these laws seemed to many as the missing link to assure growth and development. 8 In many instances these newly adopted competition laws are replicas of laws developed in the West. The problem of adopting laws that are not tailored to address local needs is that often these laws are nothing but ink on paper. Theories on legal transplant abundantly show that copy-and-pasting laws leads to inefficiency of these laws and prove their uselessness. 9 With this in mind, it is intriguing to see what the developing countries do with their antitrust laws after they adopt them, regardless of the reasons behind adopting these laws and the format that inspired their drafting. 10 The literature on antitrust in developing countries is rather meager with little research explaining the realities post adoption. Most studies focus on analyzing the adoption phenomenon and the kinds of laws adopted. In their analysis, measuring and quantifying enforcement is ignored or simply assumed to naturally follow adopting these laws. Many studies that attempt to see whether these laws are important, in terms of development, only look at the adoption as the event that should trigger economic development. These studies look at the kind of law adopted, whether it is one that addresses local needs or one that replicates the laws of the developed countries, to conclude about the effects resulting from the kind of law adopted.

Moreover, the general assumption is that developing countries do not enforce their antitrust laws, it is argued that "antitrust laws have been on the books of several developing countries for a long time, but they have done little to reduce anticompetitive behavior. This suggests that the laws have been ignored […]." 11 Whether this quote is true or not is what this paper is attempting to assess. The focus of this paper is what actually happens in developing countries after antitrust laws are adopted. The aim is simply to ask whether developing countries are capable of enforcing their competition laws or not. To answer this simple query this paper looks at antitrust enforcement activities in developing countries since the adoption of the law until 2009. Then the paper analyzes the variables affecting the potential for enforcement and non-enforcement of these laws by studying the respective national environment that resulted in the varying degrees of enforcement potential across countries. By doing that, the paper is basically analyzing the macroeconomic, political, legal and institutional environment that need to ("There are strong links between competition policy and numerous basic pillars of economic development … There is persuasive evidence from all over the world confirming that rising levels of competition have been unambiguously associated with increased economic growth, productivity, investment and increased average living standards"). 9 Michal S. Gal, The "Cut and Paste" of Article 82 of the EC Treaty in Israel: Conditions for a Successful Transplant, 9(3) EUR. J. L. REFORM 647, 648 (2007). 10 For an analysis of these questions see Dina Waked, Competition Law in the Developing World: The Why and How of Adoption and its Implications for International Competition Law, 1 GLOBAL ANTITRUST REV. 69, 70-75 (2008 REG. 209, 223 (1993-1994).

be in place to guarantee potential enforcement of these laws. To investigate the effects of these factors on potential antitrust enforcement the paper utilizes panel data estimation techniques. The aim of the paper is simply to illustrate what developing countries do with their antitrust laws after they adopt them and what affects their varying enforcement potential.

The paper is organized as follows: Section II describes the data used to measure potential antitrust enforcement in developing countries and the variables used to study what affects the intensity of enforcement potential. Section III introduces the empirical methodology. Section IV discusses the results and empirical findings. Section V concludes.

II. DATA

This article develops alternative measurements of potential antitrust enforcement using the resources of the antitrust authorities. These measurements capture the intensity of potential enforcement based on budgetary resources and staffing levels. The budgets and staffing levels of antitrust authorities in developing countries were gathered through questionnaires sent to the various authorities or through published annual reports when available. The data was collected since the adoption of the antitrust law in each country up till 2009. To assess what impacts the intensity of potential antitrust enforcement in developing countries variables across the macroeconomic, political, legal and institutional environments were assessed.

The majority of empirical studies on law enforcement focus on measuring formal enforcement. It is the most widespread method used to measure enforcement and has been dubbed measures of 'laws on the books'. This methodology uses positive law as an alternative measurement of enforcement. Writers who use 'laws on the books' look at one of two things: (1) the breadth of the law and its comprehensiveness; (2) the powers allocated in the law to the regulatory agency in charge of enforcing that law.

A. Resources-based Enforcement Measurement (Dependent Variable)

"Understandably, formal legal rules are easier for economists to code, measure, and incorporate into their regression equations, but they have little to do with reality of actual practice, particularly in developing countries." 12 In an attempt to capture the reality of actual enforcement potential in developing countries this paper diverges from the literature by focusing on antitrust enforcement data measured using budgets and staffing levels across antitrust authorities. 13 The data was collected for 40 countries, out of a total of 77 developing countries that have adopted a competition law by 2007 (see Table 1).

The method of data collection was either direct contact with the competition authority in the respective countries or collection of published antitrust data. The published data on antitrust enforcement are mainly found in the competition authorities' annual reports or OECD reports. Data collected through published materials represent only 23% of the countries studied. The rest of the data was collected by contacting the competition agencies and asking the director of the respective agency to fill a questionnaire about enforcement data on a yearly bases since the antitrust law has been adopted in his or her country. Therefore, for the 40 countries included in the research one of these criteria are satisfied: (1) published annual reports by the respective competition authorities with antitrust enforcement data is available, (2) other published resources that include actual enforcement data are found (mainly OECD or UNCTAD, CUTS, APEC reports or databases), (3) personal access to the competition authority that is willing to share enforcement data is established. 14 Both budgets and staffing levels are used as dependent variables representing potential antitrust enforcement. Budgets are normalized by the size of the economy and staffing levels are normalized by the size of the population. Both measurements indicate government commitment towards enforcement and yet each looks at this commitment from a different angle. Money and people are obviously measuring different aspects and they also have different weights to different countries. One country might find it easier or cheaper to show commitment to enforcement by well staffing its authority, especially in countries where wages are low, compared to others that prefer providing long term deep funding. Table 3 andFigures 2 and3 report the average normalized budgets and staffing levels across developing countries.

B. Explanatory Variables

After assessing the antitrust enforcement activities in developing countries, it is interesting to investigate why some are capable of implementing their competition laws while others are not. Also, it is intriguing to investigate the factors that affect the intensity of enforcement capacity.

Using inputs of the antitrust agencies as measurements for antitrust enforcement potential, the relationship with regards to the macroeconomic, political, legal and institutional environment is tested. The effects of these factors are tested on budgets and staffing levels. This investigation identifies the factors that need to be in place to assure that the antitrust authorities are capable of implementing the adopted antitrust laws.

The paper follows the footsteps of prior studies trying to link the macroeconomic environment to aspects of competition law. Most of these studies look at the relationship between these factors and adopting a competition law 15 or the effect they have on the general competition policy across countries 16 . This paper is interested in testing these same factors, identified by prior studies as the most important contributors to the antitrust environment. However, instead of looking at the relationship between the macroeconomic, political, legal and institutional factors and the adoption or the general competition environment this paper looks at the effect these factors have on actual enforcement potential, measured using enforcement inputs.

These variables fit into the following 4 categories:

Potential Antitrust Enforcement

Budgets Staffing

The effect of the macroeconomic environment on potential antitrust enforcement is tested through the relationship between the enforcement proxies and both economic development and openness to trade. Economic development is measured using the log of gross domestic product per capita in current US$. Openness to trade is measured by the log of total imports of goods and services in current US$. 17

The influence of the political environment on potential antitrust enforcement is tested using political stability and corruption. Political stability is a measurement developed by Kaufman et al. and reported in the Worldwide Governance Indicators Database. 18 It measures the likelihood that the government will be destabilized or overthrown by

Macroeconomic Environment Economic Development Openness to Trade

Political Environment Political Stability Corruption

Institutional Environment

Competition Authority Independence and Power

Comprehensiveness of Competition Law

Legal Environment

Law and Order Effectiveness of Anti-Monopoly Policy unconstitutional or violent means. 19 The corruption measurement used here is developed in the International Country Risk Guide and is concerned with actual or potential corruption in the form of excessive patronage, nepotism, job reservations, 'favor-forfavors', secret party funding, and suspiciously close ties between politics and business. 20 With regards to the legal environment, law and order in a country is measured using the index collected in the International Country Risk Guide that assesses the strength and impartiality of the legal system and the popular observance of the law. 21 The legal environment also tests for the relationship between enforcement and the effectiveness of the antitrust policy, a subjective measurement developed by the World Economic Forum in the Global Competitiveness Reports. 22 This variable is often used as a proxy for antitrust enforcement, but is used here as an explanatory variable to investigate the relationship between potential enforcement and the perceived and subjective legal environment surrounding competition policy established in the previous years in a given country.

Finally, the effect of the institutional environment on potential antitrust enforcement is tested by looking at the comprehensiveness of the competition law and the formal independence of the competition authority. Comprehensiveness of the competition law is measured using the index developed by Hylton and Deng, scoring the provisions of each country's antitrust law and summing the score into a formal index. 23 To account for the formal independence and power of the antitrust authority, a measurement call de jure independence developed by Stephan Voigt is used. 24 Table 2 lists a detailed description of all variables used in the empirical study, their sources and measurement methodology.

19 Id. 20 The PRS Group, International Country Risk Guide, available at http://www.prsgroup.com/ICRG_Methodology.aspx.

21 Id.

22 World Economic Forum, The Global Competitiveness Report (1997-2009). 23 Keith N.

III. EMPIRICAL METHODOLOGY

To investigate whether developing countries are capable of enforcing their competition laws or not the paper compares normalized budgets per billion US$ of GDP and staffing levels per million of population across the developing countries. The results of this comparison are reported in Table 3.

To test the influence of macroeconomic, political, legal and institutional factors on the intensity of antitrust enforcement in developing countries the paper utilizes a panel estimation model. The use of panel data achieves two goals: First, it controls for permanent unobserved heterogeneity in the cross-sectoral model, when controlling for country fixed effects. Second, it allows the study of dynamic cross-sectoral populations.

The model specifications are as follows:

Y it = α i + β i X it-1 + ε it ,
where Y it is the value of enforcement of the respective country i at time t, α i is a constant, X it-1 is the lagged vector of explanatory variables, and ε it is the error term.

The dependent variable is potential antitrust enforcement proxied by two variables: budgets and staffing levels. The independent variables include all macroeconomic, political, legal and institutional factors stated above to explain the varying intensities of enforcement potential. Although several factors can be found to describe each of the macroeconomic, political, legal and institutional environments, it is known that these factors are highly correlated and the effect of one factor could be an interpretation of other factors.

The independent variables are tested for multi-collinearity by investigating the Variance Inflation Factors (VIF) for the variables. All VIFs and their degrees of collinearity (tolerance levels) are found to be of acceptable levels. All the independent variables are lagged one year, assuming that the effects of the environment take time to impact the level of enforcement.

The Hausman test is used to indicate whether the fixed or random effects model is more suitable given the data at hand.

To explore the robustness of the results alternative models are tested to show the consistency of the findings. First, pooled sample OLS modeling is used. Then year fixed effects are added to control for autocorrelation across time. Then both year and country fixed effects are included in the regressions. Finally, instead of fixed country effects random country effects are used, thereby utilizing the GLS model (Tables 5 and6 report the results).

IV. DISCUSSION AND EMPIRICAL RESULTS

A. Measuring Potential Antitrust Enforcement

Measuring inputs of antitrust authorities is also a measure of potential enforcement, as it is assumed that the higher the budgets and staffing levels the higher the probability of enforcement. Collecting budgets and staffing levels of the various antitrust authorities is a burdensome activity, which explains the lack of many studies using this methodology to measure legal enforcement. One of the few studies, by Jackson and Roe 25 , using a resource-based methodology to measure legal enforcement inspired the assumptions and methodology of this paper. Despite the difficulty in collecting these input data, conducting an empirical study using a resource-based methodology is not impossible.

Staffing levels are easier to collect than budgets, the latter being reluctantly shared or not shared at all. Therefore, the number of observations for staffing levels is slightly higher than the number of observations for budgets collected here.

The assumptions made in studies using budgets and staffing are as follows. First, "[h]igher budgets and greater staffing allow the regulator to examine allegations of wrongdoing, to write rules carefully, to conduct market surveillance and review filings, and to act more often to remedy, prevent and punish wrong-doing." 26 Second, even "a not-very-independent regulator with a high budget and strong staffing indicates that political and market authorities have given the agency the go-ahead to enforce […] [the] rules"." 27

The limits to using inputs as measurements of enforcement include the following: Higher budgets and more staffing do not always allow a regulator to do all the things listed above. It is arguably pending certain formal powers that the agency is equipped with, for example, a well-funded and well-staffed agency may simply not have the authority, by law or its executive regulation, do impose fines, write rules, examine allegations etc. Therefore, these input measurements, of staffing levels and budget values, should be studied together with formal powers allocated to the respective regulatory body under study, which is done in this paper. Here, measures for formal power, covering the comprehensiveness of the law and the independence of the antitrust authority, are added in the regression analyses to account for the effect these institutional variables have on potential antitrust enforcement.

Another limitation is that high budgets and strong staffing do not necessarily indicate political and market support to such a regulator. This funding and staffing could be merely put in place to make the regulator seem to be powerful. This is especially true for nations where the setting up of different regulatory bodies has been orchestrated by international bodies or donor institutions. Not only that, the funding often comes from these institutions themselves. It is often the case, that players such as the World Bank, the International Monetary Fund or the like, require developing countries to set up regulatory 25 Jackson and Roe, supra note 13. 26 Id. at 210. 27 Id.

bodies in various fields to assure compliance to international standards and rules adopted by the receiving countries. In this case, the budgets and staffing levels may be, like the presence of the law itself, a mere "window dressings" that the countries put in place to please and signal compliance to the mandating authority. And if the budgets are donations, so-to-speak, they may even say nothing at all about where the political will lies. However, this critique may be unfounded, if the budgets remain high over a certain period as it would make little sense, even in terms of persuading and pretending power, to keep money idle in authorities that do nothing. Therefore, it may be true, that high budgets and well staffed regulators that maintain such budgets and staffing over a longer period are indeed signaling political will.

Moreover, ample resources do not mean that they will be used wisely or that they are deployed for the purposes for which they were dedicated. 28 This means that measuring resources is not a measure of enforcement efficiency, only one of potential enforcement. Also, crony-oriented appointees or lazy bureaucrats may result in nothing being done to enforce the laws. 29 Despite the various shortcomings of input measurements they do have their merit. They are especially informative once they are investigated in tandem with output measurements. 30 They also offer themselves for comparative analysis between different countries once the budgets are normalized by GDP and the staffing levels are normalized by population.

B. Do Developing Countries Enforce their Antitrust Laws?

Table 3 presents the results of antitrust enforcement potential measured using average budgets and staffing. From the table one can see that all developing countries included in this study have the potential to enforce their competition laws with varying intensities. The only countries that have no potential to enforce their competition laws are India, Mauritius and Syria.

The average staffing level of all developing countries is around 11 staff per million of population. Figure 2 illustrates the variation between average staffing levels across countries. The graph shows that the variation in staffing levels is quite stark between developing countries. The majority of developing countries have staffing levels ranging from 2 to 18 per million of population. Some countries, like Barbados, Panama and Estonia highly exceed the mean levels. Both Barbados and Panama have competition authorities that are staffed at more than 7 fold the average staffing levels across developing countries. On the other hand, countries like Peru, Indonesia, Egypt, Tunisia, Kenya and Argentina are far blow the average levels of staffing with numbers blow 1 per million of population.

28 Id. at 211. 29 Id. 30 Id at 237. (Jackson and Roe acknowledge this fact by arguing that it is more useful to collect information on the actual enforcement activities undertaken, such as how many cases prosecuted each year, how many sanctions imposed and with what level of monetary penalty, how many criminal convictions obtained, etc.)

The average budget of competition authorities across developing countries is around $55,500 per billion US$ of GDP. Figure 3 ranks developing countries according to their average budgets. From the graph one can see that the variation between countries in terms of budgets is less stark that their variation in terms of staffing. The majority of developing countries have budgets below $50,000 per billion GDP. Few countries have average budgets slightly higher than $50,000 and only 3 countries are far above this level affecting the mean budget level across the developing world. These are Turkey, Panama and Papua New Guinea. The with the lowest budgets are Kenya, Argentina and Peru.

Comparing Argentina with Panama, both upper middle economies, one can easily see that Panama reports 79 regulators staffing their competition authority per million of population whereas Argentina has only 1 staffer per million of population. Similarly, the budget of the competition authority in Panama is around $285,000 per billion of GDP whereas the budget of the Argentinean competition authority is roughly equal to $4,600 per billion of GDP. This comparison shows that the competition authority in Panama is more intensely staffed and better funded compared to its counterpart in Argentina. It can indicate that the government of Panama has put a stronger commitment to competition enforcement in Panama than Argentina. It also shows that the Panamanian competition authority will have more resources to actively enforce their competition law compared to Argentina.

On average staffing levels vary by 20 regulators per million population between countries and budgets vary by almost $86,000 per billion US$ of GDP. This raises the question of why these competition authorities have such varying resources across the developing world.

While comparing potential enforcement, it is important to take into account the actual size of the population and GDP. Countries with large populations will by default have a smaller ratio of staff over population. Similarly, countries with higher GDP will have a smaller ratio for their budgets over GDP. For example, comparing Papua New Guinea with Egypt, both considered low-income economies, the former has an average budget per million population of 8.65 and the latter of 0.58. One explanation of this difference is that Papua New Guinea is dedicating more resources to their antitrust authority compared to Egypt, hence indicating stronger commitment to potential antitrust enforcement. Another explanation could be that because Papua New Guinea's population is much smaller than Egypt's, the former being around 6 million and the latter being close to 80 million, their relative potential enforcement measurement proxied by staffing shows drastic differentiation. Nevertheless, only by normalized staffing and budgets can one compare different countries. Whether this translates into more actual enforcement or not needs to be further investigated and the validity can only become apparent when measurements collecting actual enforcement outputs, such as decisions and convictions, are combined with input data.

Table 4 reports the results of the regressions. The Hausman test recommends the use of country random effects instead of fixed effects and therefore the model used in Table 4 is GLS. The random effects are more relevant here seeing that the numbers of observations per country are relatively low. Also, using random effects allows the analysis to include the institutional environment variables that all lack temporal variation.

The significance and effect of the results does not change much when fixed effects are used instead (this can be seen in Table 5 which presents alternative regression models to test for robustness).

Column (1) includes the variables representing the macroeconomic, political and legal environment. Column (2) adds the institutional variables to the analysis. The reason for this sequential addition is that the numbers of observations for the institutional variables are lower than the other variables and thereby reducing the total observations and countries included in the regressions.

Starting with the variables representing the macroeconomic environment, the results in Table 4 show that economic development, measured using GDP per capita, is positive and significantly related to both staffing and budget levels of antitrust authorities in developing countries. This relationship is maintained even in the sample including the institutional variables. This means that the more developed a country the more its antitrust authority will be funded and staffed. This is rather intuitive and is in line with the literature arguing that the poorer the country, the less it is willing to invest its scare resources on antitrust enforcement. 31 Poor countries would rather save these scarce resources to be spent on more pressing issues. A famous quote is worth noting: "Exporting antitrust to Eastern Europe is like giving a silk tie to a starving man. It is superfluous; a starving man has much more immediate needs. And if the tie is knotted too tightly he won't be able to eat what little there is available to him." 32 The positive and significant effect on both budgets and staffing levels emphasizes the influence of economic development on both these proxies of potential antitrust enforcement. With regards to openness to trade, measured using imports of goods and services, the relationship is negative and significant with regards to both budgets and staffing levels. This means that the more a country imports the lower the potential antitrust enforcement will be. In other words, the higher the imports the lower the resources a developing country is willing to dedicate to its antitrust enforcement authority. This highlights the positive effects of imports on the functioning of the market place by effectively increasing competition and putting competitive pressure on firms to abide by the antitrust laws. Whether imports in reality reduce the anticompetitive behavior of local firms is not necessarily answered by the results of this regression. The results only confirm that governments will dedicate fewer resources to antitrust enforcement the higher the import levels are. This is a very interesting finding that affirms that potential enforcement is reduced in relationship to increased importation. It could also be read to mean that developing countries in order to attract more imports will reduce their antitrust enforcement potential so not to scare imports away by appearing to be an aggressive country with respect to antitrust enforcement. This might confirm the "race to the bottom" made about developing countries in the face of imports and foreign direct investments. Whether the results of this regression can be read to mean this negative connotation or just simply the positive effect imports have on disciplining local firms is impossible to decide. Only when looking at actual enforcement together with potential enforcement can one attempt to settle this issue.

Turning to the relationship between the political environment and antitrust enforcement, the results show that political stability in a developing country positively and significantly affects the potential antitrust enforcement. This result is, however, only maintained when looking at the staffing levels as a proxy of potential antitrust enforcement. This indicates that countries prone to revolutions and upheavals are less willing to commit to antitrust enforcement. This is also an intuitive result, seeing that countries that are plagued with political instability are probably indifferent with regards to antitrust enforcement.

The second variable representing the political environment is corruption. As the measurement of corruption increases, the level of corruption that is present in a country decreases. Corruption seems to be the only measurement that treats both variables capturing potential antitrust enforcement differently. The relationship between corruption and budget intensity is significantly negative, whereas it is positive and significant with regards to staffing. However, the effect of corruption on staffing levels disappears once the explanatory variables are increased to account for the institutional environment. Yet it stays negative and significant once these variables are added to the budgetary levels. The fact that the significance of corruption with regards to staffing is not robust once the explanatory variables are increased might indicate that this relationship is not really remarkable. Also the magnitude of the effect of corruption on staffing is much lower than on budgets. Focusing on the relationship with regards to budget intensities the negative and significant relation to corruption means that countries with less corruption allocate less budgetary resources to their antitrust authorities, i.e. are less concerned with potential antitrust enforcement. This shows that corruption might be related to either more actual or perceived antitrust violations hence requiring more commitment towards enforcement. Therefore, countries with higher levels of corruption will be more committed towards antitrust enforcement than countries with low corruption. Now turning to the legal environment, the effects of the strength and impartiality of the legal system and law observance is not related to potential antitrust enforcement. This is rather counterintuitive as it is often argued that the imperfections in the legal systems of developing countries are responsible for the lack of antitrust enforcement. This result shows that antitrust enforcement is not necessarily linked to the effectiveness of the legal system. As for the measurement used as a proxy for the antimonopoly policy in a country, these are not robust across the regressions as well. Only after the inclusion of more explanatory variables does the relation between the antimonopoly policy and budget levels become significantly positive. This relation means that the more the subjective legal environment surrounding competition policy in a developing country is considered effective the higher the potential antitrust enforcement will be. This is an intuitive consequence showing that the competition policy as perceived by people plays a minor role, not as important as other factors discussed before, in affecting the intensity of antitrust enforcement.

Finally, to consider the role played by the institutional environment in affecting the intensity of antitrust enforcement the results of the comprehensiveness of the competition law and the formal independence and power of the antitrust authority are assessed. The formal independence of the competition authority is not related to enforcement intensity. However, the comprehensiveness of the competition law only positively impacts the staffing levels. This means that the more comprehensive a law is, the more the authority will be endowed with resources to help it to enforce this law. This may be read to mean that the more comprehensive a competition law the more seriously antitrust enforcement will be taken. A comprehensive law can therefore signal an increasing commitment towards enforcement measured by more resources being dedicated towards antitrust enforcement.

To explore the robustness of the results for each proxy of potential antitrust enforcement, the paper suggests alternative measures in which the control variable of panel data are loosened. For instance, pooled sample OLS removes country effects (fixed and random) and most of the results maintain their significance across the models. Tables 5 and6 present the results of the robustness checks for both proxies of potential antitrust enforcement respectively. The robustness check does not include the institutional variables as they lower the observations, however running all alternative regressions with the extended variables holds the results constant as the ones reported in Tables 5 and6.

V. CONCLUSION

Whether developing countries enforce their competition laws or not has been a question that much of the literature on antitrust in developing countries has been struggling to answer. The assumption often made is that most developing countries never enforce their antitrust laws. Due to lack of data on antirust enforcement in developing countries this assumption has influenced much of what is written on the topic. The few that have taken up this question have focused on formal enforcement measurements, e.g. the comprehensives of the antitrust laws, independence of the competition authorities, or a subjective measurement on the effectiveness of competition policy to assess enforcement in developing countries. To offer an alternative to these measurements this paper collects resources of antirust authorities to address this question.

This alternative proxy of antitrust enforcement developed in this paper is focusing on potential antitrust enforcement by collecting in an original dataset information on budgets and staffing levels of antitrust authorities across a sample of 40 developing countries. The data clearly challenge the assumption of non-enforcement by illustrating that the majority of developing countries dedicate resources towards antitrust enforcement. This can be assumed to mean that actual enforcement does takes place. The leap from potential enforcement to actual enforcement is easier to make than from formal enforcement. Nevertheless, measuring enforcement outputs, such as decisions rendered and sanctions imposed would only enrich the conclusions made here. Also, if output variables are examined together with the input measurements presented in this paper, this will allow a more thorough comparison between counties. It will be interesting to investigate whether countries with higher budgets and more staffing actually enforce their laws more or not. This would reaffirm the effectiveness of measuring enforcement using a proxy based on potential enforcement variables.

After establishing that developing countries have the necessary potential to enforce their antirust laws the paper turns to assess why the potential for enforcement varies across the countries studied. The most important elements that affect antitrust enforcement capabilities were categorized into macroeconomic, political, legal and institutional factors. The relationship between these factors and potential antirust enforcement was examined using panel data estimation techniques. The results of the regressions show that economic development is strongly correlated with the resources dedicated to antitrust enforcement across developing counties. This affirms that the higher the economic development the higher the level of potential antirust enforcement. Moreover, the results show that imports also play an important role in antirust enforcement. The higher the imports the lower the resources allocated to antirust authorities. This is an interesting correlation and can be either be understood to mean that imports positively affect the marketplace by increasing competition and hence lowering the potential of antitrust enforcement or that developing countries, to attract more imports, lower their potential antitrust enforcement. The results also show that the higher the level of corruption the more resources are dedicated towards antitrust enforcement. This entails that countries plagued with high levels of corruption are also more concerned with potential antitrust enforcement. Furthermore, political stability is also found to have positive relation with regards to staffing levels at competition authorities. Finally, the effects of the legal and institutional environments on potential antirust enforcement are less significant across the regressions. The results of these influences on potential enforcement are significant across alternative statistical models.

The analysis in this paper can be extended in several directions. First, adding output variables to the input variables used here would enrich the study and improve the quality of the antitrust enforcement proxy developed. Second, alternative proxies measuring antirust enforcement could be compared with the ones developed here to assess how they relate to other. Third, the effectiveness of antitrust enforcement in developing countries could be studied by assessing the relationship between the enforcement proxies and antitrust outputs, such as improved competition intensities. These and other extensions of the paper await further research.

FIGURE 1 :

 1 FIGURE 1: TIME-LINE OF ADOPTING COMPETITION LAWS IN THE DEVELOPING WORLD

FIGURE 2 :

 2 FIGURE 2: AVERAGE STAFFING PER MILLION OF POPULATION

7

  See The Euro-Med Association Agreements, supra note 5. 8 See Organization for Economic Cooperation and Development (OECD), Implementing Competition Policy in Developing Countries in Promoting Pro-Poor Growth: Private Sector Development 39 (2006).

TABLE 1 :

 1 BREAKDOWN OF ALL DEVELOPING COUNTRIES WITH A COMPETITION LAW BY 2007

	##	Country	Region	Year of Adoption	Income Group 2	##	Country	Region	Year of Adoption	Income Group 2
	1 Albania*	Europe	1995 Lower middle	40 Mali	Africa	1992 Low

TABLE 3 :

 3 POTENTIAL ENFORCEMENT DATA -RESOURCE-BASED RESULTS

	Country	Year of Adoption	Years since first enforcement data available	Average Staffing per million of Population	Average Budget per billion US$ of GDP
	Albania	1995	5	7.59	49,882
	Argentina	1980	12	0.97	4,642
	Barbados	2002	5	109.70	n/a
	Bosnia & Herzegovina	2001	4	5.09	50,544
	Bulgaria	1998	7	15.55	46,883
	Chile	1973	14	3.57	27,600
	Costa Rica	1994	14	2.68	13,667
	Croatia	1995	12	6.22	33,904
	Czech Republic	1991	16	8.93	26,366
	Egypt	2005	3	0.58	21,540
	El Salvador	2006	3	4.09	76,496
	Estonia	1993	15	26.19	n/a
	Honduras	2005	3	2.79	60,787
	Hungary	1996	11	11.01	60,295
	India	2003	7	0.00	0
	Indonesia	1999	9	0.45	14,169
	Jamaica	1993	16	6.60	68,842
	Jordan	2004	5	1.56	n/a
	Kenya	1988	18	0.83	483
	Latvia	1998	8	18.72	54,428
	Lithuania	1999	10	17.61	47,466
	Macedonia	2000	8	6.57	58,519
	Mauritius	2003	7	0	0
	Mexico	1992	15	1.76	21,026
	Mongolia	1993	5	12.71	34,307
	Panama	1996	13	71.86	284,853
	Papua New Guinea	2002	6	8.65	452,192
	Peru	1991	14	0.21	5,911
	Poland	1990	12	6.33	30,926
	Romania	1996	11	7.03	41,149
	Russia	1991	10	13.09	23,030
	Serbia	2005	3	2.30	38,208
	Slovakia	1994	14	12.64	34,386
	South Africa	1979	9	1.85	37,616
	Sri Lanka	1987	4	n/a	n/a
	Syrian Arab Republic	2007	3	0	0
	Tunisia	1991	17	0.61	16,686
	Turkey	1994	9	4.48	147,377
	Ukraine	1993	5	16.87	58,202
	Uzbekistan	1996	11	14.70	n/a
	Zambia	1994	4	2.18	n/a
	Mean		9.20	10.86	55,497
	Median		9.00	5.66	34,386
	Minimum		3.00	0.00	0
	Maximum		18.00	109.70	452,192
	Standard Deviation		4.55	20.08	85,643

Table 4 :

 4 EFFECTS ON POTENTIAL ANTITRSUT ENFORCEMENT -GLS REGRESSION MODELS Significance levels: * 10%, ** 5%, *** 1%. Standard errors are in parentheses.

		Log of Budget per billion US$	Log of Staffing per million of
			of GDP		Population
		(1)	(2)	(1)	(2)
	Log of GDP per Capita	0.773*** (0.232)	0.663** (0.334)	0.097 (0.156)	0.382** (0.189)
	Log of Imports	-0.392** (0.194)	-0.559** (0.262)	-0.370** (0.159)	-0.717*** (0.17)
	Political Stability	0.067 (0.178)	0.002 (0.228)	0.304*** (0.099)	0.348*** (0.123)
	Corruption	-0.177*** (0.065)	-0.382*** (0.148)	0.076** (0.032)	0.068 (0.046)
	Law and Order	-0.103 (0.095)	-0.038 (0.117)		(0.054)
	Effectiveness of Antimonopoly	0.185	0.308**	0.071	0.096
	Policy	(0.115)	(0.14)	(0.059)	(0.067)
	Comprehensiveness of		0.113		0.201***
	Competition Law		(0.089)		(0.065)
	Competition Authority Formal		1.833		0.479
	Power and Independence		(2.597)		(1.902)
	Constant	13.554*** (3.605)	14.862*** (4.662)	8.780*** (2.896)	11.115*** (3.125)
	Observations	139	94	154	103
	Number of Countries	28	17	30	18
	Note:

Table 5 :

 5 TESTING ALTERNATIVE REGRESSION MODELS USING BUDGET INTENSITIES Log of Budget per billion US$ of GDP Significance levels: * 10%, ** 5%, *** 1%. Standard errors are in parentheses.

		Pooled Sample OLS	OLS with year fixed effects	OLS with both fixed effects year and country	GLS
		(1)	(2)	(3)	(4)
	Log of GDP per Capita	0.631*** (0.199)	0.673*** (0.209)	1.463*** (0.452)	0.773*** (0.232)
	Log of Imports	-0.123 (0.121)	-0.14 (0.125)	-1.441** (0.558)	-0.392** (0.194)
	Political Stability	0.087 (0.225)	0.109 (0.236)	0.158 (0.207)	0.067 (0.178)
	Corruption	-0.272** (0.13)	-0.356** (0.155)	-0.201*** (0.067)	-0.177*** (0.065)
	Law and Order	0.074 (0.108)	0.073 (0.112)	-0.126 (0.104)	-0.103 (0.095)
	Effectiveness of	0.097	0.117	0.213*	0.185
	Antimonopoly Policy	(0.147)	(0.152)	(0.122)	(0.115)
	Constant	8.048*** (2.12)	8.283*** (2.244)	33.110*** (10.556)	13.554*** (3.605)
	Observations	139	139	139	139
	R-squared	0.189	0.207	0.231
	Number of Countries			28	28
	Note:

Table 6 :

 6 TESTING ALTERNATIVE REGRESSION MODELS USING STAFFING LEVELS Significance levels: * 10%, ** 5%, *** 1%. Standard errors are in parentheses.

		Staffing per million of population
		Pooled Sample OLS	OLS with effects year fixed	OLS with both fixed effects year and country	GLS
		(1)	(2)	(3)	(4)
	Log of GDP per Capita	0.579*** (0.189)	0.529*** (0.199)	-0.002 (0.212)	0.097 (0.156)
	Log of Imports	-0.087 (0.104)	-0.078 (0.108)	-0.374 (0.251)	-0.370** (0.159)
	Political Stability	0.908*** (0.208)	0.939*** (0.217)	0.234** (0.099)	0.304*** (0.099)
	Corruption	-0.197 (0.12)	-0.173 (0.143)	0.076** (0.031)	0.076** (0.032)
	Law and Order	-0.037 (0.103)	-0.038 (0.106)	-0.026 (0.048)	-0.038 (0.049)
	Effectiveness of	-0.520***	-0.537***	0.093	0.071
	Antimonopoly Policy	(0.135)	(0.14)	(0.057)	(0.059)
	Constant	1.49 (1.87)	1.653 (1.945)	10.612** (4.643)	8.780*** (2.896)
	Observations	154	154	154	154
	R-squared	0.439	0.446	0.475
	Number of Countries			30	30
	Note:

John C. Coffee, Jr., Law and The Market: The Impact of Enforcement, 156 U. PA. L.REV. 229, 244 (2007REV. 229, 244 ( -2008)).

Measuring budgets and staffing levels to proxy enforcement is inspired by Howell E. Jackson and Mark J. Roe, Public Enforcement of Securities Laws: Resource-based Evidence, 93 J. FIN.ECON. 207 (2009).

In many instances some enforcement data were initially collected from published reports yet when contact with the respective competition authority was made, these data were shared to be complemented, reviewed or confirmed.

Ajit Singh and Rahul Dhumale, Competition Policy, Development And Developing Countries, 7 T.R.A.D.E. Working Papers (November 1999).

Paul E. Godek, One U.S Export Eastern Europe Does Not Need,REGULATION 15, 21 (1992).

Nesta, Ludovic DiBiaggio and Maryam Nasiriyar. I am also very grateful to the assistance I received while working at SKEMA Business School (Sophia Antipolis,

Source

Collected through questionnaires sent to the various antitrust authorities or available mainly in published annual reports by the authority.

Collected through questionnaires sent to the various antitrust authorities or available mainly in published annual reports by the authority.

Description/Methodology

The budget of antitrust authorities divided by the country's GDP. Size of the antitrust authorities staff divided by a countries population in million.

The log of GDP per capita in current US$.

The log of total imports of goods and services in current US$.

Political stability and absence of violence/terrorism. This is an assessment of corruption within the political system. The measure is more concerned with actual or potential corruption in the form of excessive patronage, nepotism, job reservations, 'favor-forfavors', secret party funding, and suspiciously close ties between 6. The higher Law are assessed separately, with each sub-component comprising zero to three points. The Law sub-component is an assessment of the strength and impartiality of the legal system, while the Order sub-component is an assessment of popular observance of the law. Thus, a country can enjoy a high rating -3 -in terms of its judicial system, but a low rating -1 -if it suffers from a very high crime rate of if the law is routinely ignored without effective sanction (for example, widespread illegal strikes). Maximum points 6.

This indicator is based on a survey of participants in each country asked to rate, on a scale from 1 (lowest value) to 7 (highest value), whether anti-monopoly policy promotes competition.

A measurement of the comprehensiveness of a law, where the presence of certain provisions in a law is used to construct an index. The minimum possible total index score is 0 and the maximum is 30 De jure independence by looking at whether the agency is subjected to the supervision of government, whether it has direct access to court, whether the leading member of the competition agency are appointed though the executive, the length of the term for competition officers, whether the salaries are set by the executive, whether the agency receives instructions by the executive, and finally whether the agency is required to publish its decisions. These are 13 different variables that cover formal independence of the competition agency.


Simone De Liberato

Cristiano Ciuti

Quantum theory of electron tunneling into intersubband cavity polariton states

Through a non-perturbative quantum theory, we investigate how the quasi-electron excitations of a two-dimensional electron gas are modified by strong coupling to the vacuum field of a microcavity. We show that the electronic dressed states originate from a Fano-like coupling between the bare electron states and the continuum of intersubband cavity polariton excitations. In particular, we calculate the electron spectral function modified by light-matter interactions and its impact on the electronic injection of intersubband cavity polaritons. The domain of validity of the present theoretical results is critically discussed. We show that resonant electron tunneling from a narrowband injector can selectively excite superradiant states and produce efficient intersubband polariton electroluminescence.

I. INTRODUCTION

Cavity quantum electrodynamics in the strong coupling regime is presently the subject of many fascinating investigations in several interesting systems, including ultracold atoms 1 , Cooper pair quantum boxes 2 and semiconductor nanostructures 3 . In the strong coupling regime, the eigenstates of a cavity system are a coherent mixing of photonic and electronic excitations. This occurs when the light-matter interaction, quantified by the so-called vacuum Rabi frequency, is dominant with respect to loss mechanisms for the cavity photon field and for the electronic transition.

Recently, the strong coupling regime has been demonstrated also between a planar microcavity mode and an intersubband transition in a doped semiconductor quantum well. The normal modes of such a system are called intersubband cavity polaritons 4,5,6,7,8,9,10,11,12 . The active electronic transition is between two conduction subbands, where a dense two-dimensional electron gas populates the lowest one. Large vacuum Rabi frequencies can be achieved thanks to the giant collective dipole associated to the dense electron gas and even an unusual ultra-strong coupling regime can be reached 9,10,13 .

Electroluminescence experiments in microcavityembedded quantum cascade devices 14,16 have recently demonstrated that it is possible to obtain intersubband cavity polariton emission after resonant electrical excitation even at room temperature (and even a lasing mechanism has been proposed 15 ).

A fundamental question to address is how the strong interaction with the microcavity vacuum field modifies the quasi-electron states in the quantum well and how the electron tunneling is affected. In this paper, we present a quantum theory to investigate such fundamental problems. We show that the electronic eigenstates originate from a Fano-like coupling between the bare injected electron and the continuum of cavity polariton modes. Our theory demonstrates that resonant electron tunnelling from a narrow-band injector contact can selectively excite polaritonic states leading to ultraefficient polariton electroluminescence.

The paper is organized as follows: in Sec. II, we introduce the general formalism, by presenting the quantum Hamiltonian, the approximations behind it and by introducing the concept of the electron spectral function in the non-interacting case. In Sec. III we present the calculations leading to the spectrum of the Hamiltonian and to the determination of the electron spectral function in the light-matter interacting case. In Sec. IV we calculate the tunneling coupling and the radiative lifetime of the excited states and present the resulting electroluminescence spectra. Conclusions are drawn in Sec V.

II. GENERAL FORMALISM

To describe the system to investigate, we consider the following quantum Hamiltonian:

H = k ω 1 (k)c † 1,k c 1,k + k ω 2 (k)c † 2,k c 2,k + q ω c (q)a † q a q + k,q χ(q)a q c 1,k c † 2,k+q + k,q χ * (q)a † q c 2,k+q c † 1,k , (1)
where

ω 1 (k) = 2 k 2 2m ⋆ and ω 2 (k) = ω 12 + 2 k 2
2m ⋆ are the energy dispersions of the two quantum well conduction subbands as a function of the in-plane wavevector k, m ⋆ being the effective mass. The corresponding electron creation operators are c † 1,k and c † 2,k . ω c (q) is the frequency dispersion of the cavity photonic mode and a † q is the corresponding photon creation operator. Due to the selection rules of intersubband transitions, we omit the photon polarization, which is assumed to be Transverse Magnetic (TM). Being all the interactions spin-conserving, we can omit the electron spin. For simplicity, we consider only a photonic branch, which is quasi-resonant with the intersubband transition, while other cavity modes are supposed to be off-resonance. The interaction between the cavity photon field and the two electronic conduction subbands is quantified by the coupling constant

χ(q) = ω 2 12 d 2 12 ǫ 0 ǫ r L cav Sω c (q) q 2 (π/L cav ) 2 + q 2 , ( 2
)
where c is the light speed, d 12 is the intersubband transition dipole , ǫ r is the cavity dielectric constant and S is the sample area. Here, we have considered the simple case of a λ/2-cavity of lenght L cav , where π/L cav is the cavity photon quantized vector along the growth direction. The geometrical factor q 2

(π/Lcav ) 2 +q 2 is due to the TM-polarization nature of the intersubband transition. Here, for simplicity, we are considering the case of just a single quantum well coupled to the cavity quantum field. Notice that in the Hamiltonian in Eq. ( 1) the antiresonant terms of the light-matter interaction have not been included. Therefore, here we can describe the strong coupling for the two subband system, but not all the peculiar features of the ultrastrong coupling regime 8,9,10,13 .

If one is interested in describing the dynamics of the microcavity under optical excitation, it is possible to use an effective Hopfield Hamiltonian with bosonic operators associated to the intersubband polaritons, which are the elementary optical excitations of the system. If, instead, one is interested in studying microscopically how the electronic injection into to such a microcavity system is modified by non-perturbative light-matter excitation, it is necessary to work with the full fermionic Hamiltonian in Eq. ( 1). In fact, electrical excitation occurs through injection of (fermionic) carriers: the dynamics must include the (bosonic) optical excitations and the electronic (fermionic) excitations at the same level. As well known in the theory of quantum transport [START_REF] Datta | Quantum Transport: Atom to Transistor[END_REF] , if we wish to study the tunneling injection of one electron at low temperature, we have to determine the electron spectral function, defined as:

A + j (k, ω) = ζ | ζ|c † j,k |F N | 2 δ(ω -ω ζ ) , (3)
where |F N is the N-electron Fermi sea ground state times the vacuum state for the cavity photon field and j = 1, 2 is the conduction subband index. The index ζ labels the excited (N+1)-electron eigenstates and ω ζ are the corresponding eigenenergies. Note that even if in the present paper we consider the case of zero temperature, the theory can be applied as far as K B T is much smaller than the Fermi energy of the two-dimensional electron gas.

As apparent from Eq. ( 3), the electron spectral function is the density of quasi-electron states, weighted by the overlap with the bare electron state c † j,k |F N . In other words, it is the many-body equivalent of the single electron density of states. This is the key quantity affecting the electron tunneling and can be non-trivially modified by interactions like in the case of superconductors. For a non-interacting electron gas, c † 1,k |F N and c † 2,k |F N are eigenstates of the Hamiltonian and thus all the other eigenstates are orthogonal to them. Therefore the non-interacting spectral functions are

A + 1 (k, ω) = δ(ω -ω 1 (k))θ(k -k F ), (4)
A + 2 (k, ω) = δ(ω -ω 2 (k)), (5)
where k F is the Fermi wavevector. θ(x) is the Heaviside function and its presence is due to Pauli blocking:

c † 1,k |F N = 0 for k < k F .

III. SPECTRAL FUNCTION WITH LIGHT-MATTER INTERACTIONS

As seen in the previous section, in the non-interacting case, the electron spectral function is just a Dirac delta. Physically, this means that an electron with wavevector k can be injected in the subband j = 1, 2 only with an energy equal to the bare electron energy ω j (k) and that such excitation has an infinite lifetime. By contrast, interactions can profoundly modify the nature of electron excitations and therefore produce qualitative and quantitative changes of the electron spectral functions. In the case of a weakly interacting electron gas, the spectral function has a broadened "quasi-electron" peak: the spectral broadening is due to the finite lifetime of the electronic excitation. In the case of a strongly interacting electron gas (like in the case of superconductors) the electron spectral function can be qualitatively different from the non-interacting gas. Here, we are interested in how the nature of the quasi-electron excitations is modified by the strong coupling to the vacuum field of a microcavity. In particular, we assume that the light-matter interaction is the strongest one. We will provide here a nonperturbative theory to determine the dressed electronic excitations in such a strong coupling limit and their corresponding spectral function. All other residual interactions will be treated as perturbations. The consistency and limit of validity of such a scheme will be discussed in the next section, where the theoretical results are applied.

In the interacting case, it is easy to verify that c † 1,k |F N is still an eigenvector of the Hamiltonian in Eq. ( 1) and thus the first subband spectral function A + 1 (k, ω) is still given by Eq. ( 4). Instead for the electrons in the second subband we have to distinguish between two cases: k well inside or outside the Fermi sea. In the first case, an electron in the second subband can not emit a photon because all the final states in the first subband are occupied (Pauli blocking), hence the spectral function will be given by the unperturbed one (Eq. ( 5)). Well outside the Fermi sea, an injected electron can emit and the spectral function will be modified by the light-matter interaction. A smooth transition between the two cases will take place for |kk F | of the order of the resonant cavity photon wave-vector q res , where ω c (q res ) = ω 12 . Being the ratio q res /k F typically very small, of the order of 10 -2 (see FIG. 1: Sketch of the dynamical coupling between quantum states in a microcavity-embedded quantum well (QW) containing a two-dimensional electron gas (2DEG). In the ground state, the first subband is doped with a dense 2DEG (bold lines at the bottom of the dispersions). Black dots represent bare electrons, while white dots denote holes in the 2DEG. The dashed cones depicts the possible final states for an electron radiatively relaxing from the second to the first subband by emission of a cavity photon. The ground state with N electrons is the standard Fermi sea |FN . The injection (e.g., through electron tunneling) of an additional electron in the second subband creates the state |C = c † 2,k |FN , which, in presence of light-matter interaction, is not an eigenstate. Spontaneous emission of a cavity photon couples the |C state to the states |A, q = a † q c † 1,k-q |FN . Reabsorption of the emitted cavity photon can couple back to the |C state or to the states

|B, q, k ′ = c † 2,k ′ +q c 1,k ′ c † 1,k-q |FN
. Spontaneous emission couples the |B states back to |A states or to states of the form |D, q, q ′ , k

′ = a † q ′ c † 1,k ′ +q-q ′ c 1,k ′ c † 1,k-q |FN .
Being the relevant cavity photon wavevectors very small compared to the Fermi wavevector, spontanous emission can occur only on narrow emission cone in momentum space. Due to the small probability of photon absorption by electrons on the border the Fermi sea, we can neglect |D states and assume that the system always jumps from |B states to |A states. Thus the relevant dynamics takes place only between the states in the shaded region. We can thus neglect the other marginal states while diagonalizing the light-matter Hamiltonian.

Ref. [18]), we can safely consider an abrupt transition at the Fermi edge.

For evaluating A + 2 (k, ω) for k > k F we need to find all the (N+1)-electron eigenstates that have a nonzero overlap with c † 2,k |F N . In order to do this we notice that the Hamiltonian in Eq. ( 1) commutes with the number of total fermions

NF = j=1,2 k c † j,k c j,k ,
the total in plane wave-vector operator

K = j=1,2 k k c † j,k c j,k + q qa † q a q ,
and the excitation number operator

Q = k c † 2,k c 2,k + q a † q a q .
Hence the eigenstates |ζ of H can be also labeled by the corresponding eigenvalues N ζ , K ζ and Q ζ . We will thus identify an eigenstate of H in the subspace

( NF = N, K = K, Q = Q) as |N, K, Q, ζ
(N + 1, K + k, Q); c 1,k |N, K, Q, ζ in (N -1, K -k, Q); c † 2,k |N, K, Q, ζ in (N + 1, K + k, Q + 1); c 2,k |N, K, Q, ζ in (N -1, K -k, Q -1).
Having |F N quantum numbers (N,0,0) the state c † 2,k |F is thus in the subspace labeled by the quantum numbers (N + 1, k, 1). We can limit ourselves to diagonalize H in this subspace, which is spanned by vectors of the form:

(i) c † 2,k0 N j=1 c † 1,kj |0
, where |0 is the empty conduction band state and

N j=1 k j = k -k 0 ; (ii) a † q0 N +1 j=1 c † 1,kj |0 with N +1 j=1 k j = k -q 0 .
For a large number of electrons, the exact diagonalization of the Hamiltonian in this subspace is an unmanageable task.

Here, we show that by a judicious approximation, we can considerably simplify the diagonalization problem, keeping the relevant non-perturbative physics. Namely, we claim that the elements of the (N + 1, k, 1) subspace can be well approximated by vectors of the form

|N + 1, k, 1, ζ =    µ ζ c † 2,k + q   α ζ (q) a † q c † 1,k-q + |k ′ |<kF β ζ (q, k ′ )c † 2,k ′ +q c 1,k ′ c † 1,k-q      |F N . (6)
To understand the origin of our approximation, let us consider the time evolution picture sketched in Fig. 1. Suppose that initially the system is in its ground state |F N . After injection of one bare electron, the state of the system is

|C = c † 2,k |F N .
If k is well inside the Fermi sphere, as we said before, it is Pauli blocked and can not radiatively relax into the first subband. Instead, when k > k F , the injected electron can radiatively decay, emitting a photon and falling into the first subband. After the first emission the state will have the form

|A, q = a † q c † 1,k-q |F N .
If the cavity system is closed and only the light-matter interaction is considered, the emitted photon will be eventually reabsorbed. The system can evolve back to the state |C or into one vector of the form

|B, q, k ′ = c † 2,k ′ +q c 1,k ′ c † 1,k-q |F N .
If k ′ is well inside the Fermi sea, when the second subband electron decays, the only avaiable final state in the first subband will be the one with wavevector k ′ , that is the system will go back to state |A, q . If k ′ is on the border of the Fermi sea, on the contrary, the system can evolve into a state of the form |D, q, q ′ , k

′ = a † q ′ c † 1,k ′ +q-q ′ c 1,k ′ c † 1,k-q |F N .
The probability of ending in any of the |D, q, q ′ , k ′ states is negligible. In fact, the probability for k ′ to be near enough to the border of the Fermi sea for allowing an emission to electronic states with k > k F is proportional to the ratio q res /k F ≪ 1. Hence, the diagonalization problem can be simplified and we can thus look for vectors of the form shown in Eq. ( 6).

In such a subspace, spanned by {|C , |A, q , |B, q, k ′ , |B, q, k ′′ , . . . , |A, q ′ , |B, q ′ , k ′ , |B, q ′ , k ′′ , . . . }, H has the following matrix representation:

H N +1,k,1 =         ω 2 (k) v(q) v(q ′ ) v(q ′′ ) • • • v(q) T M (q) 0 0 • • • v(q ′ ) T 0 M (q ′ ) 0 • • • v(q ′′ ) T 0 0 M (q ′′ ) . . . . . . . . . . . . . . . . . .        
where M (q) is the Hamiltonian matrix block in the subspace spanned by {|A, q , |B, q, k ′ , |B, q, k ′′ , . . . }, that effectively describes the system in presence of one photon with a well defined wavevector q

M (q) =       ω c (q) + ω 1 (|k -q|)) χ * (q) χ * (q) • • • χ(q) ω 2 (|k ′ + q|) -ω 1 (k ′ ) + ω 1 (|k -q|) 0 • • • χ(q) 0 ω 2 (|k ′′ + q|) -ω 1 (k ′′ ) + ω 1 (|k -q|) . . . . . . . . . . . . . . .      
, where v(q) = [χ(q), 0, 0, . . . ]. Since the typical wavevector q of the resonantly coupled cavity photon mode is much smaller than k F , we can perform the standard approximation ω 2 (|k The non-radiative relaxation rate 1/τ nr,k,ζ has been taken equal to 0.005ω12. In all panels, the dashed-dotted lines are the frequency dispersions ω±(q) of the two cavity polariton branches. In the first panel the solid line represents the edge of the light cone [START_REF] Anappara | [END_REF] . ω 1 (k ′ ) = ω 12 . This way, we can exactly diagonalize each of the M (q) blocks. As expected from the theory of optically excited polaritons 9 , by diagonalizing the matrix M (q) we find two bright electronic states (i.e., with a photonic mixing component)

′ + q|) -ω 1 (k ′ ) ≃ ω 2 (k ′ ) -
|±, q = (ω ± (q) -ω 12 )|A, q + χ(q) k |B, q, k (ω ± (q) -ω 12 ) 2 + |χ(q)| 2 N , (7)
with energies ω 1 (k) + ω ± (q), where

ω ± (q) = ω c (q) + ω 12 2 ± ω c (q) -ω 12 2 2 + N |χ(q)| 2 .
(8) Note that ω ± (q) are the energies of the two branches of intersubband cavity polaritons 9 .

The other orthogonal states are dark (no photonic component), with eigenvalues ω 2 (k) = ω 1 (k) + ω 12 and eigenvectors

|l, q = k β l (q, k)|B, q, k √ N ( 9
)
where the β l (q, k) are such that k β l (q, k) = 0 and

k β l (q, k)β * l ′ (q, k) = δ l,l ′ .
Since l, q|H c † 2,k |F N = 0, the dark states |l, q are also eigenstates of the matrix H N +1,k,1 and do not contribute to the electron spectral function, because they have zero overlap with the state |C = c † 2,k |F N = 0. In contrast, this is not the case for the bright eigenstates of each block M (q), as we find:

±, q|H c † 2,k |F N = χ(q) * (ω ± (q) -ω 12 ) (ω ± (q) -ω 12 ) 2 + |χ(q)| 2 N = J * ± (q) .
(10) Therefore, the representation of H in the subspace {|C , |+, q , |-, q , |+, q ′ , |-, q ′ , . . . } reads

H ′ N +1,k,1 =         ω 1 (k) + ω 12 J * + (q) J * -(q) J * + (q ′ ) J * -(q ′ ) • • • J + (q) ω 1 (k) + ω + (q) 0 0 0 • • • J -(q) 0 ω 1 (k) + ω -(q) 0 0 • • • J + (q ′ ) 0 0 ω 1 (k) + ω + (q ′ ) 0 • • • J -(q ′ ) 0 0 0 ω 1 (k) + ω -(q ′ ) • • • . . . . . . . . . . . . . . . . . .        
.

Hence, here we have demonstrated that the bare electron state c † 2,k |F N is coupled to the continuum of the polariton modes with all the different wavevectors q. Since the polariton frequencies ω ± and the coupling J ± depend only on the modulus of q, we can further simplify the diagonalization problem by introducing the 'annular' bright states

|±, q = 1 √ Lq |q|=q |±, q , (11)
where L = √ S and 2π/L is the linear density of modes in reciprocal space. All annular states are coupled to |C . Instead, all the orthogonal linear combina-tions of |±, q (with |q|= q) are uncoupled and therefore do not contribute to the electron spectral func-tion. The matrix representation of H in the subspace {|C , |+, q , |-, q , |+, q ′ , |-, q ′ , . . . } reads

H ′′ N +1,k,1 =           ω 1 (k) + ω 12 J * + (q) √ Lq J * -(q) √ Lq J * + (q ′ ) √ Lq ′ J * -(q ′ ) √ Lq ′ • • • J + (q) √ Lq ω 1 (k) + ω + (q) 0 0 0 • • • J -(q) √ Lq 0 ω 1 (k) + ω -(q) 0 0 • • • J + (q ′ ) √ Lq ′ 0 0 ω 1 (k) + ω + (q ′ ) 0 • • • J -(q ′ ) √ Lq ′ 0 0 0 ω 1 (k) + ω -(q ′ ) . . . . . . . . . . . . . . . . . . . . .           (12)
Hence, in the subspace (N + 1, k, 1), we have found that eigenstates of H with a finite overlap with the bare electron have the form

|N + 1, k, 1, ζ = µ ζ c † 2 k |F N + q,σ=± λ ζ,σ,q |σ, q . (13)
The coefficients µ ζ and λ ζ,σ,q as well as the corresponding energy eigenvalues ω ζ can be calculated though a numerical diagonalization of the matrix in Eq. ( 12). In conclusion, the spectral function of the second subband reads

A + 2 (k, ω) = ζ |µ ζ | 2 δ(ω-ω ζ )θ(k-k F )+δ(ω-ω 2 (k))θ(k F -k). (14)
In Fig. 2, we show numerical results using a vacuum Rabi frequency Ω R (q res ) = |χ(q res )| √ N = 0.1ω 12 . As it appears from Eq. ( 14), the broadening of the spectral function is intrinsic, being associated to the continuum spectrum of frequencies ω ζ corresponding to the dressed electronic states and given by the eigenvalues of the infinite matrix in Eq. (12). At each frequency ω ζ , the magnitude of the spectral function is given by the spectral weight |µ ζ | 2 , depending on the overlap between the dressed state |N + 1, k, 1, ζ and the bare electron state

|C = c † 2,k |F N .
As shown in Eq. ( 12), the electronic eigenstates of the system are given by the Fano-like coupling between the bare electron state and the continuum of cavity polariton excitations. Indeed, the pronounced dip around ω = ω 12 in the spectral function is a quantum interference feature, typical of a Fano resonance 19 .

As we said before the sharp transition in Eq. 14 between k > k F and k < k F is only a consequence of the approximations we made of neglecting the border of the Fermi sea and the effect of the temperature. In a real case both effects will tend to smooth the transition, the first on an energy scale of the order of 2 kF qres m * and the second on an energy scale of K B T .

IV. TUNNELING COUPLING, LOSSES AND ELECTROLUMINESCENCE

The states |N + 1, k, 1, ζ have been obtained by diagonalizing the Hamiltonian (1), which takes into account only the coupling between the two-subband electronic system and the microcavity photon quantum field. If, as we have assumed, the light-matter interaction is the strongest one, all other residual couplings can be treated perturbatively. These residual interactions include the coupling to the extracavity fields, the interaction with contacts, phonon and impurity scattering as well as Coulomb electron-electron interactions 20 .

The states |N + 1, k, 1, ζ can be excited for example by resonant electron tunneling from a bulk injector or an injection miniband. If V tc k is the tunneling coupling matrix element between the state |F and c † 2,k |F induced by the coupling with the injector we have, using the Fermi golden rule, the following injection rate:

Γ inj (k, ζ) = 2π |µ ζ | 2 |V tc k | 2 ρ inj (ω ζ )n F (ω ζ ), (15)
where ρ inj (ω) is the density of electronic states inside the contact and n f (ω) its Fermi distribution. ρ inj (ω)n f (ω) determines the spectral shape of the injector. µ ζ comes from Eq. ( 14) and represents the electron spectral weight.

It is worthwhile to notice that the formula in Eq. 15 is quite independent from the model of injector considered. All the relevant informations are contained in the coupling strenght V tc k and the spectral shape ρ inj (ω)n f (ω). Any form of scattering, including in-plane wavevector non-conserving interactions or non-resonant injection, will simply give a different (and possibly broadened) injector spectral shape.

The finite transmission of the cavity mirrors is responsible for a finite lifetime for the cavity photons and consequently for the dressed states |N + 1, k, 1, ζ . By using the Fermi golden rule and a quasi-mode coupling to the extracavity field, we find that the radiative lifetime τ r,k,ζ reads:

1 τ r,k,ζ = 2π q,qz |α ζ (q)| 2 |V qm q,qz | 2 δ( ω ζ -ω q,qz )θ(k -k F ),
where V qm q,qz is the quasi-mode coupling matrix element, ω q,qz the extracavity photon frequency and α ζ (q) = A, q|N + 1, k, 1, ζ as defined in Eq. ( 6). Having calculated the tunneling injection rate and the radiative lifetime for the different states, we are able to evaluate the electroluminescence spectra. It is convenient to introduce the normalized photon emission distribution corresponding to each eigenstate |N + 1, k, 1, ζ , namely

L(q, ζ) = N qz |α ζ (q)| 2 |V qm q,qz | 2 δ( ω ζ -ω q,qz ), (16)
where the normalization N is fixed by imposing q L(q, ζ) = 1. The number of photons with in-plane wave-vector q and frequency ω emitted per unit time is

N ph (q, ω) = 1 π k,ζ Γ inj (k, ζ)L(q, ζ) 1/τ r,k,ζ (ω -ω ζ ) 2 + (1/τ r,k,ζ + 1/τ nr,k,ζ ) 2 , (17)
where the last factor accounts for the Lorentzian broadening due to radiative and non-radiative processes. τ nr,k,ζ is the non-radiative lifetime of the electronic excitations and Γ inj (k, ζ) is given by Eq. ( 15). Fig. 3 reports representative electroluminescence spectra in the case of a broadband (panel a) and narrowband (panel b,c,d) injector. In the broadband case, the emission is resonant at the intersubband cavity polariton frequencies (dashed lines) and it is significant in a wide range of in-plane wavectors 16 . In contrast, in the case of narrowband electrical injector our theory shows that the photon in-plane momentum and the energy of the cavity polariton emission can be selected by the resonant electron tunneling process, in agreement with what suggested by recent experiments 14 .

In free-space, the quantum efficiency of electroluminescent devices based on intersubband transitions is poor (≈ 10 -5 in the mid-infrared) due to the slow radiative recombination of long wavelength transitions. In the microcavity case, the efficiency of the emission from an excited state |N + 1, k, 1, ζ is given by (1 + τ r,k,ζ /τ nr,k,ζ ) -1 . Being 1/τ nr,k,ζ essentially proportional to the matter component of the excitation and 1/τ r,k,ζ to its photonic fraction, we have found that it is possible to obtain a quantum efficiency approaching unity by selectively injecting electrons into dressed states with a high photonic fraction. In particular, this is achievable by avoiding injection resonant with the central peak of the electron spectral function in Fig. 2, which corresponds to states with strong overlap with the bare electron state.

In the present theory, we have not considered the role of electronic disorder, which is known to break the inplane translational invariance. However, in the limit of large vacuum Rabi energies (i.e., significantly larger than the energy scale of the disorder potential), the inhomogenous broadening is expected to have a perturbative role.

Let us point out clearly that in order to achieve a high quantum efficiency, it is necessary to have a considerably narrow spectral width for the injector, on the order of a small fraction (10 -2 ) of the intersubband transition energy ω 12 . This is essential in order to be able to inject electrons selectively into the superradiant states, while avoiding both the peak associated to the dark excitations at the bare electron energy and the states with k < k F that can not radiatively decay. In the experiments in Ref. 14, the spectral width of the injector (a heavily doped superlattice) is comparable to the polariton vacuum Rabi frequency and hence such selective excitations of the superradiant states cannot be reached. In order to have an injector with narrower spectral width, several electronic designs could be implemented. For example, one can grow a "filter" quantum well between the superlattice injector and the active quantum well: resonant electron tunneling through the intermediate quantum well can significantly enhance the resonant character of the excitation. Moreover, for a given injector, improved microcavity samples with larger vacuum Rabi frequency would allow the system a more resonant excitation of the superradiant electronic states.

V. CONCLUSIONS

In conclusion, we have determined in a nonperturbative way the quasi-electron states in a microcavity-embedded two-dimensional electron gas. Such states originate from a Fano-like coupling between the bare electron state and the continuum of cavity polariton excitations. We have proven that these states can be selectively excited by resonant electron tunneling and that the use of narrow-band injector may give rise to efficient intersubband polariton electroluminescence. Our theoretical work shows that the strong coupling to the vacuum electromagnetic field can modify significantly the fundamental electron injection processes.

  , where the index ζ now runs over all the eigenstates of the subspace The states obtained by applying electron creation or destruction operators on the eigenstates |N, K, Q, ζ are still eigenstates of NF , K and Q. The state c † 1,k |N, K, Q, ζ is in the subspace labeled by the quantum numbers

FIG. 2 :

 2 FIG. 2: Electron spectral function A + 2 (k, ω) for the second subband , for all k > kF . The spectral function, defined in Eq. (3), is the density of quasi-electron states, weighted by the overlap with the bare electron state.The integral of the spectral function over the frequency ω is by construction equal to one. Coupling parameter: ΩR(qres) = |χ(qres)| √ N = 0.1ω12.

FIG. 3 :

 3 FIG.3: Extracavity electroluminescence spectra N ph (q, ω). Panel (a): the case of a broadband electrical injector (bandwidth equal to ω12, centered at ω = ω12). The other panels show the results for a narrow-band injector (width 0.05ω12) centered respectively at ω = ω12 (b), 1.2ω12 (c) and 0.8ω12(d).The non-radiative relaxation rate 1/τ nr,k,ζ has been taken equal to 0.005ω12. In all panels, the dashed-dotted lines are the frequency dispersions ω±(q) of the two cavity polariton branches. In the first panel the solid line represents the edge of the light cone[START_REF] Anappara | [END_REF] .

VI. ACKNOWLEDGEMENTS

qWe thanks Iacopo Carusotto, Raffaele Colombelli, Alberto Santagostino, Luca Sapienza, Carlo Sirtori, Yanko Todorov and Angela Vasanelli for discussions.