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Essays on economic inequality, income taxes, and intergenerational mobility Gutierrez Cubillos, Pablo 2020

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Essays on economic inequality, income taxes, andintergenerational mobilitybyPablo Gutie´rrez CubillosBA (Honours), University of Chile, 2010MA (Honours), University of Chile, 2014A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTSFORTHE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSDOCTORAL STUDIES(Economics)The University of British Columbia(Vancouver)December 2020c© Pablo Gutie´rrez Cubillos, 2020The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the dissertation entitled:Essays on economic inequality, income taxes, and intergenerational mobilitysubmitted by Pablo Gutierrez Cubillos in partial fulfillment of the requirements for thedegree of Doctor of Philosophy in EconomicsExamining Committee:Prof. Kevin Milligan, Economics, University of British Columbiaco-SupervisorProf. Thomas Lemieux, Economics, University of British Columbiaco-SupervisorProf. Nicole Fortin, Economics, University of British ColumbiaSupervisory Committee MemberProf. Claudio Ferraz, Economics, University of British Columbia UniversityUniversity ExaminerProf. Sanghoon Lee, Sauder School of Business, University of British Columbia UniversityUniversity ExaminerAdditional Supervisory Committee Members:Prof. Terry Moon, Economics, University of British Columbia UniversitySupervisory Committee MemberiiAbstractThe first chapter provides the first consistent estimates of intergenerational earnings mo-bility in Chile, based on administrative records that link a child’s and their parent’searnings from the formal private labour sector. We estimate that the intergenerationalearnings elasticity is between 0.288 and 0.323, whereas the rank-rank slope is between0.254 and 0.275. We find significant non-linearities in the intergenerational mobility mea-sures, where intergenerational mobility is very high in the bottom 80% of the parents’distribution but with extremely high intergenerational persistence in the upper part ofthe earnings distribution. In addition, we find remarkable heterogeneity in intergener-ational mobility at the regional level, where Antofagasta, a mining region, is the mostupwardly-mobile region. Finally, we estimate significant differences across municipalitiesin the Metropolitan Region, where our estimates suggest that the place of residence makesa significant difference in intergenerational mobility for children of upper-class families,while it is less relatively important for children of lower- and middle-class families.The second chapter proposes a new methodology to value retained earnings asincome by transforming them into accrued capital gains and develops a parametric proce-dure to impute corporate retained earnings to households. We use this approach to esti-mate income inequality for Canada using household survey data, and aggregate retainedearnings information from national accounts. We show that including retained earningsby transforming it into accrued capital gains increases income inequality in Canada andchanges the trend in income inequality, exhibiting more consistency with the decline incapital income after the Great Recession.The third chapter investigates consequences of top-distribution undercoverage onthe Gini coefficient. It shows that not correcting for underreporting and nonresponse atthe top does not necessarily result in an underestimated Gini coefficient. In addition, thispaper proposes a Gini approximation based on the Atkinson approximation to correct forunderreporting at the top. Under plausible assumptions, the approximation proposed forcorrecting underreporting at the top is near exact.. To evaluate this methodology, thispaper uses Chile and Canada as examples where we include undistributed business profitsiiito measure income inequality.ivLay SummaryI present three essays on economic inequality and intergenerational mobility in the Cana-dian and Chilean context. In the first chapter, with co-authors, we study study earningsintergenerational mobility for Chile. For this, we use a novel administrative dataset. Weshow that earnings intergenerational mobility is non-linear, with very high mobility forthe bottom 80 percent and very high persistence for the upper tail. In the second chapter,I study the effects of corporate retained earnings on income inequality in Canada. I showthat including corporate retained earnings for the measurement of income inequality canchange levels and trends of income inequality. Finally, in the third chapter, I study theGini coefficient in the context of underreporting and nonresponse in the upper tail of theincome distribution. A Gini approximation is proposed and studied.vPrefaceChapter 1 constitutes joint work with Juan Dı´az and Gabriel Villarroel. The researchpresented in Chapter 1 is covered by UBC Behavioural Research Ethics Board Certificatenumber H20-02952. Chapters 2 and 3 are original, unpublished and independent workby the author, Pablo Gutierrez Cubillos. In chapter 1 co-authors were equally involvedin all stages of the research project, including the identification of the research question,review of the literature, preparation and analysis of the data, and writing of the paper.viContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviiiIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Intergenerational mobility in Chile . . . . . . . . . . . . . . . . . . . . . . 91.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Development of the income intergenerational mobility literature . . . . . . 121.2.1 Intergenerational mobility of income, the case of developing countries 141.2.2 Intergenerational mobility of income, regional differences . . . . . . 151.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.1 Information on labour earnings . . . . . . . . . . . . . . . . . . . . 151.3.2 Information on child-parent linkage . . . . . . . . . . . . . . . . . . 171.3.3 Measurement of earnings . . . . . . . . . . . . . . . . . . . . . . . . 17vii1.3.4 Comparison between unemployment insurance program dataset andENE survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.5 Information on child residential address . . . . . . . . . . . . . . . . 191.4 Intergenerational mobility for Chile . . . . . . . . . . . . . . . . . . . . . . 201.4.1 Traditional indicators of intergenerational mobility . . . . . . . . . 201.4.2 More on non-linearities . . . . . . . . . . . . . . . . . . . . . . . . . 291.4.3 Robustness checks . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.5 Geographic variation in intergenerational mobility: the case of Chileanregions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.5.1 Chilean regional context . . . . . . . . . . . . . . . . . . . . . . . . 401.5.2 Intergenerational earnings mobility at the regional level . . . . . . . 411.6 Geographical variation in intergenerational mobility within the Metropoli-tan region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501.6.1 The Metropolitan Region . . . . . . . . . . . . . . . . . . . . . . . . 511.6.2 Estimates of intergenerational mobility . . . . . . . . . . . . . . . . 511.6.3 Geographic correlations and mobility across the Metropolitan region 551.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 Income inequality, taxes, and undistributed corporate profits: evi-dence from Canada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.2 Measure of personal income derived from retained earnings . . . . . . . . . 652.2.1 Retained earnings and ownership of the firm . . . . . . . . . . . . . 662.2.2 Retained earnings and the marginal investor . . . . . . . . . . . . . 702.2.3 Retained earnings and transaction costs . . . . . . . . . . . . . . . 722.2.4 Summary of the contexts used in the valuation and imputation ofretained earnings and drawbacks of the methodology . . . . . . . . 732.3 Imputation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.3.1 Overview of the imputation procedure . . . . . . . . . . . . . . . . 742.3.2 Estimation of ηx and w¯ . . . . . . . . . . . . . . . . . . . . . . . . . 772.3.3 Estimation of ηh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79viii2.3.4 Estimation of ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.3.5 Estimation of hi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802.4 Estimation of inequality measures with imputed corporate undistributedprofits for Canada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802.4.1 Data and definition used . . . . . . . . . . . . . . . . . . . . . . . . 812.4.2 Inequality measures including accrued capital gains . . . . . . . . . 852.5 Discussion of the methodology and assumptions . . . . . . . . . . . . . . . 892.5.1 Contrasting the parametric estimation using a capitalization approach 892.5.2 Evaluation of the ranking preservation assumption . . . . . . . . . . 932.5.3 The effect of changing pˆ . . . . . . . . . . . . . . . . . . . . . . . . 942.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953 Gini and undercoverage at the upper tail: a simple approximation . . 973.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.2 Gini coefficient and undercoverage at the top . . . . . . . . . . . . . . . . . 1003.2.1 Underreporting at the top . . . . . . . . . . . . . . . . . . . . . . . 1013.2.2 Nonresponse at the top . . . . . . . . . . . . . . . . . . . . . . . . . 1033.2.3 Nonnegative underreporting and nonresponse: the joint case . . . . 1053.3 A Gini approximation for undercoverage at the top . . . . . . . . . . . . . 1063.3.1 An approximation as a solution for underreporting at the top . . . 1063.3.2 A Gini approximation as a solution for nonresponse at the top . . . 1093.3.3 A Gini approximation for underreporting and nonresponse at the top1103.3.4 The underreporting vs the nonresponse approximation . . . . . . . 1113.3.5 Montecarlo simulation . . . . . . . . . . . . . . . . . . . . . . . . . 1133.4 An extension of the Atkinson approximation in the case of nonresponse . . 1153.5 Empirical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.5.1 Application. Income inequality and undistributed business profits. . 1183.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127ixA Appendix for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141A.1 Data appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141A.2 Additional regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144A.3 Penn parade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145A.4 Additional Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148A.5 More on non-linearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152A.6 Why imputation-based IGE estimates may fail: the importance of admin-istrative data use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153B Appendix for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160B.1 A stochastic model for income process . . . . . . . . . . . . . . . . . . . . 160B.1.1 Effect of the inclusion of retained earnings in the stochastic incomemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163B.2 Standard error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 164B.2.1 Estimation of standard errors . . . . . . . . . . . . . . . . . . . . . 164B.3 Proof of proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165C Appendix for chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168C.1 Tables for empirical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 168xList of Tables1.1 Representativity of the unemployment insurance program dataset. . . . . . 161.2 Comparison of earnings between our dataset and ENE for individuals be-tween 28-33 years old. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3 OLS estimates of the intergenerational earnings elasticity for our baselinelinkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.4 OLS estimates of the intergenerational earnings elasticity for female children 221.5 OLS estimates of the intergenerational earnings elasticity for male children 221.6 OLS estimates of the rank-rank correlation for our baseline linkage . . . . . 251.7 OLS estimates of the rank-rank correlation for our female children . . . . . 261.8 OLS estimates of the rank-rank correlation for our female children . . . . . 261.9 Transition matrix of parental earnings quintiles to child earnings quintiles . 271.10 Decile Transition Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.11 91st to 100th parental percentile to 91st to 100th child percentile transitionmatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.12 Estimations of IGE and rank-rank slope for different years where parentalearnings were measured. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.13 Estimates of IGE and rank-rank slope for different child ages. . . . . . . . 381.14 Estimates of IGE and rank-rank slope using different years to averageparental earnings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.15 Intergenerational mobility indicators for different Chilean regions. . . . . . . . . . . 441.16 Correlation between mobility measures and socio-economic characteristics . 572.1 Value of retained earnings (θ) given different contexts . . . . . . . . . . . . 742.2 Totals used in the estimation of income inequality . . . . . . . . . . . . . . 842.3 Parameter estimatets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85xi3.1 Montecarlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114A.1 Linkage units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141A.2 Educational linkages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142A.3 Age distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142A.4 Sex distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143A.5 Period robustness checks for IGE . . . . . . . . . . . . . . . . . . . . . . . 144A.6 Age robustness checks for IGE . . . . . . . . . . . . . . . . . . . . . . . . . 145A.7 Period robustness checks for rank-rank correlation . . . . . . . . . . . . . . 146A.8 Age robustness checks for rank-rank correlation . . . . . . . . . . . . . . . 146A.9 Descriptive statistics of the UIP database . . . . . . . . . . . . . . . . . . . 148A.10 Regional information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150A.11 Intergenerational mobility indicators by municipality in the Metropolitanregion. “Santiago” refers to the municipality, not the city. . . . . . . . . . . 151A.12 Results from simulated exercise 1 . . . . . . . . . . . . . . . . . . . . . . . 156A.13 Results from simulated exercise 1 with additional variance . . . . . . . . . 157A.14 Results from simulated exercise 1 with additional variance and more pre-diction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158C.1 Data for Canada. µ in current Canadian Dollars . . . . . . . . . . . . . . . 168C.2 Data for Chile. µ in current Chilean Pesos . . . . . . . . . . . . . . . . . . 168xiiList of Figures1.1 Expected child ranking conditional on parental ranking . . . . . . . . . . . 241.2 International comparison of expected child earnings ranking conditional tothe parental earnings ranking . . . . . . . . . . . . . . . . . . . . . . . . . 281.3 conditional (on parental deciles) child earnings distribution . . . . . . . . . 331.4 onditional (on parental percentiles in the top decile) child earnings distri-bution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.5 Unconditional quantile and conditional quantile estimates of the regressionslope of log child earnings vs log parental earnings . . . . . . . . . . . . . . 361.6 Expected child ranking conditional on parental national ranking for 4 dif-ferent regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421.7 Heat maps for absolute upward mobility in Chilean regions . . . . . . . . . 461.8 Heat maps for relative mobility in Chilean regions . . . . . . . . . . . . . . 471.9 Heat maps for circle of poverty p11 transition probability for Chilean regions. 481.10 Heat maps for circle of privilege p55 transition probability for Chilean regions. 491.11 Gatsby curve Chilean regions . . . . . . . . . . . . . . . . . . . . . . . . . 501.12 Heat maps for absolute upward mobility and relative mobility indicatorsfor Metropolitan region municipalities. . . . . . . . . . . . . . . . . . . . . 521.13 Heat maps for circle of poverty p11 and circle of privilege p55 transitionprobabilities for Metropolitan region municipalities. . . . . . . . . . . . . . 531.14 Gatsby curve Metropolitan Region municipalities . . . . . . . . . . . . . . 552.1 Lorenz curves for market income and parametric imputed income . . . . . 862.2 Gini coefficient with and without parametric imputed income . . . . . . . . 882.3 Top 10% and top 1% with and without parametric income. . . . . . . . . . 892.4 Lorenz curves of capitalized income vs market income . . . . . . . . . . . . 91xiii2.5 Gini coefficients of capitalized income, parametric imputed income andmarket income . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922.6 Top 10% and top 1% of capitalized income, parametric imputed incomeand market income. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932.7 Share of the income before pˆ and after pˆ . . . . . . . . . . . . . . . . . . . 942.8 Top 10% and top 1% of capitalized income, parametric imputed incomeand market income (with pˆ and p = 0.99) . . . . . . . . . . . . . . . . . . . 953.1 Relative Lorenz curves with underreported income at the top . . . . . . . . 1023.2 Lorenz curve in a context of nonresponse at the top . . . . . . . . . . . . . 1043.3 Relative Lorenz curves in a context of underreported income and nonre-sponse at the top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.4 Relative Lorenz curves of both underreporting and nonresponse problems . 1123.5 Corrected vs uncorrected Gini coefficient for Chile . . . . . . . . . . . . . . 1203.6 Corrected vs uncorrected Gini coefficient for Canada . . . . . . . . . . . . 121A.1 Pen parade for parental earnings . . . . . . . . . . . . . . . . . . . . . . . . 147A.2 Pen parade for child earnings . . . . . . . . . . . . . . . . . . . . . . . . . 147A.3 Regions of Chile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149xivAcknowledgementsDoing a PhD can feel like a Greek epic. Sometimes you think that everything is lost (evenyourself!), but then, a Deus ex Machina event happens and suddenly you are finishingyour thesis without even realizing it.My Deus ex Machina, or god from the machine, was the good guidance I receivedduring the process, so the first person that I would like to thank is my supervisor KevinMilligan for his unflinching support, adamant confidence, and excellent advice throughoutthis process. Likewise, I wouldn’t be at this point if it wasn’t for Nicole Fortin, who alwayspushed me to be a better researcher and who was the person that motivated me to writethe first chapter of this thesis. I also cannot forget thanking Thomas Lemieux for his goodresearch advice and for the valuable time we spent discussing my naive research ideas,and Terry Moon for his precise guidance. Also, I would like to thank to David Green,Craig Riddell, and Erik Snowberg for giving me the opportunity to work as an RA, andMaureen for all the good advices and help in administrative decisions.Doing a PhD is not a process one undertakes alone. As such, I would like tothank to my mother Isabel and my father Antonio for their emotional support. Evenwhen I called them late or frustrated about the process, they were there for me. I alsowant to thank t´ıa Eli and Verito for their tremendous and altruistic support during thepandemic. Their encouragement was key to finish this research. Additionally, I wouldlike to thank Iris for her good advice and patience during the pandemic.Also, I would like to thank to my friends for their motivation and support. Firstof all, I would like to thank Juanito. We started the PhD application process together andwe supported each other during different stages of this process. This comradery lasteduntil the very end, as the first chapter of the thesis was co authored with him. Second, Iwould like to thank to JF ”El pelao” for his partnership and honest friendship. Despitebeing classmates in the same PhD cohort and roommates for two years our relationshipwas characterized by supportiveness rather than combativeness. Even now we still supporteach other with the collateral damage that a PhD might bring (aka. sindrome del gato).Moreover, I would like to thank Pablito Troncoso. His encouragement, friendship andxvoptimism were and are valued.Another person that I would like to thank is Javier (aka. compan˜ere) for hisgenuine friendship and support. Also, I would like to thank Victor ”El maestrito”, hewas the very first person I met in Vancouver. We later met in Paris where he providedme with accomodation for more than 10 days. He is both a remarkable researcher andeven better person. In addition, I would like to thank the support of the 1st year PhDgang, aka. Don Davide and Dona Maria, Anand, Vinicius and Sev. All the good dinnersand company will never be forgotten. Moreover, help and friendship from Luchito is veryappreciate. I would also like to thank my classmates: Neil, Jeff, Jasmine, Hao, Hugh andJustin.Another key part of this process is the relationship with your co-authors (men-tors). In this regard, I would like to thank to Claudio for his confidence, motivation andtrust on my process. The same goes with Eugenio, my former supervisor, now co-author.Thank you for your motivation and for the good advice. I would also like to extendmy thanks to Ramo´n who introduced me to the research business and who lent me hisfriendship and support during the process. In addition, I would like to thank Juan PabloTorres-Martinez, for his patience and for being a prime example of what scholarship trulymeans, and Pablo Tapia for his encouragement and motivation. Moreover, the relation-ship with my young co-authors is also important, I am really grateful to Gae¨lle, Nacho,Vania and Mancu for their patience and willingness to work with me.The PhD process is a journey. During this journey I lived in Ottawa for 18months. Here I worked for the Canadian government where I received tremendous supportfrom Eric, Christine, Andrew, Patrick, Jacinthe, Carlos, Danny, George, Brian, Rebekha,Jen and Wendy (for her patience and diligence in the RDC).Even back at home there are friends and people that help you along this process.I am indebted to Tito, Mito, Dieguito, Sergio, Didi, Isabel and Pipo for their friendshipand support and all the teachers that help me during my formation such as Antonio, Luis,et al.The PhD process is also an immigration process, it is about understanding a newxviculture while also missing your own one. So, I would like to thank to all my friends thatI met during this process, Mario, Roni, Rolo, Karla, Mati, Seba, Carito, Chedo, Mari,Tata et al. Many thanks for making me feel like at home.Finally, I would like to thank to Andrea for her love, her loyal and unconditionalsupport, even in the darkest hours, and for making this journey all the more exiting andmeaningful.xviiA mi madre, quien siempre me ha motivado a seguir adelante a pesar de lasdificultades. Sin su apoyo, ejemplo de vida y visio´n no existir´ıa esta tesis. Y tambie´n ami padre por siempre confiar en mi y por tener la sabidur´ıa de escuchar y seguir a mimadre.xviiiIntroductionIn this thesis, I present three essays on economic inequality and intergenerational mobilityin the Canadian and Chilean context. The first chapter studies earnings intergenerationalmobility for Chile by estimating the correlation between parent’s and Children’s earningsusing a novel administrative dataset. The second chapter estimate income inequalityin Canada adding corporate retained earnings to households. To do this, I develop avaluation and an imputation methodology. The third chapter study the Gini coefficientin the context of underreporting and nonresponse at the upper tail. We study a verysimple and useful Gini approximation to solve those data issues and we provide twoempirical examples, for Canada and Chile.In the first chapter, along with co-authors, we study intergenerational mobilityin Chile by building a new and unique data set after assembling three administrative datasources. We obtain information on labour earnings of children and their parents from2002 to 2019 from the database of the Chilean government’s unemployment insuranceprogram (UIP). We link children and their parents using administrative records providedby the Civil Registry Office. We obtain the place of residence of a child when she wasbetween 13 and 18 years old from administrative records at the Ministry of Education.To the best of our knowledge, this is the first work that uses administrative informationto estimate intergenerational mobility for a non-advanced economy.We estimate intergenerational earnings mobility at national level. We find thatit is highly non-linear in Chile, and that it is extremely mobile for the bottom 80 percentof the earnings distribution, even more mobile than in advanced economies such as theUS and Canada. But it is also highly persistent for the upper decile of the earnings1distribution, much more so than for any advanced economy. This result resembles whatBratsberg et al. (2008) finds for comparing the Nordic countries with the US and UK.We also estimate intergenerational earnings mobility at the regional level. This isof particular interest for a country like Chile, where the climate and economic conditionsare significantly heterogeneous across its geography. We find that the most mobile regionis Antofagasta, which is a miner-intensive region located in the north of the country.This result is in line with the findings for developed economies (Australia and Canada).Meanwhile, the least mobile region is Araucan´ıa, where about a third of the populationis ethnic Mapuche (an indigenous population) - the highest proportion of any region inChile.Finally, we estimate intergenerational mobility across different municipalities forthe Santiago Metropolitan Region. This region contains the nation’s capital, Santiago,one of the cities with a better quality of life in South America. We find that Santiagois extremely heterogeneous in upward mobility, circles of poverty and circles of privilege.In particular, there is a cluster of rich municipalities where the conditional probabilitythat child stays in the fifth quintile given that the parent was in the fifth quintile of theirearnings distribution is higher than 0.7. Those rich municipalities are quite similar interms of upward mobility.We also make a methodological contribution; we use for the first time tools toestimate intergenerational mobility at the top of the distribution such as RIF regressionsand Kernel conditional densities. In addition, we estimate the Gatsby curve for Chile andSantiago using two measures of intergenerational mobility: absolute intergenerationalmobility and relative intergenerational mobility. We show that the Gatsby curve could bevalid for a persistence indicator but not for an absolute mobility indicator. Meaning thatinequality could be related with persistence at the top instead of mobility at the bottom.Of course, there is a vast body of literature from economists trying to learn aboutsocial mobility from administrative records in advanced economies. For the United States,there is a series of articles that are based on a project by Raj Chetty, Nathaniel Handrenand others, who use administrative tax data to estimate the intergenerational elasticity of2income.1 For example, the work of Chetty et al. (2014) studies how social mobility variesthrough geographic zones called community zones in the US. For Canada, the literature onintergenerational income mobility starts with the seminal work of Corak and Heisz (1999),a pioneering paper in the use of administrative data to study intergenerational mobilityof income. More recently, Corak (2019) studies intergenerational mobility in Canadautilizing census data and analyzing data at various geographic levels. Europe has alsoproduced some interesting literature in this regard. For instance, Acciari et al. (2019) usetax data to investigate how intergenerational mobility varies geographically for Italy, as doGu¨ell et al. (2015) for social mobility at smaller geographical units in Italy, which Heidrich(2015) also does for Switzerland. Most of these works for developed countries show thatdisaggregated geographical measures of intergenerational mobility provide evidence ofsignificant heterogeneities across locations that are hidden in country-level estimates.In the case of Chile, our work does not emerge in a vacuum. Over the last twodecades, some papers have made progress in understanding social mobility by using sur-vey data. For example, Nu´n˜ez and Miranda (2010, 2011) study intergenerational incomemobility by using the Two-Sample Two-Stage Least Squares (TSTSLS) methodology de-veloped by Bjo¨rklund and Ja¨ntti (1997). Sapelli (2013) provides evidence on changesin the intergenerational mobility of education through time, using several cross-sectionsurveys. Meanwhile, Torche (2005) analyzes the intergenerational mobility of educationbased on survey data, and Celhay et al. (2010) focus on the study of intergenerationalmobility of income and schooling for the period 1996-2006 using longitudinal surveys.The only paper that uses administrative records to capture a specific dimension of inter-generational mobility in Chile is the work of Zimmerman (2019). Based on a regressiondiscontinuity design, this article exemplifies the lack of upward mobility by showing thatstudying at an elite university has a positive effect on obtaining a managerial positionwith high income in the labour market, but only for those with a high-level socioeconomicbackground who had studied at an elite private school. In part, this study quantifies theimportance of contact networks in the generation of inequality in Chile.1In this paper, we make the distinction between earnings, for which the source is wages, and income,for which the sources are wages and financial asset income. Our study is developed with earnings due tothe available dataset.3The second chapter studies the effect of corporate retained earnigns on incomeinequality. Thus, the use of retained earnings allows us to have an accrual measure ofcapital gains. However, the money inside the firm has a different value from that outsidethe firm. If an agent wants to get the money out of the company, he or she has to paypersonal taxes (depending on the tax system he or she may receive a tax credit for thecorporate taxes paid by the firm). The financial market may adjust for those future taxesdecreasing the capital gains generated by this retained earning. That is, the tax systemshould be taking into account to measure the income from capital gains associated withretained earnings. In addition, if we consider that the marginal investor is a foreigner, asBoadway and Bruce (1992) states, the domestic tax system is irrelevant for the marginalinvestor. Thus, it is crucial to identify what is the right tax rate that should be used tovalue retained earnings as capital gains. The same is true if there are some transactioncosts to get the money out of the firm. Thus, the first contribution of this work is todevelop a conceptual framework that analyzes the effect that the ownership of the firm,the tax system and the equilibrium in capital markets have on the capital gain generatedvia retained earnings.We do not have access to administrative data. To overcome this limitation, wepropose a parametric methodology to impute corporate retained earnings to families usinghousehold survey microdata and aggregate national account information. This procedureis based on the exponential-Pareto model established by Dragulescu and Yakovenko (2000)and Silva and Yakovenko (2004).2 In addition, this method follows the spirit of Jenkins(2017) and Hundenborn et al. (2018). It uses survey data for a fraction p of the populationand aggregate national account data as an additional source for the remaining 1− p.3 Toevaluate the pertinence of this parametric imputation methodology, we compare it witha non-parametric imputation procedure a capitalization approach similar to that applied2Others studies that use an exponential distribution for the bottom part of the income distributionare Banerjee et al. (2006) and Jagielski and Kutner (2013).3The use of survey data is justified because in some countries it is not mandatory for individualsmaking certain income levels to file a tax declaration; thus, data that are just generated only from taxdeclarations may have a bias in the lower tail of the income distribution. Also, some developing countriesdo not have another reliable data source than a household survey, or it is politically difficult to get accessto administrative data.4by Saez and Zucman (2016).4 This method contributes to an extensive literature thatestimate income and wealth inequality using parametric methods such as Kleiber andKotz (2003), Chotikapanich, Griffiths and Rao (2007), Clementi and Gallegati (2016),among many others.The valuation and imputation methodology is tested empirically for Canada,where we impute corporate retained earnings and then use the generated data to computeincome inequality measures. To do so, we use the Survey of Consumer Finances (SCF)for 1984 and the Survey of Financial Security(SFS) for 1999, 2005, 2012 and 2016. Thejustification for using these surveys is that they can be harmonized, allowing us to make acomparison with the capitalization method used by Saez and Zucman (2016) because thosesurveys have rich information on assets in addition to income.5 The inequality measuresestimated here are not as precise as those estimated by Saez and Veall (2005), Veall (2012)and Wolfson et al. (2016). Despite this, it has the value of correcting a household surveyfor a form of under-reporting such as Burkhauser et al. (2011), Bourguignion (2018),Blanchet, Flores and Morgan (2018) among many others.Empirically, we find that the inclusion of corporate retained earnings and itsmeasure as accrued capital gains increase the estimated measure of income inequality andthat this also affects the trend in income inequality. Indeed, for 2005, the share of thetop 1% increases by 4.5 percentage points (from 7.8% to 12.3%), and the Gini coefficientincreases by 4.4 points (from 47.5 to 51.9) which implies higher income inequality thanin 2012 and 2016; this was not the case before accrued capital gains were considered.Those results are robust to the method used to impute corporate retained earnings. Inthis context, this work contributes to a broader literature that study Canadian incomeinequality such as Saez and Veall (2005), Fortin et al. (2012), Lemieux and Riddell (2015),Milligan and Smart (2015), Wolfson et al (2016), Green, Riddell and St-Hilaire (2016),among many others.4One reason to establish another imputation procedure is that, as Kopczuk (2016) states, “In anenvironment with a low rate of return, a small bias in the estimated rate of return has large consequenceson the estimations of wealth inequality.”5Dividends could also be used to impute corporate retained earnings. However, as Alstæaeter et al.(2017) shows, this is not a good procedure to impute retained earnings because (i) retained earningsand dividends move in different directions, (ii) mechanically, the imputations results are not adequate inperiods in which the aggregated retained earnings are negative.5Chapter 3 studies the effects of two types of undercoverages at the top of theincome distribution: underreporting (i.e., missing income) and nonresponse (i.e., missingpeople). Underreporting occurs when individuals in a population report less income orwealth than they earn (e.g., tax evasion, top coding, information omissions in householdsurveys). On the other hand, nonresponse occurs when individuals in a population areunrecorded in the data source (i.e., truncated data; e.g., people not submitting theirhousehold surveys or not declaring taxes). Bourguignon (2018), Lustig (2018), Blanchet,Flores and Morgan (2018) recently studied these two missing-information types. Theirworks discuss how adjustments for these biases affect income-inequality measures. Inparticular, Lustig (2018) develops a taxonomy to differentiate the different types of un-dercoverage at the upper tail. Bourguignon (2018) shows, in a didactic manner, howdifferent adjustments in the upper tail affect the income distribution. In particular, heargue that the adjustments of the original data relies on three key parameters: i) Howmuch is to be allocated to the top of the distribution; ii) how broad should the top b;iii) what share of the population should be added to the top. Finally, Blanchet, Floresand Morgan develops a novel methodology to find the point where tax data describesbetter the income distribution than survey data. Their method can be used to correct forunderreporting and nonresponse at the top.In this chapter we depart from Bourginon (2018) and Blanchet, Flores and Mor-gan, instead of studying the whole income distribution, we only study the effects un-derreporting and nonresponse in the Gini coefficient. Our first contribution is that wedemonstrate that not correcting for underreporting and nonresponse at the top does notnecessarily result in an underestimated Gini coefficient.To correct the Gini coefficient for undercoverage at the top, Atkinson (2007) pro-poses a simple and pragmatic approximation. He uses household-survey information andtax data, and he approximates the Gini index as G = G1−p(1 − Sp) + Sp, where G1−p isthe Gini coefficient computed from a household survey representative of a population’spoorest 1 − p percent, and where Sp is the income share owned by the population’s topp percent (e.g., the share of the top 1%) and computed from income-tax data. Alvaredo(2011) further develops this procedure and analytically derives and extends his formula,6proposing an exact Gini decomposition to be used when a p proportion of the populationis not well measured in a data source but is better measured in another source. However,Alvaredo’s decomposition requires additional information: it depends on (i) the value ofp and (ii) the income distribution within the 1 − p population.6 Some scenarios lackinformation on either of these elements (e.g., measuring either income inequality addingundistributed profits or tax-haven wealth7). A modified version of the Atkinson approx-imation can be used under such information scarcity and thereby can correct the Ginicoefficient.Thus, this chapter’s second contribution is that it proposes a simple approxima-tion of the Gini coefficient in the case of underreporting at the top which is a slightlymodified version of the traditional Atkinson approximation without the necessity of know-ing the size of the top p. In addition, the approximation’s analytical bias is computed. Inaddition, we show that the bias is higher when the traditional Atkinson approximation isused for solving nonresponse and underreporting instead of the new adjusted formula tocorrect underreporting. It shows, numerically, that the proposed approximation is nearexact when used to correct the Gini coefficient for underreporting but may be heavilyupward biased for correcting the Gini coefficient for nonresponse. That is, in order to usethe underreporting methodology we need to first correct for missing people at the top.Thus, this paper’s third contribution is to propose and apply a methodologyfor estimating the missing proportion at the upper tail. Thus, we can estimate an un-derreporting and nonresponse corrected Gini coefficient by estimating the proportion ofnonrespondants and then apply the underreporting correction. It applies this method-ology to two countries: Chile and Canada, and corrects the income Gini coefficient byadding undistributed business profits, a source of capital income underreported in house-hold surveys, and administrative-tax-declaration data. Indeed, as Smith, Yagan, Zidarand Zwick (2019) argue “a primary source of top income is private “pass-through” busi-ness profit, which can include entrepreneurial labour income for tax reasons”, thus some6As was discussed by Cowell and Flachaire (2015) and Higgins, Lustig and Vigorito (2018) estimatingthis p is a major challenge.7Issues that are tremendously relevant for inequality measurement, see for instance, Alstadsæter etal. (2017, 2018).7part of labour income is transformed into capital income and left inside the firm. Indeed,some tax reforms induces to keep business income inside the firm whereas others generateincentives to take out profits as dividends (see for instance the 2003 US dividend tax re-form). Thus, not accounting for undistributed business income when we measure incomeinequality could lead to artificial changes that bias levels and trends of income inequalityestimates. Thus, the methodology developed here could be used to estimate level andtrends of income inequality that are robust to tax changes and tax avoidance behaviour.8Chapter 1Intergenerational mobility in Chile1.1 IntroductionThis paper asks whether the association between parents’ and their child’s earnings inChile varies with parental earnings level and children’s place of residence. Chile is aninteresting case study not only due to having made significant progress in its economicdevelopment in the last three decades (reaching a GDP per capita of US$ 16,143 in 2018,IMF, 2018) but also because it is one of the countries with the most unequal incomedistribution in the world. It has a Gini index of 0.477 points (World Bank, 2017), andthe fraction of the country’s total income received by the richest 10% of the population isextremely high (37.1%) when compared to the OECD average of 24.7% (OECD, 2018).Moreover, conservative estimates suggest that the share of total income that the richest1% take is 15%, while less conservative estimates establish it at 22-26% (Fairfield andJorrat, 2016; Flores et al. 2019).Under what conditions an unequal society can be tolerated is a subject of long-standing debate, especially in Chile. Supporters of meritocracy argue that economicinequality can be legitimated in a society if income differences stem from differences inreward for talent, hard work and skill, but not due to luck or transmission of advantages.According to this view, income inequality should not be tolerated in a society with less so-cial mobility and greater transmission of privileges or disadvantages from parent to child,9where children born in poverty (richness) remain in poverty (richness) in their adulthood,regardless of their skills or efforts. In part, the de-legitimization of income inequality isone of the main causes behind the social outbreak that occurred in Chile in October 2019,when the perception of unfairness in the distribution of income and privileges provokedlower and middle classes to take to the streets to express their indignation with the currentsituation. In this context, understanding social mobility in Chile is crucial to disentanglethe origins of its current levels of economic inequality.In this paper, we study intergenerational mobility in Chile by building a new andunique data set after assembling three administrative data sources. We obtain informationon labour earnings of children and their parents from 2002 to 2019 from the database ofthe Chilean government’s unemployment insurance program (UIP). We link children andtheir parents using administrative records provided by the Civil Registry Office. Weobtain the place of residence of a child when they were between 13 and 18 years old fromadministrative records at the Ministry of Education. To the best of our knowledge, this isthe first work that uses administrative information to estimate intergenerational mobilityfor a non-advanced economy.We estimate intergenerational earnings mobility at the national level. We findthat it is highly non-linear in Chile, and intergenerational mobility is very high for thebottom 80 percent of the earnings distribution, and exceed the rate of intergenerationalmobility in advanced countries such as the US and Canada. But earnings are also highlypersistent for the upper decile of the earnings distribution, much more so than for anyadvanced economy.1 We can summarize this finding as: intergenerational churning, andsocio-economic uncertainty, for the masses contrasts with secure inherited privilege forthe elite.We also estimate intergenerational earnings mobility at the regional level. This isof particular interest for a country like Chile, where the climate and economic conditionsare significantly heterogeneous across its geography. We find that the most mobile regionis Antofagasta, which is a minering intensive region located in the north of the country.1This result resembles what Bratsberg et al. (2008) find when comparing the Nordic countries withthe US and UK.10This result is in line with the findings for developed economies (Australia and Canada).Meanwhile, the least mobile region is Araucan´ıa, where about a third of the populationis ethnic Mapuche (an indigenous population) - the highest proportion of any region inChile.Finally, we estimate intergenerational mobility across different municipalities forthe Santiago Metropolitan Region. This region contains the nation’s capital, Santiago,one of the cities with a better quality of life in South America. We find that Santiagois extremely heterogeneous in upward mobility, circles of poverty and circles of privilege.In particular, there is a cluster of rich municipalities where the conditional probabilitythat child stays in the fifth quintile given that the parent was in the fifth quintile of theirearnings distribution is higher than 0.7. Those rich municipalities are quite similar interms of upward mobility.We also make a methodological contribution, we use for the first time tools toestimate intergenerational mobility at the top of the distribution such as RIF regressionsand Kernel conditional densities. In addition, we estimate the Gatsby curve for Chile andSantiago using two measures of intergenerational mobility: absolute intergenerationalmobility and relative intergenerational mobility. We show that the Gatsby curve could bevalid for a persistence indicator but not for an absolute mobility indicator. Meaning thatinequality could be related with persistence at the top instead of mobility at the bottom.Of course, there is a vast body of literature from economists trying to learn aboutsocial mobility from administrative records in advanced economies. For the United States,there is a series of articles that are based on a project by Raj Chetty, Nathaniel Handrenand others, who use administrative tax data to estimate the intergenerational elasticity ofincome.2 For example, the work of Chetty et al. (2014) studies how social mobility variesthrough geographic zones called community zones in the US. For Canada, the literature onintergenerational income mobility starts with the seminal work of Corak and Heisz (1999),a pioneering paper in the use of administrative data to study intergenerational mobilityof income. More recently, Corak (2019) studies intergenerational mobility in Canada2In this paper, we make the distinction between earnings, for which the source is wages, and income,for which the sources are wages and financial asset income. Our study is developed with earnings due tothe available dataset.11utilizing census data and analyzing data at various geographic levels. Europe has alsoproduced some interesting literature in this regard. For instance, Acciari et al. (2019) usetax data to investigate how intergenerational mobility varies geographically for Italy, as doGu¨ell et al. (2015) for social mobility at smaller geographical units in Italy, which Heidrich(2015) also does for Switzerland. Most of these works for developed countries show thatdisaggregated geographical measures of intergenerational mobility provide evidence ofsignificant heterogeneities across locations that are hidden in country-level estimates.In the case of Chile, our work does not emerge in a vacuum. Over the last twodecades, some papers have made progress in understanding social mobility by using sur-vey data. For example, Nun˜ez and Miranda (2010, 2011) study intergenerational incomemobility by using the Two-Sample Two-Stage Least Squares (TSTSLS) methodology de-veloped by Bjo¨rklund and Ja¨ntti (1997). Sapelli (2013) provides evidence on changesin the intergenerational mobility of education through time, using several cross-sectionsurveys. Meanwhile, Torche (2005) analyzes the intergenerational mobility of educationbased on survey data, and Celhay et al. (2010) focus on the study of intergenerationalmobility of income and schooling for the period 1996-2006 using longitudinal surveys.The only paper that uses administrative records to capture a specific dimension of inter-generational mobility in Chile is the work of Zimmerman (2019). Based on a regressiondiscontinuity design, this article illustrates the lack of upward mobility by showing thatstudying at an elite university has a positive effect on obtaining a managerial positionwith high income in the labour market, but only for those with a high-level socioeconomicbackground who had studied at an elite private school. This study, in part, quantifies theimportance of contact networks in the generation of inequality in Chile.1.2 Development of the income intergenerational mo-bility literatureAre the children of the poor doomed to stay poor? Are the children of the rich destined tostay rich? How difficult is it for someone who was born poor to belong to the middle class12during her adulthood? These questions have been addressed at the international level,where there is vast literature on intergenerational income mobility. Ja¨ntti and Jenkins(2015) and Corak (2013) summarize the historical results in this literature. Corak andHeisz (1999) were the first to use high-frequency administrative data on the income ofparents and children in adulthood in their seminal study on intergenerational mobility inCanada.3 This study was so innovative and ahead of its time that it took 15 years forliterature to replicate this study for other countries. In fact, thanks to the developmentof computer science and generalization in the use of administrative data, the literature ofintergenerational mobility has been given a new lease of life. The works of Chetty et al.(2014), Chetty et al. (2017), and Chetty et al. (2018a, 2018b) have extensively studiedintergenerational mobility in the United States using the same type of data.Undoubtedly, the novelty of these studies is in the data used, which mainly cor-respond to confidential high-frequency administrative data that cover a sufficiently longperiod and link the income of the parents with the adult income of their children. Theadvantage of administrative data is that they do not have the traditional problems presentin household surveys. In fact, traditional household surveys in general are not longitudi-nal but cross-sectional, which makes it difficult to obtain information on the income ofthe parent and child in adulthood. In addition, household surveys have problems such assampling, self-reporting and non-response, and it is known that non-response rises as therespondent’s income increases (Bollinger et al., 2018).Understanding the intergenerational mobility of income in the United States hasbeen tremendously important in understanding the generation of inequality. There is aseries of articles that are based on a project by Raj Chetty, Nathaniel Handren and others,who use administrative tax data to estimate the intergenerational elasticity of income.The work of Chetty et al. (2014) studies geographic zones called community zones. Theabovementioned investigation by Chetty and others differentiate between absolute andrelative intergenerational mobility, which has been of interest to both politicians and re-searchers. The Canadian literature on intergenerational income mobility starts with the3Others important studies on intergenerational mobility for Canada are Fortin and Lefebvre (1998),and Simard-Duplain and St-Denis (2020)13seminal work of Corak and Heisz (1999), pioneering in the use of administrative datato study intergenerational mobility of income. More recently, Corak (2019) studied in-tergenerational mobility in Canada, using census data and analyzing intergenerationalmobility within Canada at a geographic level. Acciari, Polo and Violante (2019) investi-gate intergenerational mobility for Italy by taking tax data, also analyzing what happensgeographically. Finally, this literature has also progressed in Europe, mostly based in theNordic countries. Ja¨ntti (2006) illustrates very well the use of these data. Also, thereare the studies for Switzerland by Heidrich (2015) and Gu¨ell et al. (2015) for Italy. Bothstudies are at the provincial and inter-country levels.1.2.1 Intergenerational mobility of income, the case of develop-ing countriesResearch on intergenerational mobility of income in developing countries faces additionalcomplications. Having longitudinal data that gather parents and children is very difficult(Daude and Robano, 2015, Neidho¨fer, 2019, Neidho¨fer et al., 2018) due to the limitationof household surveys and/or the difficulty of accessing administrative data.One way to address the limitations of the data is to restrict the analysis to childrenand parents living in the same household or to impute an income for the parents basedon multiple waves of a household survey. For example, Lambert et al. (2014) studiesintergenerational mobility in Senegal and Torche (2014) summarizes intergenerationalmobility in Latin America from studies that have used surveys as a primary source ofinformation.Recently, progress has been made to investigate intergenerational mobility usingcensus data from 26 African countries (Alesina et al., 2019) and for the regions of India,Asher et al. (2018). In this context, our research project will be pioneering in LatinAmerica because it uses administrative data, which is the way in which the frontierliterature is studying intergenerational mobility.141.2.2 Intergenerational mobility of income, regional differencesRecent literature has concentrated on studying the regional differences that exist withincountries.4 They find that regional intergenerational income mobility behaves differentlyamong countries. Chetty and Corak find differences among regions, where there arecertain territories that have less intergenerational mobility than other parts. However,for Switzerland, Heidrich (2015) does not find many differences. In the Chilean case,Nu´n˜ez and Miranda (2011) find that the intergenerational mobility of income is higher inSantiago compared to the Chilean average. Inequality has been studied at the regionallevel in Chile. However, how regional intergenerational mobility varies in Chile has notbeen studied.1.3 Data1.3.1 Information on labour earningsWe obtain the information on labour earnings of children and their parents from thedatabase of the UIP in Chile. The UIP is a benefit that covers all employees in the privatesector over 18 years old and with a formal contract, whether fixed-term or permanent.Participation in the scheme is mandatory for all contracts started after September 2002and voluntary for contracts started before that date. This means that these administrativerecords contain the monthly labour earnings of all employed workers over the age of 18 whoinitiated a work-under-contract relationship in the private sector from October 2002 toDecember 2019. This data set also includes the workers with labour contracts establishedprior to October 2002 who voluntarily joined the UIP. It is worth mentioning that thisdata set excludes workers with training contracts, workers under the age of 18, domesticworkers, pensioners, self-employed or own-account workers, and public sector employees.Table 1.1 provides information on the proportion of workers covered by the UIPover several years. As can be seen, due to the voluntary retroactive nature of the UIP4See Chetty et al. (2014), Chetty et al. (2018a, 2018b), Corak (2019), Gu¨ell et al. (2015), Heidrich(2015) Connolly et al. (2018).15policy, the coverage rate for formal contract workers was below 50% in 2003 and 2004. Inthe following years, this coverage rate significantly increased, attaining 65% in average in2005-2007 and 80% in 2012. Part of the 20% of formal contract workers still not coveredby the UIP in 2012 are public sector employees, who are covered under a similar butseparate scheme. Table 1.1 also shows information on workers covered by the UIP as aproportion of the total labour force. Initially, the labour force coverage rate was 42% inaverage for the years 2003-2007, which rapidly converged to 65% in 2012. The 35% notcovered by the UIP in 2012 is explained by public sector employees, the unemployed, andinformal workers.5Table 1.1: Representativity of the unemployment insurance program dataset.Year Total UIPD W ENE Coverage W LF ENE Coverage LF2003 1349.5 3672.7 36.7% 5119.1 26.3%2004 1849.5 3806.3 48.6% 5286.1 34.9%2005 2337.8 3987.4 58.6% 5438.7 43.0%2006 2701.3 4166.4 64.8% 5442.2 49.6%2007 3103.1 4360.3 71.1% 5555.5 55.8%2008 3309.2 4583.5 72.2% 5762.4 57.4%2009 3419.8 4500.1 76.0% 5839.9 58.6%2010 3742.4 4908.1 76.2% 6210.1 60.2%2011 4050.4 5146.7 78.7% 6448.8 62.8%2012 4286.4 5360.2 80.0% 6520.0 65.7%This dataset is compared with the information of the ENE (Encuesta Nacional de Empleo) questionnaireadministred by the goverment statistics agency in Chile (INE-Instituto Nacional de Estadisticas). W ENErefers to the total number of formal employees recorded by ENE and LF ENE is the total labour force(formal and informal) recorded by ENE. The information regarding ENE numbers is from Sehnbruch andCarranza (2015). Units are measured on thousands5As we can see, this dataset converges to a coverage rate of 80% of the formal workers but only to65% for the total labour force. This is in part because this dataset has limited coverage for the (cont’d)unemployed. Sehnbruch (2006) and Ruiz-Tagle and Sehnbruch (2010) argue that this is because a largeproportion of unemployed register by ENE previously worked in the informal sector.16We must acknowledge that the low formal contract workers’ coverage rate duringthe first years of the data (56% in average in 2003-2007) is a concern for our analysisbecause —as explained below— it impacts how we model permanent parental earningsfor our baseline sample. To assess the plausibility of our findings, we perform a robustnessexercise. We frame our analysis using data for years with a higher formal contract workers’coverage rate to construct the permanent parental earnings.1.3.2 Information on child-parent linkageWe link children and their parents using administrative records provided by the CivilRegistry Office (CRO). In Chile, the CRO registers all births, deaths, and marriages. It isa legal requirement in Chile that all births must be registered in the CRO, each of whichis backed by a birth certificate. This birth certificate contains the information on thechild and the parents given at the time of registration. We use the information providedfor all the birth certificates in Chile to build the pairs of children and parents includedin the UIP database.6 In our baseline analysis, the sample of children is composed ofindividuals that were 28-33 years old in 2018, while the sample of parents are individualsthat were 42-87 years old in 2018.1.3.3 Measurement of earningsOur administrative records have information on labour earnings in the formal privatesector, excluding any form of capital income for the workers covered by the UIP. In ourbaseline sample, we measure parental earnings as the 5-year average of monthly earningsfor months worked in the formal private sector between 2003 and 2007. For example, if aparent records 30 months worked within a 5-year period, the measure of earnings used isthe total income in those 5 years divided by 30. In our baseline sample, we only considerparents that worked at least 6 months in the formal private sector during 2003-2007. Ifboth parents worked more than 6 months in the period, we consider the average parental6Families are ever changing, so the parenting person or persons at any point in time may not be thebirth parents.17earnings as the sum of parental earnings divided by two, in line with Chetty et al. (2014)and Corak (2019).Our measure of parental earnings excludes the zeros because a zero in our dataset does not mean that the individual has no earnings, since he/she could be earning asa public employee, in the informal sector, or in the formal private sector but not coveredby the UIP, especially in its earlier years.As with the parents, we measure child earnings in our baseline sample as the five-year average of monthly earnings for worked months in the formal private sector between2014 and 2018. In our baseline analysis, we consider children that worked at least sixmonths in the formal private sector in 2014-2018. This measure of child earnings not onlyexcludes the zeros for the same reasons as for their parental earnings, but also becausechildren may start participating in the private formal labour market in their late 20s,giving a series of months with earnings preceded by a series of zeros corresponding to notbeing in the labour market.To minimize the noise provoked by low earners due to the uncertainty surroundingthe low earnings registered with the UIP, we only consider children and parents who onaverage earn more than half the minimum wage.7 In our baseline sample, we have 505,524parent-child links.1.3.4 Comparison between unemployment insurance programdataset and ENE surveyIn Chile, 29.6 percent of the population works the informal sector. One potential issuefor our dataset is that only contains information on private formal earnings. To see howdifferent are the percentiles including all workers, we compare the earnings percentilesgenerated by our dataset and the Encuesta Nacional de Empleo (ENE).7Half the minimum wage for children is $133,000 in 2019 Chilean pesos (measured from 2014 to 2018)and $103,000 in 2019 Chilean pesos for parents (from 2003 to 2007). Using CASEN 2017 information,14.1 percent of the population were under the minimum wage.18Table 1.2: Comparison of earnings between our dataset and ENE for individuals between28-33 years old.Percentile UIP ENE1% 152,889 170,613.65% 218,433 231,84010% 263,508 250,90225% 343,076 330,00050% 490,707 451,62475% 767,851 700,00090% 1,173,052 1,003,60995% 1,544,161 1,304,69299% 237,1979 2,500,000This dataset is compared with the information of the ENE (Encuesta Nacional de Empleo) questionnaireadministred by the goverment statistics agency in Chile (INE-Instituto Nacional de Estadisticas). W ENErefers to the earnings percentiles for all workers – formal, informal and self employed. Units are in 2018Chilean pesos.Table 1.2 compares our dataset earnings percentiles with ENE dataset percentilesfor 2018. We can see that percentiles similar using the whole population and types ofsector and the formal private sector.1.3.5 Information on child residential addressWe link the pairs of child and parental earnings with the residential address of the childwhile attending 12th grade in school. We obtain this information from administrativerecords provided by the Ministry of Education of Chile. If the child’s residential addresswhile attending 12th grade is not available, we use the most recently-available residentialaddress while she was enrolled from 7th to 11th grade in school (when the child is 13-1819years old).8 We end up with 93.95% of the children’s sample linked to their residentialaddress.1.4 Intergenerational mobility for ChileWe begin our empirical analysis by characterizing the relationship between parental andchild earnings at the national level. We present a set of baseline estimates of relativeintergenerational mobility and then evaluate the robustness of our estimates to alternativesamples.1.4.1 Traditional indicators of intergenerational mobilityIntergenerational earnings mobilityOne of the most commonly used measures of intergenerational mobility is the intergener-ational earnings elasticity, i.e., the effect that a 1 percent increase in the parental earningshas over their child’s earnings. In our work, we estimate the intergenerational elasticityof earnings rather than of income because our dataset only contains information on wagesand not on financial asset income. We measure this elasticity by estimating the followingequation:yci = α + βypi + ǫi, (1.1)where yci is the earnings of child i in logarithms, ypi is the earnings of that child’s parentsin logarithms, and β is the intergenerational earnings elasticity. This parameter is equaltoβ =cov(ypi , yci )var(ypi )= ρ · sd(yci )sd(ypi ), (1.2)where ρ is the intergenerational earnings correlation, and sd(yci ) and sd(ypi ) are the stan-dard deviation of child and parental log earnings, respectively. To prevent any attenuationbias, we measure child and parental earnings as the 5-year average of earnings.8We also estimate our results by making the geographic link from 5th to 12th grade. The results aresimilar.20Table 1.3: OLS estimates of the intergenerational earnings elasticity for our baselinelinkage(1) (2) (3) (4)yp 0.288*** 0.297*** 0.311*** 0.323***(0.001) (0.001) (0.002) (0.002)Constant 9.506*** 9.426*** 9.298*** 9.193***(0.016) (0.018) (0.021) (0.027)Observations 505,524 416,818 282,979 173,683R-squared 0.091 0.098 0.108 0.117Standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1Earnings are measured as average earnings over the months where a children-parents pair report positiveearnings over the studied 5-year period. We keep individuals that appear at least 6 times with positiveearnings in the dataset with average earnings greater than half of the corresponding minimum wage.Columns (1) to (4) report results for male and female children. (1) considers individuals with at least 6months of positive earnings, (2) considers individuals with at least 12 months of positive earnings, (3)considers individuals with at least 24 months of positive earnings and (4) considers individuals with atleast 36 months of positive earnings.Table 1.3 summarizes our estimates for intergenerational earnings elasticity (IGE),i.e., the OLS estimates of the regression slope of the log child earnings on log parentalearnings. Columns (1) to (4) report results for male and female children: (1) considersindividuals with at least 6 months of positive earnings (our baseline sample); (2) considersindividuals with at least 12 months of positive earnings; (3) considers individuals withat least 24 months of positive earnings; and, (4) considers individuals with at least 36months of positive earnings.Our baseline estimation for IGE equals 0.288. With our most restrictive sample—individuals with at least 36 months of positive earnings—, this estimate equals 0.323.This means that an increase of 10 percent in parental earnings implies, on average, anincrease of between 2.88 and 3.23 percent in their child’s earnings.9Tables 1.4 and 1.5 estimate the IGE for female and male children respectively.9This estimate is lower compared with previous estimates in the Chilean literature. Nunez andMiranda(2010,2011), and Celhay et al. (2010) estimate an elasticity between 0.5 and 0.6. Our differences can beexplained by the kind of data used and the method implemented to estimate IGE. Appendix C discussesthis point in detail.21Table 1.4: OLS estimates of the intergenerational earnings elasticity for female children(1) (2) (3) (4)yp 0.300*** 0.307*** 0.315*** 0.326***(0.002) (0.002) (0.003) (0.003)Constant 9.253*** 9.209*** 9.169*** 9.086***(0.024) (0.026) (0.032) (0.042)Observations 222,397 178,916 116,182 68,644R-squared 0.103 0.111 0.119 0.128Standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1Earnings are measured as average earnings over the months where a children-parents pair report positiveearnings over the studied 5-year period. We keep individuals that appear at least 6 times with positiveearnings in the dataset with average earnings greater than half of the corresponding minimum wage.Columns (1) to (4) report results for male and female children. (1) considers individuals with at least 6months of positive earnings, (2) considers individuals with at least 12 months of positive earnings, (3)considers individuals with at least 24 months of positive earnings and (4) considers individuals with atleast 36 months of positive earnings.Table 1.5: OLS estimates of the intergenerational earnings elasticity for male children(1) (2) (3) (4)yp 0.282*** 0.294*** 0.314*** 0.329***(0.002) (0.002) (0.002) (0.003)Constant 9.655*** 9.529*** 9.313*** 9.175***(0.022) (0.024) (0.028) (0.036)Observations 283,127 237,902 166,797 105,039R-squared 0.087 0.094 0.107 0.117Standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1Earnings are measured as average earnings over the months where a children-parents pair report positiveearnings over the studied 5-year period. We keep individuals that appear at least 6 times with positiveearnings in the dataset with average earnings greater than half of the corresponding minimum wage.Columns (1) to (4) report results for male and female children. (1) considers individuals with at least 6months of positive earnings, (2) considers individuals with at least 12 months of positive earnings, (3)considers individuals with at least 24 months of positive earnings and (4) considers individuals with atleast 36 months of positive earnings.22Our results suggest that female children are slightly less intergenerationally mobile thanmale children.Rank-rank correlationAnother measure of intergenerational mobility that has become extremely popular is rank-rank correlation. This correlation measures the effect that an increase of a percentile inthe parental earnings distribution has over the child earnings distribution. One of thearguments to use rank-rank correlation is that the rankings on the earnings distributionare determined at earlier ages and are difficult to change throughout the age distribution.We measure this correlation by estimating the following equation by OLS:rci = αr + βrrpi + ǫi, (1.3)where rci is the ranking of i-th child in the national distribution of child earningsby cohorts, rpi is the ranking of i-th child’s parent on the national distribution of parentalearnings, and βr is the rank-rank correlation.10 This correlation is an indicator of relativemobility that compares the maximum influence of parental ranking on expected childranking. In addition, αr is a measure of absolute mobility because it states the expectedranking that a child would have if her parents belong to the bottom of the parentalearnings distribution.10Note that we compute the ranking of the whole cohort of children and parents, regardless of whetherthey are linked.23Figure 1.1: Expected child ranking conditional on parental ranking30405060708090Mean Child Ranking 0 20 40 60 80 100Parent rankingAuthors calculationsWe estimate the expected child ranking non parametrically using a simple average. Rankings werecomputed over the national distribution. For children we compute the cohort ranking, and for parentswe compute the ranking of people 42-87 years old (in 2018).Figure A.3 presents a binned scatter plot of the mean percentile rank of childrenversus their parents’ percentile rank. This graph illustrates a nonparametric estimationof the conditional expectation of a child’s rank given her parents’ rank (E[rci |rpi = p]). Aswe can see, the relationship between parental ranking and child ranking is close to a linearfunction until the 80th parental percentile, while for parental percentiles higher than 80it is highly non-linear with an increasing gradient as the parental ranking increases.Table 1.6 presents our estimates for the rank-rank slope. To measure the per-centile rank of the children, we consider their rankings in the distribution of child earningswithin their birth cohorts. In the same way, we compute the percentile rank of the par-ents from their positions in the distribution of parental earnings in the baseline sample.Based on the child and parental percentile ranks, the rank-rank slope estimate is the OLSestimate of the regression slope of the percentile rank of a child on the percentile rankof her parents. As before, columns (1)-(4) in Table 1.6 present the results for 6 (baselinesample), 12, 24, and 36 months of positive earnings. The rank-rank correlation is between0.254 and 0.275, that is, the maximum expected difference in child earnings rankings that24Table 1.6: OLS estimates of the rank-rank correlation for our baseline linkage(1) (2) (3) (4)rp 0.254*** 0.261*** 0.270*** 0.275***(0.001) (0.001) (0.002) (0.002)Constant 37.397*** 38.668*** 40.859*** 43.368***(0.080) (0.089) (0.110) (0.141)Observations 505,524 416,818 282,979 173,683R-squared 0.064 0.068 0.073 0.078Standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1Earnings are measured as the average earnings over the months in which an individual reports positiveearnings over 5 years. We keep children-parents linkages that appear at least 6 times with positiveearnings in the dataset with average earnings greater than half of the corresponding minimum wage.Columns (1) to (4) report results for male and female children. (1) considers individuals with at least 6months of positive earnings, (2) considers individuals with at least 12 months of positive earnings, (3)considers individuals with at least 24 months of positive earnings and (4) considers individuals with atleast 36 months of positive earnings.depends on parental ranking is between the 25th and 28th child earnings percentiles.Table 1.7 show the rank-rank correlation estimates only for female children, andTable 1.8 show rank-rank correlation estimates for male children. Comparing femaleand male, results show that the rank-rank correlation is higher for female children. Thisindicates that for females, parental ranking is more persistent than for males. In addition,absolute mobility, measured as the constant of each regression, is higher for males thanfor females, which means that male children of poor parents are expected to locate in ahigher ranking than female children of poor parents.Quintiles transition matricesThese child and parental earnings rankings also allow us to estimate the quintile transitionprobabilities. These probabilities are defined by the conditional probability that a childis in quintile m (with m = 1, 2, 3, 4, 5) of the child earnings distribution given that herparent is in quintile n (with n = 1, 2, 3, 4, 5) of the parental earnings distribution.In the intergenerational mobility literature, there are three probabilities that are25Table 1.7: OLS estimates of the rank-rank correlation for our female children(1) (2) (3) (4)rp 0.278*** 0.285*** 0.293*** 0.300***(0.002) (0.002) (0.003) (0.004)Constant 31.669*** 33.234*** 35.762*** 38.362***(0.121) (0.138) (0.176) (0.233)Observations 222,397 178,916 116,182 68,644R-squared 0.075 0.079 0.083 0.088Standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1Earnings are measured as the average earnings over the months in which an individual reports positiveearnings over 5 years.We keep children-parents linkages that appear at least 6 times with positive earningsin the dataset with average earnings greater than half of the corresponding minimum wage. Columns(1) to (4) report results for male and female children. (1) considers individuals with at least 6 monthsof positive earnings, (2) considers individuals with at least 12 months of positive earnings, (3) considersindividuals with at least 24 months of positive earnings and (4) considers individuals with at least 36months of positive earnings.Table 1.8: OLS estimates of the rank-rank correlation for our female children(1) (2) (3) (4)rp 0.239*** 0.247*** 0.258*** 0.264***(0.002) (0.002) (0.002) (0.003)Constant 41.682*** 42.504*** 44.118*** 46.312***(0.103) (0.114) (0.139) (0.175)Observations 283,127 237,902 166,797 105,039R-squared 0.060 0.064 0.070 0.076Standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1Earnings are measured as the average earnings over the months in which an individual reports positiveearnings over 5 years. We keep children-parents linkages that appear at least 6 times with positiveearnings in the dataset with average earnings greater than half of the corresponding minimum wage.Columns (1) to (4) report results for male and female children. (1) considers individuals with at least 6months of positive earnings, (2) considers individuals with at least 12 months of positive earnings, (3)considers individuals with at least 24 months of positive earnings and (4) considers individuals with atleast 36 months of positive earnings.26broadly studied: i) the circle of poverty, defined by the probability that, given parents whobelong to the bottom quintile, the child will also belong to the bottom quintile. We denotethis probability as p11; ii) the circle of privilege, defined by the probability that, givenparents who belong to the top quintile, the child will belong to the top quintile. We denotethis probability as p55; and, iii) the rags to riches, defined by the probability that, givenparents who belong to bottom quintile, the child will belong to the top quintile. We callthis probability p15. Notice that p11 and p55 are measures of intergenerational persistencethat provide evidence on transmission of disadvantages and advantages, respectively; whilep15 is a measure of upward intergenerational mobility.Table 1.9: Transition matrix of parental earnings quintiles to child earnings quintilesChild quintile1 2 3 4 5Parental quintile 1 0.271 0.235 0.204 0.170 0.1202 0.236 0.235 0.213 0.186 0.1303 0.206 0.223 0.220 0.200 0.1504 0.171 0.193 0.215 0.223 0.1985 0.112 0.125 0.161 0.226 0.376Quintiles are measured using earnings and the baseline dataset. Rows refer to parental quintile andcolumns to child quintiles.Table 1.9 shows the matrix of quintile transition probabilities using our baselinesample. As can be seen in Table 1.9, p11 is equal to 0.271 meaning that a child whoseparents belong to the bottom quintile has an observed probability of 27.1 percent ofremaining in the bottom quintile; p55 is equal to 0.376, which means that a child whoseparents belong to the top quintile has a probability equal to 37.6 percent of remaining inthe top earnings quintile; and p15 is equal to 0.120 which means that the probability thata child whose parents belong to the bottom quintile will herself belong to the top quintileis 12 percent.Our results suggest that there is some persistence of parental earnings because27p55 and p11 are higher than 0.2, which is the value of a transition probability, assumingthat parental-child transitions are random. We also find that p55 > p11 meaning thatpersistence is higher at the top of the distribution than at the bottom. Notice that thetransition probabilities of the first 4 quintiles are relatively similar and close to randomtransitions; however, our results reveal that the main departure from randomness occursat the top quintile where there is a notorious intergenerational earnings persistence.International comparison with the US and CanadaTo put our analysis in perspective, we can compare Figure A.3 with findings for the USand Canada. As reference, we use the results in Chetty et al. (2014) for the US, andthe findings in Corak (2019) for Canada. Notice that, whereas for Chile we use earningsinformation, the works of Corak (2019) and Chetty et al. (2014) use income information.11Figure 1.2: International comparison of expected child earnings ranking conditional tothe parental earnings ranking30405060708090Mean Child Ranking 0 20 40 60 80 100Parent rankingChile CanadaUSWe estimate the expected child ranking non parametrically using a simple average. Rankings werecomputed over the national distribution. We compute the cohort ranking for children and for parents wecompute the ranking of people between 42 and 87 years old (in 2018). Information for Canada is fromCorak (2019) and for the US is from Chetty et al. (2014).11Studies show that income is more persistent than earnings, especially at the bottom of the distribu-tion. Thus, our results for Chile can be interpreted as a lower bound for persistence.28Figure 1.2 shows that Chile has a flatter gradient until the 80 percent in parentalincome/earnings. This evidence suggests that Chile is more mobile than Canada and theUS in parental income/earnings until the 80th percentile. Remarkably, after the 80thparental percentile, Figure 1.2 also shows that the relationship between parental andchild earnings in Chile becomes much steeper than those in the US and Canada. Thisgraphical analysis suggests that intergenerational earnings mobility for Chile is much morenon linear than the results found by the US and Canada.1.4.2 More on non-linearitiesThe previous graphical analysis suggests that the relationship between parental and childearnings in Chile is highly non-linear, even more so than in the US and Canada, with theparticularity of displaying significant intergenerational mobility until the 80th parentalearnings quintile but a notorious degree of persistence of privileges (transmission of ad-vantages from parent to child) at the top of the earnings distribution.To better understand this finding, we perform three empirical exercises. First,we show the estimates of the transition probabilities for the top decile and percentiles.Second, we estimate the conditional distribution of child earnings given a parental decile(percentile), for different parental deciles (percentiles). Finally, we again estimate theIGE equation but, instead of using OLS, we use conditional and unconditional quantileregressions.Decile and percentile intergenerational transition matricesWe now present decile transition probabilities. These estimates allow us to gain deeperunderstanding on how the child earnings distribution behaves within quintiles —especiallyfor children with parents in the top quintile. Table 1.10 shows the matrix of deciletransition probabilities.As can be seen in Table 1.10, the transition matrix —excluding the row withthe 10% richest parents— shows a somewhat intergenerationally-mobile context, withall the transition probabilities roughly close to 10%, as we would expect under random29Table 1.10: Decile Transition MatrixChild deciles1 2 3 4 5 6 7 8 9 10Parental deciles 1 0.158 0.136 0.125 0.114 0.104 0.098 0.086 0.073 0.065 0.0422 0.125 0.123 0.118 0.114 0.107 0.099 0.094 0.085 0.075 0.0593 0.122 0.126 0.124 0.114 0.111 0.102 0.095 0.085 0.072 0.0504 0.109 0.115 0.118 0.114 0.110 0.104 0.101 0.091 0.079 0.0605 0.106 0.107 0.112 0.117 0.112 0.108 0.102 0.093 0.083 0.0616 0.096 0.104 0.108 0.109 0.110 0.111 0.106 0.099 0.091 0.0667 0.086 0.093 0.100 0.104 0.109 0.112 0.110 0.107 0.100 0.0808 0.078 0.084 0.087 0.095 0.102 0.108 0.114 0.115 0.117 0.1009 0.067 0.069 0.073 0.080 0.091 0.099 0.110 0.127 0.140 0.14310 0.044 0.042 0.044 0.051 0.059 0.071 0.092 0.122 0.173 0.30130transition from parent to child. However, given the parental earnings top decile, we noticethat the dynamic of the transition probabilities is significantly different. For instance, theprobability of persistence in privilege p1010 is equal to 0.3. In contrast, the probability ofpersistence in poverty p11 is close to a half of p1010, suggesting that the transmission ofadvantages (circle of privilege) is twice as persistant as the transmission of disadvantages(circle of poverty).We now study p1010 in depth by showing the probabilities associated with transi-tions from parental percentiles to child percentiles, for percentiles from 91 to 100. Table1.11 summarizes this information.As can be seen in Table 1.11, the transition probabilities for children whoseparents belong to the 91st to 95th percentiles of the parental earnings distribution arerelatively similar, while the probability of persistence at the top percentile, p100,100, issignificantly higher compared to the rest of transition probabilities presented in Table1.11. This means that the top percentile is even more persistent than the rest of the 10thdecile. In sum, this analysis provides evidence supporting a high persistence at the top,which increases as long as parental earnings increase.Conditional distribution of child earnings, given parental decilesAnother way to understand the association between child and parental earnings is byestimating the conditional distribution of child earnings, given parental earnings f(yc|yp).Thus, instead of just observing a change in the mean, we can study variations in the entiredistribution. To do this, we perform kernel estimations of the conditional distribution ofchild earnings, given parental deciles.31Table 1.11: 91st to 100th parental percentile to 91st to 100th child percentile transition matrixChild percentiles91 92 93 94 95 96 97 98 99 100Parental percentile 91 0.019 0.018 0.017 0.019 0.020 0.019 0.017 0.019 0.019 0.02392 0.020 0.017 0.019 0.020 0.019 0.020 0.018 0.018 0.022 0.02193 0.019 0.021 0.020 0.019 0.024 0.019 0.025 0.023 0.021 0.02394 0.021 0.022 0.021 0.022 0.028 0.025 0.022 0.021 0.024 0.02595 0.024 0.022 0.018 0.027 0.025 0.026 0.029 0.031 0.028 0.02796 0.026 0.025 0.026 0.026 0.028 0.026 0.032 0.034 0.033 0.03997 0.024 0.025 0.032 0.033 0.035 0.038 0.042 0.043 0.050 0.05698 0.024 0.023 0.026 0.029 0.035 0.038 0.042 0.042 0.045 0.05799 0.026 0.029 0.030 0.029 0.032 0.043 0.039 0.047 0.055 0.066100 0.027 0.035 0.035 0.029 0.045 0.051 0.056 0.060 0.068 0.10532Figure 1.3: conditional (on parental deciles) child earnings distribution0.2.4.6.8 Frequency12 13 14 15 16Log child earningsParents decile 1 Parents decile 2Parents decile 3 Parents decile 4Parents decile 5 Parents decile 6Parents decile 7 Parents decile 8Parents decile 9 Parents decile 10Conditional log Childrens’ earnings distribution on parental earningsThis figure estimates conditional (on parental deciles) child earnings distribution, using kernel to estimatechild earnings distribution. We use the Epanechnikov method to estimate optimal bandwith.Figure 1.3 shows the conditional distribution of the logarithm of child earningsgiven that parents belong to a particular earnings decile, for earnings deciles from 1 to10. As can be seen in Figure 1.3, roughly speaking, the conditional distributions of childearnings are unchanged between parental decile 1 and 7. After decile 8, it tends to move.Indeed, conditional on parents belonging to the top decile, the conditional distribution oflog child earnings is significantly shifted to the right. This evidence is consistent with ourprevious findings of transmission of privileges, since it suggests that it is more likely forchildren whose parents belong to the top earnings decile to obtain higher earnings. Ascan be also seen in Figure 1.3, the conditional distribution of log child earnings for topparental earnings has a higher variance than conditional on lower parental earnings. Insum, this analysis supports the idea that for children in the bottom and middle part of theearnings distribution, parental earnings do not affect their own distribution of earnings;however, child earnings located at the top of their distribution are dramatically affectedby parental earnings.33Figure 1.4: onditional (on parental percentiles in the top decile) child earnings distribution0.2.4.6 Frequency12 13 14 15 16Log child earningsParents percentile 91 Parents percentile 92Parents percentile 93 Parents percentile 94Parents percentile 95 Parents percentile 96Parents percentile 97 Parents percentile 98Parents percentile 99 Parents percentile 100Conditional log Childrens’ earnings distribution on parental earningsThis figure estimates conditional (on parental percentiles in the top decile) child earnings distribution,using kernel to estimate child earnings distribution. We use the Epanechnikov method to estimate optimalbandwith.Figure 1.4 presents the estimation of the conditional distribution of log childearnings, given parents that belong to a specific percentile, for percentiles from 91 to100. As can be seen in Figure 1.4, while the conditional distribution of child earnings isquite similar for those with parents in percentiles 91 to 99, it is starkly different when wecondition by parents belonging to the top 1 percent. This evidence supports our findingthat the relationship between parental and child earnings is highly non-linear, even at thetop parental distribution where this relationship becomes significantly more positive.12Unconditional and conditional quantile regressionsThe two previous empirical analyses provide evidence on non-linearities in the relationshipbetween parental and child earnings when conditional on parental earnings. In other12This result is in line with Zimmerman’s (2019) findings. Zimmerman (2019) shows that studying in anelite college only increases the probability of belonging to the top managerial positions (obtaining higherearnings) if the student attends a top private high school, and he also shows that it is more likely thatparents that belong to the top 1 percent can afford the tuition costs of private schools. Thus, Zimmerman’sfindings are one component of this persistence at the top where the transmission of privileges from parentto child would be through paying the tuition costs for attending a top private high school.34words, as long as parental earnings increase, there is a higher persistence of child earnings,especially at the top of the parental earnings distribution. However, we can also studythe non-linearities in this relationship, conditional on child earnings. Specifically, givena child earnings percentile, is the effect of an increase in parental earnings stronger? Weanswer this question by using quantile regressions.The IGE is estimated using OLS as the expected percent change in average childearnings, given an increase of 1 percent on the average parental earnings. Additional infor-mation regarding the relationship between parental and child earnings can be obtained byestimating the effect of a change in parental earnings on any other distributional momentsof child earnings other than the mean.We can estimate, for instance, the effect of an increase in parental earnings on themedian, the 75th percentile, or the bottom 5 percentiles of the child earnings distribution.The magnitude of those effects would allow us to understand more in depth where inthe child earnings distribution an increment of the parental earnings can improve theiroutcome. We obtain these effects by fitting quantile regressions. Following the works ofFirpo, Fortin and Lemieux (2009) and Baltagi and Ghosh (2017), there are two effects ofan increase in parental earnings over the quantile distribution of child earnings: a “betweeneffect” and a “within effect”. The between effect is defined by the increase in expectedchild earnings due to an increase in parental earnings, while the within effect is givenby the change in child earnings variance associated with a change in parental earnings.Relying on these works, a conditional quantile regression allows us to estimate the betweeneffect, and an unconditional quantile regression is useful to estimate both effects. Thus, theanalysis of both methodological instruments would allow us to understand more about theobserved non-linearities in the association between parental and child earnings estimatedso far.13 Figure 1.5 presents the estimates of the unconditional quantile and conditionalquantile regressions in our applications.13Appendix B explains with more detail the relationship between conditional and unconditional quantileregressions.35Figure 1.5: Unconditional quantile and conditional quantile estimates of the regressionslope of log child earnings vs log parental earnings0.1.2.3.4.5.60 10 20 30 40 50 60 70 80 90 100Unconditional quantile regressions Conditional quantile regressionsEarnings Intergenerational ElasticityUnconditional quantile and conditional quantile estimates of the regression slope of log child earnings vslog parental earnings for the 1st to the 99th child earnings percentiles. Unconditional quantile regressionsare estimated using the RIF methodology developed by Firpo, Fortin and Lemieux (2009).As can be seen in Figure 1.5, the unconditional quantile effect is lower than theconditional quantile effect until the 65th child percentile. This suggests that for the first65 percentiles, the between effect is mitigated by the effect that an increase in parentalearnings has over the child earnings variance. Meanwhile, for higher percentiles, thebetween effect is reinforced by the within effect, since an increment of parental earningsincreases the variance of child earnings. These findings can be interpreted as that, at somepoint of the child earnings distribution, there are some higher-reward job opportunitiesthat may be available that increase expected child earnings and the child earnings variancefor those who can access these work positions.14 To sum up, this analysis reveals thatintergenerational earnings mobility in Chile is high and stable for the bottom 65% of thechildren earnings distribution; however, for the rest of the population economic status ishighly persistent.14This interpretation can be related with the results found by Zimmerman (2019) because the childrenwho can access these higher-reward jobs are those with parents at the top of the earnings distribution.361.4.3 Robustness checksWe now evaluate the robustness of our estimates of intergenerational mobility to alter-native subsamples and specifications. We begin by evaluating three potential sources ofbias: coverage of the dataset in initial years of the UIP, lifecycle bias, and attenuationbias.Dataset coverageAs can be seen in Table 1.1, coverage of the unemployment insurance dataset in its firsttwo years is less than 50% of total formal workers. To see whether this low coverage rateaffects our baseline mobility estimates, we perform new estimates by considering differentwindows of years to measure permanent parental earnings.Table 1.12: Estimations of IGE and rank-rank slope for different years where parentalearnings were measured.Parental year used IGE Rank-rank slope N2003-2007 0.288 0.254 504,9902004-2008 0.288 0.256 550,6682005-2009 0.287 0.260 584,7702006-2010 0.284 0.263 607,5452007-2011 0.283 0.268 622,3392008-2012 0.281 0.270 632,8202009-2013 0.280 0.272 636,6402010-2014 0.278 0.272 638,4812011-2015 0.280 0.275 637,808Table 1.12 presents IGE and rank-rank slope estimates for different windows ofyears to build our measure of permanent parental earnings. We can see that IGE andrank-rank slope estimates do not depend on the choice of the window of years. Specifically,IGE estimates ranges between 0.278 and 0.288, whereas the rank-rank slope is between0.254 and 0.275.37Lifecycle biasPrior research has shown that measuring children’s income at early ages can understate in-tergenerational persistence in lifetime income because children with high lifetime incomeshave steeper earnings profiles when they are young (Haider and Solon, 2006, Grawe, 2006,Solon 1999). To evaluate whether our baseline estimates suffer from such lifecycle bias,we can estimate the intergenerational earnings elasticity by single child cohorts. To dothis, we study the effects of parental earnings on child earnings when children are 23 to33 years old. To be consistent with the literature (Chetty et al., 2014; Corak, 2019), wemeasure the effect of parental earnings when their children were teenagers.Table 1.13: Estimates of IGE and rank-rank slope for different child ages.Child age IGE Rank-rank N23 0.042 0.053 72,86324 0.095 0.102 81,76525 0.151 0.153 86,76726 0.193 0.185 90,24127 0.220 0.215 93,86628 0.245 0.230 96,69329 0.259 0.241 94,49230 0.285 0.256 89,28631 0.305 0.269 81,26132 0.321 0.275 75,01033 0.333 0.276 68,231Table 1.13 shows the estimates of IGE and rank-rank slope by single child cohorts.We can see that intergenerational persistence rises as child age increases. This is consistentwith Chetty et al. (2014). In particular, IGE is more affected by child cohorts than therank-rank correlation, a fact that has been discussed previously in the intergenerationalmobility literature.1515Indeed, Becker and Mincer noted that if individuals can freely choose among occupations with dif-38Attenuation biasEarnings in a single year is a noisy measure of lifetime earnings, which attenuates estimatesof intergenerational persistence (Solon, 1992). To evaluate whether our baseline estimatessuffer from such attenuation bias, we provide the estimates of the rank-rank slope, varyingthe number of years used to build our measure of permanent parental earnings.Table 1.14: Estimates of IGE and rank-rank slope using different years to average parentalearnings.Parental years used IGE Rank-rank slope N1 0.258 0.220 156,7602 0.272 0.235 273,6733 0.277 0.241 363,8054 0.284 0.248 438,3025 0.288 0.254 505,5246 0.291 0.258 559,6667 0.293 0.263 603,4818 0.293 0.267 642,1769 0.294 0.272 676,49410 0.294 0.275 708,541Table 1.14 presents the estimates of the IGE and rank-rank correlations by usingdifferent numbers of years to create the permanent parental earnings. As can be seen inTable 1.14, IGE remains somewhat stable after averaging 4 years, whereas the rank-rankslope varies slightly between 0.254 and 0.275 over 4 years.fering age/earnings profiles, an equilibrium with equality across occupations in the net present value oflifetime earnings is consistent with (indeed predicts) inequality in annual earnings (or 5 year averages ofmonthly earnings), both within age cohorts and overall. In this human capital equilibrium of equalityin net present value, age/earnings profiles cross in the early thirties. Hence, annual incomes (or 5 yearaverages thereof) are plausible indicators of inequality of lifetime income for the 30-35 age cohort, butare heavily influenced by the fanning out of age/earnings profiles at later ages.391.5 Geographic variation in intergenerational mobil-ity: the case of Chilean regionsThe previous sections suggest that the relationship between parental and child earningsvaries non-linearly with parental earnings, especially with parents at the top of the earn-ings distribution.Another source of variation of the relationship between parental and child earn-ings that has been studied in the recent literature is geographical location. The literaturehas found remarkable differences in intergenerational mobility across geographies withina country. For example, Connolly et al. (2018) find that commodity booms may beimportant drivers of intergenerational upward mobility.16. In addition, Deutscher andMazumder (2020) finds the same result for Australia. Thus, a boom of the copper pricecan impact directly wages and the labour market in geographies that are intensive in cop-per production. This finding is important for Chile because it is the main copper producerin the world by a large margin, with approximately 28% of the total world production in2018.1.5.1 Chilean regional contextChile is divided into 16 regions, the first-level administrative division of the country. Eachregion is designated by a number —from 1 to 16— and a name. Each region is dividedinto provinces, the second-level administrative division. In total, there are 56 provinces,each one divided into municipalities, the third and lowest-level administrative division.17In Table A.10, we present current information of each region. Among the 16regions, the Metropolitan Region (the 13th region) stands out as the most populatedregion in the country (in number and density), with a population of over 7.5 million in 201716Connolly et al. (2018) finds for Canada that commodity-producing provinces such as Alberta andSaskatchewan, and mid-west US states, present the highest upward mobility indicators.17Until 2007, there were only 13 regions geographically located from north to south of the country withtheir numbers in geographically sequential order, except for the Metropolitan Region, also known as the13th region, which is located roughly in the middle of the country, between the 5th and 6th regions. Inthe period 2007-2017, the 14th, 15th, and 16th regions were created after dividing into two areas the10th, 1st, and 8th regions, respectively.40(41% of Chile’s population) according to the National Institute of Statistics of Chile (INE).Significantly, this region contains the capital of Chile, the city of Santiago, which hasbeen recognized as one of the cities with the best quality of life in South America. Basedon estimates of the Central Bank of Chile (BCCh) for 2018, the Metropolitan Regionproduces 46% of Chile’s GDP, with manufacturing, services, retail, and financial servicesas principal economic activities. According to official estimates by the Government ofChile for 2017, 5.4% of the population of this region lived in poverty in 2017 and thisregion has a GDP per capita of 3%, with a Gini coefficient of 0.43.The Antofagasta Region (the 2nd region), in the northern area of the country,stands out with a production of 10% of Chile’s GDP, with the mining industry —ledby copper— as its principal economic activity. In fact, according to estimates of theBCCh for 2018, mining output represents 54% of regional production. This region had apopulation of 623,851 inhabitants in 2017 (3% of Chile’s population) according to INE.This region has the highest GDP per capita in the country —over USD 25,000—, 5.1%of its population live in poverty in 2017, and its Gini coefficient is 0.41.On the other end of the income scale in Chile, we have the AraucanA˜a Region (the9th region), in the southern part of Chile, which is the country’s poorest region in terms ofGDP per capita, with USD 6,000 per inhabitant, on average. This region contributes with3% of Chile’s GDP, with 17.2% of its population living in poverty —the highest regionalpoverty rate in the country. It’s worth noting that a third of the region’s population of994,888 (6% of Chile’s population) is of indigenous Mapuche ethnicity, which representsthe highest concentration of this community (or, indeed, of any other national indigenouspeoples) of any Chilean region.1.5.2 Intergenerational earnings mobility at the regional levelTo characterize the variation in intergenerational mobility across geographic areas withinChile, we permanently assign each child to a single region. We use the child’s residentialaddress while attending 12th grade in school. We obtain this information from adminis-trative records provided by the Chilean Ministry of Education. If the residential address41of the child when attending 12th grade is not available, we instead use the child’s mostrecent residence while she was enrolled during 7th-11th grade in school.18Measures of relative and absolute mobilityWe measure mobility at the regional level using the baseline sample and the definitionsof parental and child earnings described in Section 2. We continue to rank both childrenand parents on the basis of their positions in the national earnings distribution (ratherthan the distribution within their regions).Figure 1.6: Expected child ranking conditional on parental national ranking for 4 differentregions.30405060708090Mean Child Ranking 0 20 40 60 80 100Parent rankingMean child earnings rankingAntofagasta30405060708090Mean Child Ranking 0 20 40 60 80 100Parent rankingMean child earnings rankingMaule30405060708090Mean Child Ranking 0 20 40 60 80 100Parent rankingMean child earnings rankingAraucanía30405060708090Mean Child Ranking 0 20 40 60 80 100Parent rankingMean child earnings rankingRegión MetropolitanaMean parents earnings ranking vs mean children earnings rankingThis figure plots the expected child ranking conditional on parental national ranking for 4 differentregions. We estimate the expected child ranking non-parametrically using a simple average. Rankingswere computed over the national distribution. For children we compute the cohort ranking, and forparents we compute the ranking of people between 42 and 87 years old (in 2018).Figure 1.6 presents a binned scatter plot of the mean child rank versus parent rankfor children who grew up in the second region (Antofagasta), the seventh region (Maule),the ninth region (Araucan´ıa), and the Metropolitan region. As can be seen in Figure 1.6,18The region where a child grew up does not necessarily correspond to the region she lives in as anadult at age 28-33 in 2018.42in each region there is a linear relationship between the parental and child ranks for thebottom part of the parental earnings distribution. The higher levels of persistence at thetop of the parental earnings distribution are a common characteristic of the four regionsdisplayed in Figure 1.6. Despite this non-linearity at the top of the distribution, we relyon Chetty et al. (2014) and Acciarri et al. (2020) to characterize the relationship betweenchild rank given the parents’ rank in each region using a simple linear regression. Moreformally, we regress child rank on parental rank by region to calculate absolute upwardmobility and relative mobility by region. We define absolute upward mobility asrabsr = αr + βrE(rp|rp < 50), (1.4)where αr and βr are the intercept and the rank-rank regression slope estimatedfor region r, respectively. That is, the conditional expected child’s position on the nationalearnings distribution given that her parental earnings are below the median of the nationaldistribution. We approximate this value as rabsr = αr + βr · 25.19 In addition, we definepersistence as the conditional expectation of a child’s percentile on the national earningsdistribution given her parent belonging to the 10th decile. We measure this expression asrperr = αr + βrE(rp|rp > 90) and approximate it as rperr = αr + βr · 95.20 We complementthis analysis studying the three transition probabilities described in section 3. Specifically,we show transition probabilities p11 (circle of poverty), p15 (rags to riches), and p55 (circleof privilege).19We also estimate the absolute upward mobility coefficient using a nonparametric estimation ofE(rp|rp < 50). Results remain unchanged.20We also estimate E(rp|rp > 90) nonparametrically. Results remain almost unchanged.43Table 1.15: Intergenerational mobility indicators for different Chilean regions.Region N βr αr rabs rper p15 p11 p551 6584 0.145 47.324 50.942 61.072 0.243 0.190 0.3602 16911 0.146 54.351 58.012 68.265 0.321 0.126 0.4513 9851 0.146 49.064 52.726 62.978 0.206 0.165 0.3684 19962 0.169 42.522 46.746 58.575 0.166 0.257 0.3375 48015 0.199 37.554 42.527 56.450 0.116 0.282 0.3136 28806 0.244 36.332 42.441 59.548 0.112 0.310 0.3687 28874 0.228 35.024 40.731 56.710 0.088 0.317 0.3218 42993 0.197 38.060 42.997 56.820 0.111 0.275 0.3079 20891 0.202 34.385 39.439 53.589 0.082 0.311 0.30810 21105 0.197 36.002 40.932 54.735 0.091 0.255 0.30511 4400 0.142 38.055 41.600 51.528 0.095 0.260 0.24312 5462 0.183 39.700 44.278 57.094 0.120 0.194 0.29113 196004 0.256 39.103 45.509 63.447 0.135 0.222 0.39814 6631 0.178 38.998 43.454 55.931 0.094 0.250 0.34915 4228 0.128 46.601 49.799 58.755 0.167 0.212 0.34516 10510 0.189 37.608 42.341 55.595 0.108 0.289 0.306As can be seen in Table 1.15, there is substantial heterogeneity across regions. Forinstance, the region with the highest absolute mobility is Antofagasta, where a child whoseparents earn below the median national earnings level has an expected national ranking of54.4; whereas, for Araucan´ıa, the same child can expect to place in the 34.3(th) percentileof the child earnings distribution. In the same way, for probability p11 we estimate 0.126for Antofagasta and 0.311 for Araucan´ıa. In addition, we can notice something similar forthe rags to riches probability. For Antofagasta, p15 is equal to 0.321 and for Araucan´ıais equal to 0.082, thus a child who grew up in Antofagasta with a parent that belongs tothe bottom quintile is almost 4 times more likely to arrive to the top quintile than thesame child who grew up in Araucan´ıa. Finally, persistence is also higher in Antofagastathan in Araucan´ıa: children with parents in the top earnings quintile are more likely to44remain in the top quintile in Antofagasta than in Araucan´ıa.Figure 1.7 and 1.8 present heat maps of absolute upward mobility and relativemobility for Chilean regions. We can see that the most upwardly-mobile regions are thoselocated at the north of the country. In particular, Antofagasta is the most upwardly-mobile region. Regarding relative mobility, the least mobile region is the Metropolitanregion.Figures 1.9 and 1.10 present heat maps of circle of poverty p11 and circle ofprivilege p55 transition probabilities for Chilean regions. We can see that the regions mostpersistent in poverty are those located in the upper south area of the country, particularlyEl Maule and Araucan´ıa regions. In contrast, the most persistent regions in privileges arethose located in the north and the Metropolitan region. Thus, we corroborate Conolly etal. (2018) results by providing evidence that Antofagasta, a commodity-intensive region,presents the highest upward mobility indicators.Is there a Gatsby curve in Chile?The Gatsby curve refers to the negative relationship between income inequality and inter-generational mobility. This relationship has been extensively explored by the literature(see for instance Corak, 2013). We use the geographical variation across regions in Chileto study the Gatsby curve.45Figure 1.7: Heat maps for absolute upward mobility in Chilean regionsRegión de Arica y ParinacotaRegión de TarapacáRegión de AntofagastaRegión de Magallanes y Antártica ChilenaRegión de Aysén del Gral.Ibañez del CampoRegión de AtacamaRegión de CoquimboRegión de ValparaísoRegión Metropolitana de SantiagoRegión de Los LagosRegión de Los RíosRegión de La AraucaníaRegión del Bío−BíoRegión de ÑubleRegión del MauleRegión del Libertador Bernardo O’HigginsZona sin demarcar1000 kmNabsolute_mobility[39.4,40.8)[40.8,41.6)[41.6,42.4)[42.4,42.5)[42.5,43.2)[43.2,44.3)[44.3,46.1)[46.1,49.8)[49.8,51.8)[51.8,58]NAAbsolute mobilityA darker blue means a higher value for the indicator46Figure 1.8: Heat maps for relative mobility in Chilean regionsRegión de Arica y ParinacotaRegión de TarapacáRegión de AntofagastaRegión de Magallanes y Antártica ChilenaRegión de Aysén del Gral.Ibañez del CampoRegión de AtacamaRegión de CoquimboRegión de ValparaísoRegión Metropolitana de SantiagoRegión de Los LagosRegión de Los RíosRegión de La AraucaníaRegión del Bío−BíoRegión de ÑubleRegión del MauleRegión del Libertador Bernardo O’HigginsZona sin demarcar1000 kmNrelative_mobility[0.128,0.143)[0.143,0.146)[0.146,0.158)[0.158,0.178)[0.178,0.186)[0.186,0.197)[0.197,0.198)[0.198,0.202)[0.202,0.236)[0.236,0.256]NARelative MobilityA darker blue means a higher value for the indicator47Figure 1.9: Heat maps for circle of poverty p11 transition probability for Chilean regions.Región de Arica y ParinacotaRegión de TarapacáRegión de AntofagastaRegión de Magallanes y Antártica ChilenaRegión de Aysén del Gral.Ibañez del CampoRegión de AtacamaRegión de CoquimboRegión de ValparaísoRegión Metropolitana de SantiagoRegión de Los LagosRegión de Los RíosRegión de La AraucaníaRegión del Bío−BíoRegión de ÑubleRegión del MauleRegión del Libertador Bernardo O’HigginsZona sin demarcar1000 kmNcirl_of_poverty[0.126,0.178)[0.178,0.194)[0.194,0.217)[0.217,0.25)[0.25,0.256)[0.256,0.26)[0.26,0.278)[0.278,0.289)[0.289,0.31)[0.31,0.317]NAQ1 to Q1 A darker blue means a higher value for the indicator.48Figure 1.10: Heat maps for circle of privilege p55 transition probability for Chilean regions.Región de Arica y ParinacotaRegión de TarapacáRegión de AntofagastaRegión de Magallanes y Antártica ChilenaRegión de Aysén del Gral.Ibañez del CampoRegión de AtacamaRegión de CoquimboRegión de ValparaísoRegión Metropolitana de SantiagoRegión de Los LagosRegión de Los RíosRegión de La AraucaníaRegión del Bío−BíoRegión de ÑubleRegión del MauleRegión del Libertador Bernardo O’HigginsZona sin demarcar1000 kmNcirl_of_privileges[0.243,0.298)[0.298,0.306)[0.306,0.308)[0.308,0.313)[0.313,0.329)[0.329,0.345)[0.345,0.354)[0.354,0.368)[0.368,0.383)[0.383,0.451]NAQ5 to Q5A darker blue means a higher value for the indicator.49Figure 1.11: Gatsby curve Chilean regions35455565AUM.4 .42 .44 .46 .48 .5Gini coefficientFigure 7a: Absolute upward mobility vs gini coefficientaum = 83.826 − 90.09 gini    R2 = 24.9%.1.15.2.25.3Relative mobility.4 .42 .44 .46 .48 .5Gini coefficientFigure 7b: Relative mobility vs Gini coefficientrr = −.07729 + .54173 gini    R2 = 16.6%This figure plots the relationship between upward mobility and the Gini coefficient at the regional level.We measure the Gini coefficient using the 2017 CASEN survey, considering the total income before transferand tax variant. Those results remain unchanged when we use other income definitions to measure theGini coefficient.Figure 1.11 left reports the relationship between absolute upward mobility and theGini coefficient, while Figure 1.11 right reports the relationship between relative mobilityand the Gini coefficient. As can be seen in these figures, there is evidence of a Gatsbycurve, where more unequal regions experience less intergenerational earnings mobility.This evidence suggests the existence of a vicious circle between intergenerational mobilityand inequality.1.6 Geographical variation in intergenerational mo-bility within the Metropolitan regionWe now study the intergenerational mobility across municipalities, which are the leastaggregated geographic units in Chile. We do this analysis inside the Metropolitan Regionof Santiago —the finance and government center of Chile. It contributes with 40% percent50of Chile’s GDP, contains the capital of Santiago (the largest city in the country), and isthe most densely populated region in the country, with close to 40 percent of the totalpopulation. This allows us to estimate intergenerational mobility at municipality level.1.6.1 The Metropolitan RegionAlthough the Metropolitan Region of Santiago shows obvious signs of modernization,especially in the city of Santiago —which exhibits modern buildings and highways, asubway system, malls, and an extensive telecommunications network—, there are alsoelements that make it a residentially-segregated region, reflecting the economic inequalitythat characterizes the Chilean economy. Residential segregation in Santiago has its originin several urban planning policies dating from the 1950s that tended to create residentialareas for the lower classes (social housing) on the urban periphery of the city. This resi-dential segregation intensified because of the implementation of slum eradication policiesunder the military dictatorship during the 1980s, where inhabitants of slum neighborhoodswere relocated to social housing constructed on the periphery of the city. This policy ofbuilding social housing on the periphery continued after the return to democracy, as theproportion of social housing units in peripheral municipalities was continuously increasingand no new social housing was constructed in the upper-class municipalities.1.6.2 Estimates of intergenerational mobilityWe estimate the same measures of intergenerational mobility as for the regional case.Table A.11 in Appendix A summarizes these mobility measures by municipality.Figures 1.12 and 1.13 present color maps for intergenerational earnings mobilityon the metropolitan region There is a remarkable heterogeneity across municipalities. Forpoor municipalities such as Cerro Navia, La Pintana and San Ramo´n, absolute upwardmobility is not lower than 42, which means that chilndren whose parents belong to thebottom 50 percent of the earnings distribution, are expected to locate at least in the 42thpercentile of the children earnings distribution. In addition, persistence at the bottom andat the top probabilities are not to far from 0.2 which means that there are not markedly51Figure 1.12: Heat maps for absolute upward mobility and relative mobility indicators for Metropolitan region municipalities.San JoaquÃ-nSan MiguelSan RamónIndependenciaLa CisternaPeñalolénProvidenciaLa ReinaCalera de TangoColinaSantiagoLampaPirquePuente AltoHuechurabaSan BernardoCuracavÃ-MarÃ-a Pinto CerrillosCerro NaviaVitacuraConchalÃ-El BosqueEstación CentralLa FloridaLa GranjaLa PintanaLas CondesLo BarnecheaLo EspejoLo PradoMaculMaipúÑuñoaPedro Aguirre CerdaPudahuelQuilicuraQuinta NormalRecoletaRencaEl MontePadre HurtadoPeñaflorTalagantePaineIsla de Maipo BuinSan José de MaipoTiltilMelipillaSan PedroAlhué40 kmNrr_abs_break[41.7,43.2)[43.2,44)[44,44.6)[44.6,45.8)[45.8,46.4)[46.4,47.4)[47.4,47.7)[47.7,48.7)[48.7,50.9)[50.9,63.7]Absolute upward mobility (RM)San JoaquÃ-nSan MiguelSan RamónIndependenciaLa CisternaPeñalolénProvidenciaLa ReinaCalera de TangoColinaSantiagoLampaPirquePuente AltoHuechurabaSan BernardoCuracavÃ-MarÃ-a Pinto CerrillosCerro NaviaVitacuraConchalÃ-El BosqueEstación CentralLa FloridaLa GranjaLa PintanaLas CondesLo BarnecheaLo EspejoLo PradoMaculMaipúÑuñoaPedro Aguirre CerdaPudahuelQuilicuraQuinta NormalRecoletaRencaEl MontePadre HurtadoPeñaflorTalagantePaineIsla de Maipo BuinSan José de MaipoTiltilMelipillaSan PedroAlhué40 kmNrr_break[0.126,0.14)[0.14,0.152)[0.152,0.165)[0.165,0.179)[0.179,0.186)[0.186,0.203)[0.203,0.231)[0.231,0.248)[0.248,0.286)[0.286,0.551]Relative moblity (RM)A darker blue means a higher value for the indicator.52Figure 1.13: Heat maps for circle of poverty p11 and circle of privilege p55 transition probabilities for Metropolitan region munici-palities.San JoaquÃ-nSan MiguelSan RamónIndependenciaLa CisternaPeñalolénProvidenciaLa ReinaCalera de TangoColinaSantiagoLampaPirquePuente AltoHuechurabaSan BernardoCuracavÃ-MarÃ-a Pinto CerrillosCerro NaviaVitacuraConchalÃ-El BosqueEstación CentralLa FloridaLa GranjaLa PintanaLas CondesLo BarnecheaLo EspejoLo PradoMaculMaipúÑuñoaPedro Aguirre CerdaPudahuelQuilicuraQuinta NormalRecoletaRencaEl MontePadre HurtadoPeñaflorTalagantePaineIsla de Maipo BuinSan José de MaipoTiltilMelipillaSan PedroAlhué40 kmNp_11_break[0.145,0.174)[0.174,0.188)[0.188,0.195)[0.195,0.207)[0.207,0.216)[0.216,0.226)[0.226,0.235)[0.235,0.249)[0.249,0.259)[0.259,0.299]NAQ1 to Q1 (RM)San JoaquÃ-nSan MiguelSan RamónIndependenciaLa CisternaPeñalolénProvidenciaLa ReinaCalera de TangoColinaSantiagoLampaPirquePuente AltoHuechurabaSan BernardoCuracavÃ-MarÃ-a Pinto CerrillosCerro NaviaVitacuraConchalÃ-El BosqueEstación CentralLa FloridaLa GranjaLa PintanaLas CondesLo BarnecheaLo EspejoLo PradoMaculMaipúÑuñoaPedro Aguirre CerdaPudahuelQuilicuraQuinta NormalRecoletaRencaEl MontePadre HurtadoPeñaflorTalagantePaineIsla de Maipo BuinSan José de MaipoTiltilMelipillaSan PedroAlhué40 kmNp_55_break[0.173,0.254)[0.254,0.269)[0.269,0.286)[0.286,0.3)[0.3,0.319)[0.319,0.351)[0.351,0.37)[0.37,0.393)[0.393,0.571)[0.571,0.771]NAQ5 to Q5 (RM)A darker blue means a higher value for the indicator.53persistence. However, the rags to riches probability is lower than 0.1.On the other hand, almost all the rich municipalities in the northeast of the city,such as Las Condes, Vitacura, and Lo Barnechea, are the most persistent municipalitiesat the top, with probabilities of persistence of privileges, the conditional probability thata child is in the fifth quintile given that his parent is in the fifth quintile. Lo Barnechea(0.720), Las Condes (0.672) and Vitacura (0.723) have the highest circle of privilegeprobability of the Metropolitan region by far, the mean of which is 0.337. Thus, for achild with a parent that belongs to the highest quintile, it is highly likely that that childwill also be in the upper quintile.But the differences in absolute upward mobility with more middle-class munici-palities such as N˜un˜oa, Santiago21 or Maipu are relatively small. For instance, absoluteupward mobility in Las Condes (51.48), is very close to N˜un˜oa (51.02) and Maipu (50.28).Different is the case of Lo Barnechea, where upward mobility is very low compare tothe other rich municipalities and is closer to La Pintana, which is a poor municipality.The major differences on persistence of privileges found between the rich municipalitiesand the rest indicates that the place of residence is an important factor to explain thehigh persistence at the top of the earnings distribution. One possible explanation for thisfinding is that social connection may play an important role on persistence of privileges.The Gastby curve in the Metropolitan regionWe can study the relationship between intergenerational mobility and inequality insidethe metropolitan region.21Santiago is the name of the city and also the name of a municipality —the latter is the statisticpresented in this table. The municipality of Santiago is what inhabitants refer to as “downtown” andcontains the presidential building La Moneda.54Figure 1.14: Gatsby curve Metropolitan Region municipalities35455565AUM.3 .4 .5 .6Gini coefficientFigure 10a: Absolute upward mobility vs gini coefficient for RMaum = 46.954 − .22292 gini    R2 = 0.0%.05.15.25.35.45.55Relative mobility.3 .4 .5 .6Gini coefficientFigure 10b: Relative mobility vs Gini coefficient for RMrr = −.0399 + .6148 gini    R2 = 22.4%This figure plots the relationship between upward mobility and the Gini coefficient at municipality levelfor the Metropolitan region. We measure the Gini coefficient using the 2017 CASEN survey, consideringthe “total income before transfer and tax” variant. Those results remain unchanged when we use otherincome definitions to measure the Gini coefficient.Figure 1.14 shows the Gastby curve for the Metropolitan region. Comparing withFigure 1.13 can see that the intergenerational mobility and inequality relationship is moresteeper for persistence than for upward mobility compared with regions. In particular,upward mobility does not strongly relate with inequality in the Metropolitan region.However persistence does strongly relate with inequality. This relationship is strongerthan the regional relationship.1.6.3 Geographic correlations and mobility across the Metropoli-tan regionThe goal of this section is to take a first step toward understanding what local character-istics can account for the divergence in upward mobility across Chilean municipalities inthe Metropolitan region that we documented in Section 5. We do not claim that the cor-relations we uncover should be interpreted as causal relations, but they certainly serve to55guide future research on the deeper determinants of intergenerational mobility. A similaranalysis has been recently performed by Chetty et al. (2014) for the U.S. and by GA˜14ellet al. (2018) for Italy.To study the relationship between mobility and municipal socioeconomic charac-teristics, we start from a large set of correlates based on the literature. We use the Ginicoefficient and the share of the top 1 percent as i) measures of inequality. We correlatethe proportion of immigrants, monoparental households and the proportion of people ofindigenous ethnicity as ii) demographic characteristics. We also include municipal percapita expenditure and per capita square meters of green areas as iii) municipal ameni-ties. We include proportion of students in publicly-funded schools and proportions ofstudents in voucher schools, proportion of people with public health plans, proportion ofovercrowded households, and poverty as iv) socio economic characteristics.56Table 1.16: Correlation between mobility measures and socio-economic characteristicsIndicator βr pabsr pperr p11 p55 p15βr 1‘pabsm -0.1055 1pperm 0.8762 0.3869 1.000p11 0.1027 -0.6958 -0.242 1.000p55 0.9055 0.2206 0.947 -0.128 1.000p15 0.2505 0.8202 0.630 -0.468 0.515 1.000Gini 0.5033 -0.1066 0.415 -0.045 0.436 -0.022Share top 1 percent 0.373 -0.0289 0.332 0.018 0.356 0.055% Immigrants 0.1609 0.3693 0.328 -0.324 0.249 0.361% People with more than 18 years of schooling 0.7221 0.4134 0.870 -0.142 0.825 0.579% Monoparental households -0.2878 -0.4585 -0.489 0.100 -0.423 -0.538% Public health plan -0.3299 -0.7967 -0.692 0.554 -0.529 -0.771Per capita expenditures 0.7209 0.3973 0.861 -0.152 0.819 0.533Per capita square meters of green areas 0.4769 -0.0206 0.4322 -0.0827 0.3754 0.0761% Indigenous -0.6007 -0.3156 -0.71 0.1812 -0.7648 -0.5119% Students in publicly-funded schools -0.2817 0.0297 -0.2469 0.0989 -0.1079 -0.1222% Students in voucher schools -0.6876 -0.0576 -0.6655 -0.0804 -0.769 -0.240Poverty -0.5092 -0.3832 -0.6579 0.215 -0.636 -0.452Overcrowding -0.3367 -0.4374 -0.5242 0.2553 -0.493 -0.508To measure these socioeconomic indicators we use information from the CASEN survey and the “RegistroSocial de Hogares” dataset.Table 1.16 sheds lights on the relationship between inequality measures and theindices. The correlations with the Gini coefficient are strong with persistence measuresbut weak with upward mobility. However, an alternative measure of inequality like theshare of the top 1 percent correlates positively with persistence and absolute mobility.This is an intriguing result because the correlation between the share of the top 1 percentand the upward mobility measures is positive. This is different to what Chetty et al.57(2014) found for the USA but it is line with evidence provided by Accarci et al. (2020)for Italy.The proportion of immigrants is positively correlated with persistence at the topand upward mobility but is negatively related to persistence at the bottom. One of thestrongest correlations is the proportion of people with more than 18 years of schooling.Municipalities with more educated people tend to see more mobility, and tend to be morepersistent at the top and less so at the bottom. The proportion of monoparental house-holds correlates positively with persistence at the bottom, but negatively with persistenceat the top and absolute mobility.Both municipal per capita expenditure and per capita square meters of greenareas correlate positively with persistence at the top, upward mobility, and correlatesnegatively with persistence at the bottom. The proportion of indigenous populations cor-relates negatively with persistence at the top and upward mobility but correlates positively(albeit weakly) with persistence at the bottom. Finally, the socioeconomic characteristicscorrelate positively with upward mobility and persistence at the top but negatively withpersistence at the bottom.We also notice a weak correlation between absolute upward mobility and relativemobility. This is explained by the fact that the variance in relative mobility is highercompared to the variance in absolute upward mobility across municipalities. This meansthat there is more variance in persistence at the top than upward mobility. For theMetropolitan region, this finding supports the claim that, in terms of intergenerationalupward mobility, where to live matters more for children from richer families than forchildren from middle- and lower-earnings families.It is worth nothing that the correlation between the proportion of people withpublic health plan and intergenerational mobility indicators is very high. This meansthat the type of health that a child can benefit is a main variable that can explainintergenerational mobility in Chile. Thus, one way to improve the imputation-basedprocedures used to improve intergenerational mobility is by including this variable in theanalysis.581.7 ConclusionThis is the first paper that studies intergenerational mobility in Chile using administrativerecords. We build a data set that links parental and child earnings using information fromthe formal labour sector and the place of residence of children during their adolescence.Our analysis reveals that intergenerational mobility at the national level is significantlylower than what was estimated in previous research. However, intergenerational mobilityis extremely non-linear. We found that mobility is very high for the bottom 80 percent ofthe earnings distribution but is very persistent at the upper tail of the parental and childdistributions.In addition, Chile is a highly heterogeneous country in its intergenerational mo-bility measures at the regional level. For instance, Antofagasta, which is a mining region,has a probability of rags to riches higher than 0.3. This result is in line with what Conollyet al. (2018) founds for the US and Canada. Meanwhile, regions like Araucan´ıa or ElMaule have a circle of poverty probability higher than 0.3. It is worth digging a littledeeper in future research to understand why those regions are so persistent in poverty.We also find heterogeneity within the Metropolitan region, with municipalitieshaving a circle of privilege probability higher than 0.7, and other municipalities with acircle of poverty probability closer to 0.3. We also learn that the variance of persistenceat the top is higher than the variance of upward mobility. This means that the place ofresidence affects children of upper-earnings parents more than middle- or poor-class par-ents. Future research should focus on understanding the causes behind these differences.Although our work is descriptive in nature, it sheds lights on intergenerational mobilityin a highly unequal country that does not belong to the advanced economies.Moreover, we make a some methodological contributions. We use RIF regressionsand Kernel conditional densities to study intergenerational mobility at the top. Thosetools help usus to show that intergenerational mobility is very persistent at the top inChile. In addition, we differentiate the Gatsby curve for Chile and Santiago using twomeasures of intergenerational mobility: absolute intergenerational mobility and relativeintergenerational mobility. We show that the Gatsby curve is valid for persistence and59upward mobility for Chile but only for persistence for la Regio´n Metropolitana. Thishelp us to differentiate different mechanisms that may affect intergenerational mobilityfor Chile.This work builds on previous national literature and brings the state of researchup to the robustness of analysis seen among works in developed economies. As such, notonly does it provide more useful information for academics; it also provides an importantcounterpoint to similar works from developed economies by analyzing intergenerationalearnings mobility in a non-developed [o developing] economy in a way that can be con-trasted with the results of that literature. We believe that, by providing a clearer pictureof how intergenerational earnings mobility occurs in Chile at a regional level, this work canboth inspire further research on the matter both in Chile and other developing economies.These results can also help Chilean authorities better understand how and where to ap-ply certain related social/economic programs in order to improve their impact, as well asprovide input for drawing up and discussing proposed bills affected by this study’s results.60Chapter 2Income inequality, taxes, andundistributed corporate profits:evidence from Canada2.1 IntroductionFor the majority of individuals measuring income is straightforward this is because theirincome is equal to their earnings. However, as long as individuals start to obtain incomethough assets it is more difficult to measure it. In particular, it is known that the linebetween labour and capital is inherently imprecise for some business owners and some topearners (such as CEOs), and it is certainly possible that tax accounting differs from thecommon-language way of separating labour from capital (Kopczuk, 2016).This imprecision is even greater in the case of corporate earnings and wages ofsmall business owners. In some cases, part of the labour income is left inside the firmas retained earnings and accounted as capital income (Smith, Yagan, Zidar and Zwick,2019). This point is important in inequality measurement; ignoring retained earningsmay imply that a significant part of income is not correctly measured, and thus, theresulting inequality statistics do not reflect the true income (or wealth) distribution.11This is also shown by Flores (2018) who demonstrates that household surveys only account for the61In this context, the more recent literature on income (Piketty, Saez and Zucman, 2018;Smith, Yagan, Zidar and Zwick, 2019) inequality aims to measure all the labour andcapital income from national accounts. However, can retained earnings be included aspart of income?To answer this question we use a comprehensive definition of income based onthe Haig-Simons concept:2 the flow of resources that can be consumed while leaving thestock of wealth unchanged; or in simple terms, income is the sum of consumption and thechange in wealth. This chapter follows this definition for the case of business retainedearnings. We should notice however that retained earnings is just a part of capital stock.3This definition is important because there is a relationship between retained earn-ings and capital gains. More retained earnings mean higher future dividends, implyingan increase in the value of the firm which turns into capital gains.4 By definition, capitalgains are changes in wealth. Thus, retained earnings are an indirect source of income.Additionally, the use of retained earnings to measure capital gains is an attractivemethod to account for capital gains. In particular, there is an open debate about howone should measure capital gains. Indeed, in the income inequality literature there aretwo approaches to do this: one considers a realized approach while the other one usesan accrual approach. However, the use of realized capital gains as opposed to accruedcapital gains has been criticized for several reasons: i) It often reflects capital income thatoccurred in different years in which the gains are being measured (Armour et al. 2013;Smeeding and Thompson, 2010). ii) Capital gains obtained by individual investors in aparticular period may not be realized in the same year and therefore are omitted whenthe conventional approach of accounting for only realized capital gains is used.570% of labour income and 20% of capital income.2For more details on this definition see Haig (1921) and Simons (1938).3This chapter can only provide a partial glimpse of inequality in the Haig-Simons concept of income.In the national accounts, the capital stock includes the plant and equipment of firms, owner- occupiedhousing, the rental housing stock and publicly owned capital. For example, housing and speculativecapital gains are not included in this chapter’s inequality measurement.4This argument was developed by Lopez et al. (2016) and Gutierrez et al. (2015). They refer to thecapital gains generated in this manner as fundamental accrued capital gains.5As Burkhauser et al. (2014) argue, taxable realized capital gains are not a good proxy for the account-ing of yearly accrued capital gains because changes in the tax legislation within countries may affect thedefinition of taxable capital gains over time; thus, there exists considerable variation of income betweenyears.62Thus, the use of retained earnings allows us to have an accrual measure of capitalgains. However, the money inside the firm has a different value from that outside thefirm. If an agent wants to get the money out of the company, he or she has to paypersonal taxes (depending on the tax system he or she may receive a tax credit for thecorporate taxes paid by the firm). The financial market may adjust for those future taxesdecreasing the capital gains generated by this retained earning. That is, the tax systemshould be taking into account to measure the income from capital gains associated withretained earnings. In addition, if we consider that the marginal investor is a foreigner, asBoadway and Bruce (1992) states, the domestic tax system is irrelevant for the marginalinvestor. Thus, it is crucial to identify what is the right tax rate that should be used tovalue retained earnings as capital gains. The same is true if there are some transactioncosts to get the money out of the firm. Thus, the first contribution of this work is todevelop a conceptual framework that analyzes the effect that the ownership of the firm,the tax system and the equilibrium in capital markets have on the capital gain generatedvia retained earnings.To compute income inequality and to measure the value of retained earnings ascapital gains, one needs data about the ownership of retained earnings. Previous works(Alstæaeter et al. 2017 for Norway, Wolfson et al. 2016 for Canada, Fairfield and Jorrat2016, for Chile and Austen and Splinter, 2016 for the USA) use administrative tax data,allowing them to attribute directly retained earnings to personal taxpayers. However,obtaining access to these data is quite complicated, and it is difficult to make a compar-ative study across countries. Moreover, even with the ideal dataset, some imputation (orassumption) might still be needed to have a defined distribution of retained earnings.In this paper, we do not have access to administrative data. To overcome thislimitation, we propose a parametric methodology to impute corporate retained earningsto families using household survey microdata and aggregate national account information.This procedure is based on the exponential-Pareto model established by Dragulescu andYakovenko (2000) and Silva and Yakovenko (2004).6 In addition, this method follows the6Others studies that use an exponential distribution for the bottom part of the income distributionare Banerjee et al. (2006) and Jagielski and Kutner (2013).63spirit of Jenkins (2017) and Hundenborn et al. (2018). It uses survey data for a fractionp of the population and aggregate national account data as an additional source for theremaining 1− p.7 To evaluate the pertinence of this parametric imputation methodology,we compare it with a non-parametric imputation procedure a capitalization approachsimilar to that applied by Saez and Zucman (2016).8 This method contributes to anextensive literature that estimate income and wealth inequality using parametric methodssuch as Kleiber and Kotz (2003), Chotikapanich, Griffiths and Rao (2007), Clementi andGallegati (2016), among many others.The valuation and imputation methodology is tested empirically for Canada,where we impute corporate retained earnings and then use the generated data to computeincome inequality measures. To do so, we use the Survey of Consumer Finances (SCF)for 1984 and the Survey of Financial Security(SFS) for 1999, 2005, 2012 and 2016. Thejustification for using these surveys is that they can be harmonized, allowing us to make acomparison with the capitalization method used by Saez and Zucman (2016) because thosesurveys have rich information on assets in addition to income.9 The inequality measuresestimated here are not as precise as those estimated by Saez and Veall (2005), Veall(2012) and Wolfson et al. (2016). Despite this, it has the value of correcting a householdsurvey for a form of under-reporting such as Burkhauser et al. (2011), Bourguignon (2018),Blanchet, Flores and Morgan (2018) among many others.Empirically, we find that the inclusion of corporate retained earnings and itsmeasure as accrued capital gains increases the estimated measure of income inequalityand that this also affects the trend in income inequality. Indeed, for 2005, the share of thetop 1% increases by 4.5 percentage points (from 7.8% to 12.3%), and the Gini coefficient7The use of survey data is justified because in some countries it is not mandatory for individualsmaking certain income levels to file a tax declaration; thus, data that are just generated only from taxdeclarations may have a bias in the lower tail of the income distribution. Also, some developing countriesdo not have another reliable data source than a household survey, or it is politically difficult to get accessto administrative data.8One reason to establish another imputation procedure is that, as Kopczuk (2016) states, “In anenvironment with a low rate of return, a small bias in the estimated rate of return has large consequenceson the estimations of wealth inequality.”9Dividends could also be used to impute corporate retained earnings. However, as Alstæaeter et al.(2017) shows, this is not a good procedure to impute retained earnings because (i) retained earningsand dividends move in different directions, (ii) mechanically, the imputations results are not adequate inperiods in which the aggregated retained earnings are negative.64increases by 4.4 points (from 47.5 to 51.9) which implies higher income inequality thanin 2012 and 2016; this was not the case before accrued capital gains were considered.Those results are robust to the method used to impute corporate retained earnings. Inthis context, this work contributes to a broader literature that study Canadian incomeinequality such as Saez and Veall (2005), Fortin et al. (2012), Lemieux and Riddell (2015),Milligan and Smart (2015), Wolfson et al (2016), Green, Riddell and St-Hilaire (2016),among many others.The remainder of the paper proceeds as follows. In the next section, we developa theoretical model for valuing retained earnings as income. We discuss how the owner-ship of the firm, the marginal investor and transaction costs affect the value of retainedearnings as income. Section 3 describes the parametric imputation methodology. Section4 presents the data and estimation of income inequality before and after the parametricimputation procedure. Section 5 evaluates the performance of the parametric imputationprocedure by comparing it with a capitalization procedure and studies how reliable thestated assumptions are for building the parametric imputations. Section 6 concludes.2.2 Measure of personal income derived from retainedearningsThis section presents a conceptual and theoretical development of some issues regardingthe measure of retained earnings as income. More specifically, if a firm belongs to someindividuals, then the associated retained earnings belong indirectly to those agents. Thismoney inside the company means that in the future, there will be dividends that must bepaid to those agents, which implies an increase in the value of the firm. Thus, a capitalgain is generated, even if not yet realized.Following Burkhauser et al. (2015), Lopez et al. (2016), Gutierrez et al. (2015),Smeeding and Thompson (2010), Wolfson et al. (2016) and Piketty, Saez and Zuckman(2016), we use a comprehensive definition of income based on the Haig-Simons concept65of income.10 A capital gain is income generated via a change in wealth. However, thetransformation of retained earnings into capital gains depends primarily on the tax system.The intuition comes from the fact that the shareholders of the firm have to pay taxes totake the money out of the firm, thus a potential buyer of this firm is willing to pay thevalue of retained earnings discounting the taxes that she has to pay to get her income outof the firm. So, even if no tax is paid right away, the capital gains derived from retainedearnings would be affected by taxes.2.2.1 Retained earnings and ownership of the firmIn this section we analyze two scenarios. The first is the case of a publicly traded firm,which has previously been treated in the literature. The second is the case of a privatelyowned firm. The novelty here is to describe the case of a closely held firm where there existsthe possibility of using part of the goods inside the firm for the owner’s own consumption.In the case in which an agent is an atomistic owner, it is complicated for her to useretained earnings for her consumption. Moreover, given the law of one price, there shouldbe just one value for a unit of retained earnings for each agent.Retained earnings in publicly traded firms (corporate sector)Starting from the Haig-Simons definition of income:yt ≡ ct +∆Vt, (2.1)where yt is the total income accrued by an agent in time t, ct is the total consumption intime t, and ∆Vt ≡ Vt+1 − Vt is the change in wealth in period t.In the context of capital market equilibrium, a change in wealth (∆Vt) is equiva-lent to a change in the value of the firm. In order to obtain this value, one can start fromthe following non-arbitrage condition in the capital markets:10This concept refers to the flow of resources that can be consumed while leaving the stock of wealthunchanged with respect to the previous period. In simple terms, the Haig-Simon definition of income inperiod t is the consumption plus the change in wealth in period t.66rVt = d0t + (Vt+1 − Vt) , (2.2)where r is the interest rate, Vt is the value of the firm in t and d0t are the pre-tax dividendspaid by the firm. This equation reflects the fact that the return from investing the valueon the firm in another project (rVt) should be equal to the return on the money insidethe firm (d0t + (Vt+1 − Vt)). Now, following King (1974), one can write this equation aftertaxes:111− τs + ιτs1− τs (1− τe)rVt = dt + (Vt+1 − Vt) (1− τk), (2.3)where dt is the after-tax dividends paid by the firm, τs is the entity tax rate (corporatetax), τe is the tax faced by securities holders, τk is the capital gains tax on an accruedbasis, ι is the percentage of tax integration between the corporate tax and the incometax,12 r is the interest rate, Vt is the value of the firm in t and dt are the after-tax dividendspaid by the firm. Now with this information, we can state the following proposition.Proposition 1. In a publicly traded firm, the capital gains generated by retained earnings(Gf) are given byGf (θ(τe, τs, ι, τk), πr) = θ(τe, τs, ι, τk)πr, (2.4)where θ(τe, τs, ι, τk) =(1−τs+ιτs)[1−τe](1−τs)(1−τk)and πr are the retained profits indirectly owned by theindividual.Proof. First, we follow Gutierrez et al. (2015) using the equilibrium in capital markets:11This equilibrium condition assumes that the only value that generates retained earnings is the futuredividend that could be given today and not the value of the expected future returns that a prospectiveinvestment could yield using retained earnings as a source of financing. We do not consider either theproblem of the cost of capital or corporate financial policy because given the existence of tax credits andliquidity constraints, a unit of retained earnings could have a different opportunity cost. That is, theonly relevant comparison is what an external agent that buys the firm could do today with the moneyinside the company by taking the money out of it.12An integrated tax system is a system in which the personal tax takes account of corporate tax alreadypaid. In Canada, the system is an imputation system; that is, the idea is to “impute” the gross profitsthat an individual receives in the form of a dividend, and she has to pay taxes using her personal taxrate; however, for this amount, a tax credit applies to the taxes that the firm previously paid.671− τs + ιτs1− τs (1− τe)rVt = dt + (Vt+1 − Vt) (1− τk).Now, using the definition of dt =(1−τs+ιτs)[1−τe](1−τs)· (π (1− τs)− πr) where π is the totalprofits, and noting that in equilibrium rV = (1− τs)π, we have thatGf ≡ Vt+1 − Vt = (1− τs + ιτs) [1− τe](1− τs) (1− τk) · πr = θ(τe, τs, ι, τk) · πr.In this context, θ(τe, τs, ι, τk) is a ratio that represents the opportunity cost of themoney left inside the firm in terms of the value outside the firm. This value is decreasingin τe, that is taxes that reduce the value of the money outside the firm may negativelyaffect the value of retained earnings as income. However, θ(τe, τs, ι, τk) increases with ι,τs or τk. This is due to the fact that ι and τs are related to a tax credit that reduces thevalue of the personal tax paid on dividends. The case of τk is more complex: having themoney inside the firm implies more capital gains; thus, a higher τk also increases the taxthat the agent pays due to capital gains. Therefore, the opportunity cost of not receivingdividends increases.13Gf(θ(τe, τs, ι, τk), πr) is the amount of capital gains derived from retained earn-ings. Moreover, because of the law of one price, τe, τs, ι and τk should be equal foreach individual and equal to the highest values. Otherwise, there will be an arbitrageopportunity.Retained earnings in closely held firmsMaking the distinction between closely held firms and publicly traded companies is im-portant to understand the implications of leaving the money in the company and theconsumption possibilities derived from this money. Wolfson et al. (2016) estimate that13Leaving money in the firm as retained earnings comes with some risk that the firm will have futurelosses and the anticipated future will not fully materialize. Thus if we include this type of risk, thevaluation of retained earnings should change. However, θ reflects the opportunity cost of retained earningsrelated to dividends today, so there should be no risk included.6836% of all retained earnings were in Canadian-controlled private corporations (CCPC) in2010.14 The major distinction between closely held firms and publicly traded firms is thatin the former the owner has some opportunity to buy goods using retained earnings forher consumption (for example, a chair, a computer, or a car for transportation). However,she can not do this for all the types of goods (for example, she cannot buy a trip to Hawaiifor holiday or a luxury car). The distinction between which goods could be consumedinside the firm or outside the firm depends on the tightness of the tax administration.A tighter tax administration will require more documents to allow for some goods to bedeductible. Thus, the amount of consumption made within the firm will be lower. Forinstance, in Canada, to deduct the car consumption it is necessary to justify that eachmile is used for corporate purposes.To illustrate these implications and to understand the equivalence between goodsbought inside and outside the firm, consider a firm owner who has the problem of choosinghow much to consume between goods inside and outside the firm using only retainedearnings.15 She faces the following maximization problem:maxu(x1, x2)s.t. p1θ(τe, τs, ι, τk)x1 + p2x2 ≤ θ(τe, τs, ι, τk)πr,where x1 are the goods that she buys inside the firm, x2 are goods that she buysoutside the company and θ(τe, τs, s, z)πr is the value of retained earnings (πr) outsidethe firm in terms of forgone dividend. The amount of x2 is restricted by the tax code.Assuming that a good has the same price inside the firm or outside the firm and that thegoods bought inside the firm do not incur value added tax (VAT), we can assume thatp1 = (1− τv)p2, where τv is the VAT tax. Without loss of generality, we can normalize p2to 1. Now, if θ = 1 − τv there are no differences between buying goods inside or outsidethe firm.14A Canadian-controlled private corporation is a firm whose shares are not publicly traded and that isnot controlled by a public corporation or non-residents.15This consumption constitutes a residual consumption that is not related to other income apart fromretained earnings.69Define u∗ as the optimal level of utility. The object of interest is C(1, 1, u∗)which is the cost of getting a level of utility u∗ outside the firm. Using duality theory, itis possible to write C((1− τv) θ(τe, τs, ι, τk), 1, u∗) = θ(τe, τs, ι, τk)πr.With this in mind, define the Konu¨s (1924) true cost of living index Pk(u∗, p0, p1)as the ratio of the minimum costs of achieving the same utility level u∗ when an individualfaces two sets of prices p0 and p1.C(1, 1, u∗)C((1− τv) θ(τe, τs, ι, τk), 1, u∗) = Pk. (2.5)Then, the value of retained earnings in term of the Haig-Simons definition of income isGch(θ(τe, τs, ι, τk), τv, πr) = Pk · θ(τe, τs, ι, τk)πr. (2.6)Using equation (2.1), we define the value of retained earnings as consumption asGch −Gf which is equal to:(Pk − 1) · θ(τe, τs, ι, τk)πr. (2.7)The next lemma summarizes the finding of the previous subsection:Lemma 1. For closely held firms, the value of a monetary unit of retained earnings asincome is given by θ and for a closely held firm this value is given by Pk · θ(τe, τs, ι, τk).2.2.2 Retained earnings and the marginal investorAs we demonstrated, the value of retained earnings as income depends on the value oftaxes. Given this result, and assuming the law of one price, it is important to understandthe value of the tax rate the marginal investor is paying. This issue is even more importantfor an integrated tax system. In particular, the Canadian tax system is an imputationtax system, with a tax credit used against the personal tax paid on dividend taxation.There are important implications if the marginal investor is a foreigner or not70subject to domestic taxes. Following Boadway and Bruce (1992), Fuest and Huber (2000)and Edwards and Shevlin (2011), in a small open economy with a stock market that isfully integrated, the required rate of return on domestic shares may be determined by thebehavior of foreign rather than domestic investors. In that case, the required return onshares issued by domestic companies may not be affected by changes in domestic personaltax rules. One of the reasons for this is that the dividend tax credit applies against thepersonal income tax which is typically levied on a residence basis, which implies that thetax integration benefit is irrelevant to determine the return of a particular stock from acorporate firm. However, this view is challenged by Sørensen (2014), who argues that,even in a small open economy, not all shares are traded internationally. Hence, one mightexpect that tax relief for domestic shareholders will at least reduce the cost of capital forsmall closely-held companies controlled by one or a few domestic residents. Given thisdebate in the literature, the question about who is the marginal investor is still open.Thus, we analyze both cases: (i) the marginal investor is a foreign agent and (ii) themarginal investor is a resident.For the foreigner marginal investor, the arbitrage condition in a small open econ-omy is given by(1− τ fe )rVt = dt + (Vt+1 − Vt) (1− τk), (2.8)where τ fe is the foreigner tax rate. This implies that the value of retained earnings willbe: [1− τ fe](1− τk) πre. (2.9)In the case of a domestic marginal investor, for which the valuation will include thetax credit, the integration has to be included. Assuming an imputation tax system, thearbitrage condition will be(1− τ re )1− τs rVt = dt + (Vt+1 − Vt) (1− τk),where τ re is the tax rate on dividends applied to the residents. Using this equation,71the value of retained earnings for a resident marginal investor is[1− τ re ](1− τs) (1− τk)πre. (2.10)Quantitatively, this could be quite relevant. Following Edward and Shevlin (2011)prior to 2006, τ fe = 0.15 but for domestic investors τre = 0.46 and τs = 0.32. Using thosenumbers, the value of retained earnings inside the firm will be 0.85 for the foreign investorand 0.79 for the national investor.2.2.3 Retained earnings and transaction costsTransaction costs are a relevant component of the financial system. The genesis of thosecosts is that investors require extra compensation to cover the costs of buying and sellinga security. These transactions costs tend to be lower for more frequently traded andmore liquid stocks. These costs are both theoretically and empirically relevant. Indeed,as Fisher (1994) showed, transaction costs explain part of the equity premium puzzlereducing it from 6.2%(Mehra and Prescott, 1985) to 0.4%. Spreads are not the only costassociated with trading stocks. Equity investors must also pay brokerage commissions.For instance, Jones and Lipson (2001) find that one-way institutional fees on NYSE-listedstocks during 1997 are about 0.12% of the amount transacted. However, as Jones (2002)documented, there are two types of commissions, a proportional commission and a fixedfee.16The existence of a transaction cost implies that getting the money out of the firmis costly, that is, the opportunity cost of having the money in the firm decreases. Forinstance, if we assume a proportional transaction cost of χp, the no-arbitrage conditionin the capital markets becomes16Jones (2002) documented that for the USA between 1925 and 2002 proportional commissions rangedfrom 3% of the total transaction to 0.1% depending on the amount of the transaction, and the fixedcommission was from 3 USD to 148 USD. When summed together, transaction costs and commissionsrepresent a substantial and variable friction in trading US equities during the 20th century. The totalcosts averaged 0.84% over the 1925-2000 period.721− τs + ιτs1− τs (1− τe)rVt = dt + (Vt+1 − Vt) (1− τk)[1 + χp]. (2.11)This implies that the value of a unit of retained earnings inside the firm is givenby:(1− τs + ιτs) [1− τe](1− τs) (1− τk) (1 + χp) · πr. (2.12)In the case of a fixed transaction cost of χf per monetary unit, the non-arbitrage conditionis given by:1− τs + ιτs1− τs (1− τe)rVt = dt + (Vt+1 − Vt) (1− τk) + χf . (2.13)Then, the value of a unit of retained earnings inside the firm is given by((1− τs + ιτs) [1− τe](1− τs) (1− τk) − χf)· πr. (2.14)Combining the two cases, we arrive at((1− τs + ιτs) [1− τe](1− τs) (1− τk) (1 + χp) − χf)· πr. (2.15)As we can see, an increase of the transaction costs, decreases the value of retainedearnings as income.2.2.4 Summary of the contexts used in the valuation and impu-tation of retained earnings and drawbacks of the method-ologyThis section extends the results of Gutierrez et al. 2015 and Lopez et al. 2016. We give anextended theoretical framework to value retained earnings in the following contexts: (i)ownership (corporate vs non-corporate), (ii) marginal investor (domestic vs foreigner) and73(iii) transaction costs (transaction costs vs no transaction costs). For clarity, we presentthe following table as a summary of this section:Table 2.1: Value of retained earnings (θ) given different contextsType of assumption Yes NoCorporate Firm (1−τs+ιτs)[1−τe](1−τs)(1−τk)Pk(1−τs+ιτs)[1−τe](1−τs)(1−τk)Domestic marginal investor [1−τre ](1−τs)(1−τk)[1−τfe ](1−τk)Transaction Costs (1−τs+ιτs)[1−τe](1−τs)(1−τk)(1+χp)− χf (1−τs+ιτs)[1−τe](1−τs)(1−τk)The valuation method described so far has two important drawbacks. First, itdoes not include speculative capital gains, which are part of income. Ignoring them maygenerate a bias in the computed inequality measures. Second, this methodology relies onthe non-arbitrage condition; if the market allows for arbitrage, then this method givesan imprecise value of the capital gains generated through retained earnings. However,showing that the value of retained earnings as income is different from the value of theretained earnings is a relevant contribution to this literature. In particular, previous worksthat try to measure income inequality (Wolfson et al. 2016; Fairfield and Jorrat, 2016;and Alstæaeter et al. 2017) ignore completely the effects of the tax system in the valueof retained earnings as income. They simply add retained earnings directly to income.This implies that there could be a bias in the estimation of income inequality measures.2.3 Imputation procedure2.3.1 Overview of the imputation procedureThis section presents a parametric procedure to impute retained earnings from an aggre-gate source (national accounts totals) into a microdata source (household survey). Thisprocedure contributes to a broader literature that uses parametric methods to estimateincome and wealth inequality such as Kleiber and Kotz (2003), Chotikapanich, Griffithsand Rao (2007), Clementi and Gallegati (2016) among many others.74We start by describing the general procedure and the data sources; then weproceed to describe each stage in more detail. As a starting point, suppose that incomeincluding capital gains (hi) of a family i is given by:hi ≡ xi + ycgi , (2.16)where xi are income observed in a household survey (without including any capitalgains) and ycgi are income derived from accrued capital gains; this income is generally notobserved in microdata. Additionally, from section 2 we know that:ycgi = θ · πrei , (2.17)where πrei are retained earnings accrued to the family i. In addition, we assume that θ iscommon over the whole population, thus it is irrelevant if we use individual income dataor family income data. With this assumption and some other distributional assumptionswe can develop a parametric methodology that allows us to have an estimate of πrei onlyusing xi and∑ni πrei as inputs.Assumption 1: Minimum threshold. Corporate undistributed profits πre area function ψ : R+ →R+ that takes as input xi, withψ(xi) =0 xi ≤ w¯ψ∗(xi) xi > w¯, (2.18)where w¯ is the minimum income required to own corporate retained earnings.17Assumption 2: Ranking preservation. ψ∗(xi) is a strictly increasing andcontinuous function of xi.Assumption 3: Parametric form. Conditional on xi > w¯, xi and hi =xi + ψ(xi) are Pareto distributed, with exponents ηx and ηh, respectively. Additionally,conditional on xi ≤ w¯, xi is exponentially distributed.17This is a simplification. In real life there are some people, such as retirees, with low income and someassets.75With this set of assumptions, it is possible to find a closed form for ψ as a functionof xi, w¯, ηx and ηh.Proposition 2. If Assumptions 1-3 hold, then ψ∗(x) has a close form and is given by:ψ∗(x) = xηxηhw¯w¯ηxηh− x. (2.19)Proof. See appendix C.Proposition 2 means that the only plausible way that, conditional on x > w¯, thedistribution of x and h are Pareto, and, if assumptions 1 and 2 holds (minimum thresholdand ranking preservation) the amount of retained earnings owned by each family must bea deterministic function of income without retained earnings (the ψ function or equation(2.19)).Before describing the estimation procedure, it is essential to discuss the pertinenceand intuition of each of the assumptions stated above. Assumption 1 means that it ismandatory to earn a minimum amount of income to be an owner of retained earnings.There are some strong empirical arguments to support the idea that there is a non-trivialproportion of the population that do not hold any capital income.18Assumption 2 means that an agent (or family) that earns more income owns moreretained earnings. That is, the richer the household, the higher the amount of retainedearnings owns. Empirically, there is a positive relationship between capital and labourincome; this correlation is higher for top incomes.Assumption 3 is a parametric one; this seems slightly arbitrary. However, Paretotype 1 distribution is commonly used in the inequality literature because of its simplic-ity and precision. Also, by construction, retained earnings increase inequality, this issomething that can be justified empirically.19 Those three assumptions are necessary forgetting an appealing closed form for πre. In particular, Proposition 2 states that it is pos-sible to have an approximate measure of retained earnings simply by knowing the income18Also, it makes sense that there could be two thresholds, one for small business and another for largecorporations ownerships. In principle, our methodology can be adjusted to this case.19See for instance Wolfson et al. (2016), Fairfield and Jorrat (2016), Lopez et al. (2016) and Alstæaeteret al. (2017). Those works show that including retained earnings increase inequality.76without retained earnings.Now, we describe the procedure to estimate the parameters of the ψ function, ηh,ηx and w¯ by using household survey data and national account data.i Using household survey microdata, we fit a parametric model to estimate w¯ and ηx.ii Using the estimators generated in the previous step it is possible to compute ηhcombining household survey and the national accounts aggregates.iii With estimates of ηx, w¯ and ηh it is possible to estimate ψ(xi) for each xi asψˆ(xi|ηˆx, ηˆh, ˆ¯w).iv Now, with iii) it is possible to estimate the total income including retained earningsas hˆi = xi + θψˆ(xi|ηˆh, ηˆx, ˆ¯w)2.3.2 Estimation of ηx and w¯We assume that the true income distribution is a combination of two parametric distri-butions. When xi ≤ w¯, xi is drawn from an exponential distribution and if xi ≥ w¯,then xi is drawn from a Pareto distribution. From an empirical perspective, it is nothard to justify a Pareto distribution for the upper part of the income distribution, butfitting an exponential distribution for the bottom part is not a perfect distribution forthis. However, this distribution is frequently used in inequality research, for instance,Dragulescu and Yakovenko (2000), Silva and Yakovenko (2004), Banerjee et al. (2006),Jagielski and Kutner (2013) among many others. Also, in Appendix A, we develop aneconomic model to justify those parametric functions. With this in mind, the cumulativedistribution function for x is:F (x) =1− e−λx x ≤ w¯1− e−λw¯ + e−λw¯ (1− ( w¯x)ηx)x > w¯. (2.20)In addition, we define p = F (w¯) as the proportion of people (or families) receiving anobserved income lower than w¯. We assume that those people do not own any retained77earnings (assumption 1). Thus, we impute retained earnings using national account datafor those that have a market income higher than w¯ (the richest 1 − p proportion of thepopulation).Also, notice that the CDF in (2.20) is not differentiable in w¯. Thus, the classicaltheory of extremum estimators fails. To overcome this issue, we use a threshold modelto estimate Θ = (w¯, λ, ηx). This estimation procedure is similar to those of Coles (2001),Clauset et al. (2009) and Jenkins (2017).20 To estimate the parameters of the model, weminimize the mean square error (MSE) between the empirical cumulative distribution andthe theoretical distribution derived from the model presented in the previous subsection.MSE =n∑i=1(F emp(xi)− Fˆ (xi,Θ))2, (2.21)where F emp(xi) is the empirical CDF defined as:F emp(xi) =∑nj=1 1(xj ≤ xi)n, (2.22)and Fˆ (xi,Θ) is the estimate of the theoretical distribution from equation (2.20).Now, to estimate Θ, a finite grid (Gr) is built for w¯. For each wk ∈ Gr, we setwk = w¯. We assume that each xi ≤ wk are exponentially distributed and each xi > wk arePareto distributed. We use each sub-sample to estimate maximum likelihood estimatorsfor λ and ηx, call them λk and ηkx. Then, we compute the MSE for each (wk, λk, ηkx). Wechoose ( ˆ¯w, λˆ, ηˆx) as (wk, λk, ηkx) that minimizes the MSE.Knowing ˆ¯w, we define the estimated proportion of families who receive observedincome lower than w¯ as pˆ = F emp(ˆ¯w). This is the estimated proportion of people thatdo not own any retained earnings.20Cowell and Van Kerm (2015) argue that “In practice, however, this threshold is generally determinedheuristically, selecting by eye the amount of the upper tail that needs to be replaced by inspecting a Paretodiagram showing the linear relationship between the log of wealth and the log of the inverse cumulativedistribution function”. In this context, Clauset et al. (2009) argues in favour of a more objective andprincipled approach based on minimizing the distance between a power-law model and the empirical data.782.3.3 Estimation of ηhTo estimate ηh it is necessary to combine the total amount of retained earnings Πre (fromnational accounts) with ˆ¯w and apply assumption 3. Proposition 3 shows how.Proposition 3. Assuming that assumption 3 holds, we know w¯ and E (h|h > w¯). Then,ηh is uniquely identified.Proof. Using assumption 3 we know that if hi > w¯ then hi distributes Pareto with expo-nential parameter ηh. Thus, we can use the following relationship derived by Atkinson etal. (2011).βh =E (h|h > w¯)w¯=ηhηh − 1 . (2.23)Then, ηh =βhβh−1Now, to estimate ηh, it is sufficient to use the sample analogue estimator forE (h|h > w¯) defined by:Eˆ(h|h > ˆ¯w) =∑xi> ˆ¯whi +Πre∑ni=1 1(hi ≥ w¯). (2.24)Then, βˆh =Eˆ(h|h> ˆ¯w)ˆ¯wand ηˆh =βˆhβˆh−1.2.3.4 Estimation of ψFrom i) and ii), we have estimated values for ηx, w¯, and ηh. We just need to replace thosevalues in the ψ function. However, the ψ function is derived in a context of continuousrandom variables and microdata in household surveys is presented as a discrete randomvariable. To account for this, we need to add a normalization constant c, such that thefollowing restriction holds.Πre =n∑iψˆ(xi) (2.25)79Then, the estimator for ψ is given by:ψˆ(xi|ηˆh, ηˆx, ˆ¯w)=0 xi ≤ ˆ¯wc ·[xηˆxηˆhiˆ¯wˆ¯wηˆxηˆh− xi]xi > ˆ¯w. (2.26)with c = Πre∑xi≥ ˆ¯wxηˆxηˆhiˆ¯wˆ¯wηˆxηˆh−xi.With this, we estimate a parametric imputation function for each xi (incomeknown in a household survey).2.3.5 Estimation of hiThe final step is to estimate the total income for a family. We know xi from the surveydata and ψˆ from iii). Also, we need to transform retained earnings into accrued capitalgains, we can do this just by multiplying ψˆ by θ (the transformation rate discussed insection 3). With this, we can build an estimation for hi as:hˆi = xi + θ · ψˆ(xi|ηˆh, ηˆx, ˆ¯w). (2.27)Now, we can compute inequality measures using parametric imputed income (hˆi)as input. The following section describes the data used in this application and the in-equality measures estimated for hˆi.2.4 Estimation of inequality measures with imputedcorporate undistributed profits for CanadaIn this section, we show the estimated measures of inequality for Canada using householdsurvey data and aggregate data from national account. The two objectives of this practicalapplication are i) to show the effect of include accrued capital gains (derived from retainedearnings) on the measurement of economic inequality and ii) to apply the parametric80imputation method developed in the previous section. The results exposed here should bestudied with caution because they are not methodologically comparable with studies thatuse administrative data to estimate income inequality for Canada.21 For instance, Saezand Veall (2005), Veall (2012) and Wolfson et al. (2016) do a much more rigorous measureof inequality trends and top incomes measures. Despite this, this empirical estimationshed lights on the importance of correcting a household survey for under-reported capitalincome and the consequences on the measures of income inequality.222.4.1 Data and definition usedData usedWe use three data sources. First, we use the Survey of Consumer Finances (SCF) for1984. In that year, this survey had a supplementary questionnaire that asks wealthrelated questions at the family level. Second, we use the Survey of Financial Security(SFS) for 1999, 2005, 2012 and 2016. Those surveys measure income and wealth at thefamily level. Finally, we use annual data of the change in corporate retained earningsfrom CANSIM table 36-10-0117-01.23.21Also, there are two other sources of error, the sampling error (from the household survey) and thenon-sampling-error (that coming from the imputation process via data combination). This total errorshould be taken into account when we analyze the results stated in this section.22Burkhauser et al. (2011), Bourguignon (2018) and Blanchet, Flores and Morgan (2018) study theeffect of correcting household survey for under-reported income and non-response rates.23Following CANSIM description of corporate savings: “Retained earnings of a corporation or quasi-corporation are equal to the distributable income less the dividends payable or withdrawal of income fromthe corporation or quasi-corporation respectively. If the foreign direct investment enterprise is whollyowned by a single foreign direct investor (for example, a branch of a foreign enterprise), the whole ofthe retained earnings is deemed to be remitted to that investor and then reinvested, in which case thesaving of the enterprise must be zero. When a foreign direct investor owns only part of the equity ofthe direct investment enterprise, the amount that is deemed to be remitted to, and reinvested by, theforeign investor is proportional to the share of the equity owned. Retained earnings are equal to thenet operating surplus of the enterprise plus all property income earned less all property income payable(before calculating reinvested earnings) plus current transfers receivable less current transfers payableand less the item for the adjustment for the change in pension entitlements. Reinvested earnings accruedfrom any immediate subsidiaries are included in the property income receivable by the direct investmententerprise.” That is, this work has flow of retained earnings for Canadian residents81θ and accrued capital gains before taxesIn section 2, we showed that retained earnings could be transformed into accrued capitalgains by multiplying them by θ. This factor reflects the valuation that the financialmarket makes for a unit of retained earning inside the firm rather than the taxes that theshareholder pays. Those taxes will be paid if and only if the shareholder decides to sellher stocks. For this reason, the accrued capital gains estimated in this work are measuredpre-personal taxes. Also, the tax parameters used to obtain θ are from Milligan (2016).We use the highest personal tax rate in Canada for each year. For the sake of simplicity,we only present the result for one case, a domestic marginal investor without transactioncosts. However, the results are quite similar using any combination from Table 2.1.Income Definition.Most of our interest in inequality from a ‘social welfare’ point of view is in some definitionof post-fiscal income. However, to be consistent among the different income sources, theincome used here is pre-tax and transfers. Indeed, accrued capital gains imputed usingretained earnings are pre-tax. Thus, the income definition used here is similar to thepre-tax and transfers income used by Piketty and Saez (2003), Saez and Veall (2005) andBurkahuser et al. (2012).Also, during this application we use the Haig-Simons comprehensive definition ofincome; that is, we treat income as the sum of consumption and the change in wealthduring a defined period of analysis. We can write this as the sum of two components:family market income (x) and accrued capital gains (ycg). Family market income x isdefined as the sum of employment income (wages and salaries, net farm income and netincome from a non-farm unincorporated business and/or professional practice), invest-ment income, retirement pensions, superannuation and annuities (including those fromRegistered Retirement Savings Plans [RRSPs] and Registered Retirement Income Funds[RRIFs]) and other money income. It is equivalent to total income minus all governmenttransfer payments. It is also referred to as income before transfers and taxes. Also, mar-ket income does not include net capital gains or losses. Thus, the use of retained earnings82as a source of accrued capital gains does not generate double accounting issues.In addition we use accrued capital gains derived from retained earnings ycg as ameasure of family capital gains. That is:ycgi = θ · πrei ,where θ is the transformation rate of retained earnings to accrued capital gainsand πrei is the flow of corporate retained earnings on a given year accrued to family i.However, we do not know the distribution of retained earnings across families. For thisreason, we use the estimated retained earnings using the parametric estimator ψˆ definedin section 3. Then, the parametric imputed income (hˆi) is defined as:hˆi = xi + θ · ψˆ(xi).This income concept along with market income xi are the objects of study during thissection.Units of AnalysisThe units of analysis are the economic families of two or more individuals and unattachedindividuals. That is, the imputation is made for the family group than the individual.Because we assume that θ is common across individuals, there are no taxation discrep-ancies of using economic family instead individuals to value retained earnings as income.This assumption is consistent with the fact that we do not know the distribution of familyadjusted retained earnings. Thus, we are not able to correct neither ycgi nor hˆi for familycomposition.2424See Chanfreau and Burchard (2008) for more details about equivalence scales and family size.83Control for Total IncomeThe control for total income (H) is the total family market income (X) plus the totalaccrued capital gains imputed to families Y cg. Moreover, we do not impute all corporateretained earnings to families because, following Be´dard-Page´ et al. (2016), approximately15 percent of the Canadian stock market is controlled by pension plans funds. Thus, weimpute 85 percent of total corporate retained earnings (Πre) to families. Table 2.2 is asummary of the information used to construct the control for total income.Table 2.2: Totals used in the estimation of income inequalityYears X (2) Πre (3) Πre imp. (4) θ (5) Y cg (6) H (7) (6)/(7)SCF 1984 248,100 8,061 6,852 0.84 5,756 253,856 2.3 %SFS 1999 525,300 15,272 12,981 1.14 14,779 540,099 2.7 %SFS 2005 705,600 94,809 80,588 0.89 71,723 777,323 9.2 %SFS 2012 959,800 57,892 49,208 0.94 46,256 1,006,056 4.6 %SFS 2016 1,132,400 10,720 9,112 0.92 8,383 1,140,783 0.7%Total market income X from SFS and SCF and the annual flow of corporate retained earnings Πre fromCANSIM table 36-10-0117-01. Values for X , Πre, Y cg and H are in millions of nominal CAD. Values forθ from Milligan (2016).We can observe that the share of total corporate retained earnings as the controlfor total income changes significantly over time. For instance, in 2005 accrued capitalgains are 9.2 percent of the control for total income but in 2016 are just 0.7 percent. .Accounting period for incomeThe accounting period for market income is income generated from January 1 to December31 of the corresponding year. This is consistent with the work done by Brzozowski etal. (2010) and Davies, Fortin and Lemieux (2017) which used the SFS for inequalityanalysis. The same accounting period is used for retained earnings; we use the annualflow of corporate retained earnings starting from January 1 to December 31. The data84used for this purpose is from Statistics Canada table 36-10-0117-01 (formerly CANSIM380-0078).252.4.2 Inequality measures including accrued capital gainsEstimates of the parametric modelTable 2.3 shows the estimate of the parameters of the model presented in section 3.Table 2.3: Parameter estimatetsYear pˆ∗ ˆ¯w ηˆx ηˆh1984 0.69 32,993 2.7 2.5(0.009) (473.3) (0.047) (0.001)1999 0.78 67,448 2.78 2.54(0.009) (453.5) (0.027) (0.001)2005 0.82 87,518 2.59 1.93(0.006) (641.3) (0.037) (0.001)2012 0.76 92,507 2.29 2.04(0.008) (1,213.8) (0.046) (0.001)2016 0.76 102,542 2.23 2.22(0.007) (1,642.4) (0.036) (0.001)Standard errors in parenthesis. Note: Standard errors were generated using a semi-parametric bootstrapfollowing Cowell and Van Kerm (2015). All the estimation procedures use sample weights.From Table 2.3 we observe that roughly between 25 percent and 20 percent ofthe population had some type of corporate savings (1− pˆ∗). This proportion changes overtime; in particular, this ratio decreases between 1984 and 2005, increases by 6 percent25However, Statistics Canada measures market income as the amount earned the year before the surveyis asked. For example, for 1984, the income reflected are incomes earned between January 1 and December31 of 1983. This is a common issue in household surveys. Also, the wealth variables are measured inthe middle of the reference year (the year of the survey). Despite this issue, we use values for the actualyear for retained earnings. The rationale for this approach is that retained earnings are a consequence ofassets bought before the reference year.85between 2005 and 2012 and stays constant between 2012 and 2016. We can also observethat ηˆh (the Pareto parameter including accrued capital gains), is lower for years in whichretained earnings are more important related to market income.Estimate of the Lorenz curve, Gini coefficient and p-shares with and withoutparametric imputationWe can estimate retained earnings by replacing the estimated parameters presented inTable 2.3 along with xi into equation (2.28).26ψˆ(xi|ηˆh, ηˆx, ˆ¯w)=0 xi ≤ ˆ¯wc ·[xηˆxηˆhiˆ¯wˆ¯wηˆxηˆh− xi]xi > ˆ¯w. (2.28)Next, we can compare the Lorenz curve using market income with the Lorenzcurve estimated using the parametric imputed income hˆi.Figure 2.1: Lorenz curves for market income and parametric imputed income0.2.4.6.81Share of Income0 .2 .4 .6 .8 1Cumulative population proportion19841999200520122016 Market Income0.2.4.6.81Share of Income0 .2 .4 .6 .8 1Cumulative population proportion19841999200520122016 Parametric Imputed IncomeBased on the Survey of Consumer Finances, Survey of Financial Security and CANSIM table 36-10-0117-01.26Recall that c = Πre∑xi≥ ˆ¯w[xηˆxηˆhiˆ¯wˆ¯wηˆxηˆh−xi] .86From Figure 2.1, we observe that after the imputation of accrued capital gainsit is easier to differentiate between years; in particular, after the imputation of retainedearnings as accrued capital gains, the year that is clearly unequal is 2005 followed by2012. The latter fact is not valid for market income; it is not easy to observe usingFigure 2.1 that the unequal year is 2016. This result is consistent using other inequalitymeasures. Figure 2.2 shows the Gini coefficient using those two income definitions. For1984 the Gini coefficient after imputing accrued capital gains increases by 2.3 percentrelative to the Gini coefficient using market income, which was similar to the increase in1999 (2.8 percent). In 2005, the Gini coefficient increases by 9.3 percent (from 47.5 to51.9). However, for the following years, the increase in the Gini coefficient is lower, 4.3percent in 2012 and 0.7 percent in 2016.27 Moreover, in Figure 2.2 we observe a changein the inequality trend. There is an increase in inequality between 2005 and 2016 usingmarket income; that is, inequality increases after the Great Recession. However, usingthe parametric imputed income, inequality decreases between 2005 and 2016. This resultis consistent with the wealth inequality trend documented by Davies, Fortin and Lemieux(2017) using the same databases.27The change in the Gini coefficient can be explained using the following formula G = (1− p) ·G1−p ·S1−p+ p ·Gp ·Sp+Sp− p where p is the proportion of people that is in the top p percent Sx is the shareof the x proportion of the population and Gx is the Gini coefficient of the x percentage of the population.Thus, retained earnings affect the Gini coefficient because it changes Sp without changing G1−p thus,despite retained earnings belongs to the top part of the distribution, the Gini coefficient is affected byretained earnings.87Figure 2.2: Gini coefficient with and without parametric imputed income354045505560SCF 1984 SFS 1999 SFS 2005 SFS 2012 SFS 2016Market Income Parametric Imputed IncomeBased on the Survey of Consumer Finances, Survey of Financial Security and CANSIM table 36-10-0117-01.In addition, Figure 2.3 shows the share of the top 10 percent and top 1 percent.Qualitatively, this figure shows the same as that Figure 2.2: we observe an increase of57.7 percent in 2005 in the share of the top 1 percent after imputing accrued capital gains(from 7.8 percent to 12.3 percent) but only of 2.1 percent for 2016 (from 9.6 percent to9.8 percent).88Figure 2.3: Top 10% and top 1% with and without parametric income.05101520253035404550%SCF 1984 SFS 1999 SFS 2005 SFS 2012 SFS 2016Top 10% Market Income Top 10% Parametric Imputed IncomeTop 1% Market Income Top 1% Parametric Imputed IncomeAuthor calculations based on the Survey of Consumer Finances, Survey of Financial Security and CAN-SIM table 36-10-0117-01.2.5 Discussion of the methodology and assumptionsThe imputation procedure presented so far gives us an appealing parametric procedureto estimate retained earnings. However, there are non-trivial limitations that lead topotential weaknesses of this methodology. First, how well does this parametric approachadjusts to the real income distribution? Second, How valid is the ranking preservationassumption? Third, how precise are the estimations of p presented in Table 2.3? To answerthose questions, we contrast our estimated measures of inequality with those generatedby a non-parametric procedure: the capitalization imputation method.2.5.1 Contrasting the parametric estimation using a capitaliza-tion approachOne of the limitations of the methodology proposed is to assume a parametric exponential-Pareto model for income distribution. Indeed, the estimates for pˆ (incomes where a89Pareto distribution starts) are far away from those used by the literature (Atkinson,2017; Bourguignon, 2018; Jenkins, 2017). This cast doubts that the distribution of thetop incomes is well-described by a Pareto distribution. One way to address this limitationis to use a non-parametric procedure to impute retained earnings. In particular, we usethe capitalization imputation approach used by Atkinson and Harrison (1974) and, morerecently, by Saez and Zucman (2016).To use the capitalization method we must know a wealth source to impute anincome source. In particular, we know ownership of corporate stocks as a wealth sourceto impute the flows of retained earnings. Assume that a monetary unit of corporatestocks generates a stream of αcap retained earnings, this value is constant for the wholepopulation and the different types of corporate stocks. Then, the amount of retainedearnings imputed to a family i is equal to:πrei = αcap · vi, (2.29)where πrei are the retained earnings accrued to the family i, αcap is the capital-ization factor that is assumed constant for each i and vi is the corporate stock owned bythe family i.Summing over i both sides of equation (2.29), we have:αˆcap =ΠreV, (2.30)where αˆcap is the estimated capitalization factor, Πre =∑ni=1 πri and V =∑ni=1 vi. Then,we have,πˆcapi = αˆcap · vi, (2.31)where πˆcapi are the imputed retained earnings using the capitalization method.For our example, we use national accounts data for total retained earnings anddata from the SCF-SFS for corporate stock totals held by families. With this information,we define the capitalized income hˆcapi as the sum of market income xi and capitalizedaccrued capital gains (θ · πˆcapi ).90Figure 2.4 shows the Lorenz curves of market income and capitalized income. Weobserve that the unequal year is 2005 following by 2012. Also, comparing with Figure 2.1,the effect of including capitalized capital gains is not as evident as using the parametricimputation procedure to impute capital gains.Figure 2.4: Lorenz curves of capitalized income vs market income0.2.4.6.81Share of Income0 .2 .4 .6 .8 1Cumulative population proportion19841999200520122016 Market Income0.2.4.6.81Share of Income0 .2 .4 .6 .8 1Cumulative population proportion19841999200520122016 Capitalized IncomeBased on the Survey of Consumer Finances, Survey of Financial Security and CANSIM table 36-10-0117-01.Figure 2.5 shows the Gini coefficient of the capitalized income, parametric im-puted income, and market income. We observe that the Gini coefficient computed usingthe capitalized income confirms that the income inequality trend changes after we includea measure of accrued capital gains. On the other hand, computing inequality using cap-italized income shows lower levels of inequality than using parametric imputed income.Despite this, the differences are very small (the Gini coefficient computed using capitalizedincome is on average 1.1 percent lower than using parametric imputed income).91Figure 2.5: Gini coefficients of capitalized income, parametric imputed income and marketincome354045505560SCF 1984 SFS 1999 SFS 2005 SFS 2012 SFS 2016Market Income Parametric Imputed IncomeCapitalized IncomeAuthor calculations based on the Survey of Consumer Finances, Survey of Financial Security and CAN-SIM table 36-10-0117-01Figure 2.6 shows the share of the top 10 percent and the top 1 percent usingcapitalized income, parametric income, and market income. Again, using capitalizedincome shows the same trend than using parametric imputed income, which means thatincome inequality is decreasing after the Great Recession.Moreover, the top income shares estimated using the parametric imputed incomeare higher than the capitalized income shares (3.3 percent higher for the top 10 percentand 2.2 percent higher for the top 1 percent). This difference is considerable for 2005where the top 1 percent using the parametric imputed method is 1.3 percentage pointshigher than the same measure using capitalized income (12.3 vs. 11 percent, i.e., 10.7percent higher) and 2.9 percentage points higher for the top 10 percent (39.3 vs. 36.7,i.e., 8.4 percent higher). For 1984 and 1999, the share of the top 1 percent using thecapitalization method is slightly higher than the same share estimated using parametricimputed income. Thus, estimate inequality measures after impute accrued capital gainsusing the capitalization method gives similar results than estimating inequality measuresusing the parametric procedure developed in section 4.92Figure 2.6: Top 10% and top 1% of capitalized income, parametric imputed income andmarket income.051015202530354045%SCF 1984 SFS 1999 SFS 2005 SFS 2012 SFS 2016Top 10% Market Income Top 10% Parametric Imputed IncomeTop 10% Capitalized Income Top 1% Market IncomeTop 1% Parametric Imputed Income Top 1% Capitalized IncomeBased on the Survey of Consumer Finances, Survey of Financial Security and CANSIM table 36-10-0117-01.2.5.2 Evaluation of the ranking preservation assumptionAnother strong assumption used to build the parametric imputation methodology is thatthe ranking on the income distribution is the same before and after including accruedcapital gains or the “ranking preservation assumption”. This assumption is in practicenot true; some people are owners of corporate stock shares, but they do not receive anyincome, and other people receive a very high labour income, but they do not own anyassets. Thus, using the ranking preservation assumption could imply a bias in the es-timation of inequality measures. Indeed, we can see from Figures 2.5 and 2.6 that theinequality measures computed using the parametric imputed income are slightly higherthan using capitalized income. One way to evaluate the pertinence of the ranking preser-vation assumption is by computing the share on the total income of the 1 − pˆ percent(those who in theory are the owners of corporate retained earnings). We take pˆ from Table2.3. If the ranking preservation assumption is true, then there should be no differencesin the share of the top 1− pˆ percent using parametric imputed income or the capitalized93income.Figure 2.7 shows that the difference between the share of the top 1 − pˆ percentbetween the parametric imputed approach and the capitalization approach is small, onaverage is less than 1 percentage point.Figure 2.7: Share of the income before pˆ and after pˆ354045505560SCF 1984 SFS 1999 SFS 2005 SFS 2012 SFS 2016Market Income Parametric Imputed IncomeCapitalized IncomeBased on the Survey of Consumer Finances, Survey of Financial Security and CANSIM table 36-10-0117-01. pˆ estimates are based on Table 2.32.5.3 The effect of changing pˆOne of the lessons of Jenkins (2017) is that the threshold above which incomes are Paretois higher than often assumed. Following Jenkins, we estimate the parametric imputationprocedure assuming (1− p) is 1 percent of the population. To do so, we can fix w¯ such ap proportion of the population is driven by an exponential distribution. Figure 2.8 showsthe share of the top 10 percent and the share of the top 1 percent after assuming thatp = 0.99. We observe that the estimates for the top 10 percent are very similar betweenthe baseline parametric imputed income (with pˆ from Table 2.3) and the new imputa-tion (with p = 0.99). However, the share of the top 1 percent for this new imputationgives higher estimates than the same inequality measure computed using the baseline94parametric imputation. Thus, although our baseline estimates for p are very far fromwhat the literature had found, our inequality measures estimated using the parametricimputed methodology developed here gives results that are closer to what we estimateusing capitalized income.28Figure 2.8: Top 10% and top 1% of capitalized income, parametric imputed income andmarket income (with pˆ and p = 0.99)051015202530354045%SCF 1984 SFS 1999 SFS 2005 SFS 2012 SFS 2016Top 10% Market Income Top 10% Parametric Imputed IncomeTop 10% Capitalized Income Top 10% Parametric Imputed Income assuming p=0.99Top 1% Market Income Top 1% Parametric Imputed IncomeTop 1% Capitalized Income Top 1% Parametric Imputed Income assuming p=0.99Author calculations based on the Survey of Consumer Finances, Survey of Financial Security and CAN-SIM table 36-10-0117-01.2.6 ConclusionThis study argues in favour of using retained earnings as a measure of accrued capitalgains, clarifies the difference between left money inside the firm for closely held corpora-tions and for publicly traded firms. In particular, we make a conceptual contribution byshowing that for closely held corporations retained earnings could have a consumptionvalue in addition to the change of wealth value that the literature acknowledged. Theintuition is that goods can be bought inside the firm to satisfy personal consumption.In addition, we propose a methodology to impute corporate retained earnings28If we assume p = 0.9 or p = 0.95 the results are qualitatively the same.95to households. The convenience of this methodology is that it only needs market incomeinformation from a household survey and aggregated retained earnings information from ahousehold survey. We apply this methodology to Canada. We show that including accruedcapital gains increases measured income inequality and, more importantly, changes theobserved trend of inequality which is similar to what Davies, Fortin and Lemieux (2017)found for Canadian wealth.We compare our imputation method with the broadly used capitalization impu-tation method. We obtain similar results by using the capitalization method to imputeretained earnings. Even though the parametric imputation procedure developed here isnot perfect, it gives results that are close enough to the capitalization method. This factsuggests that in the context of scarcity of data, for example, developing countries wherewe cannot access easily to administrative data, and this procedure could be useful forimputing capital income in such contexts.Another application of the imputation methodology developed here can be usedin the context of undercoverage at the upper tail. Recent literature, such as Flores (2018),show that because of nonresponse and underreporting, household surveys only accountfor 20 percent of the capital income that appears in national account data. Thus, becausethe parametric imputation procedure developed here only need a household survey andnational account data, could be used to correct for underreporting of capital income inhousehold surveys. Future research is needed to study the imputation procedure in suchcontexts.96Chapter 3Gini and undercoverage at the uppertail: a simple approximation3.1 IntroductionThe Gini coefficient is an indispensable index to measure income inequality.1 Institutionslike the World Bank, the Economic Commission for Latin America and the Caribbean, andother important policy institutions routinely use the Gini coefficient to analyze income-distribution changes.2 Given its popularity, this index should be measured properly;however, data used to compute this coefficient are mainly derived from household surveys,which typically undercover the upper part (top) of the income distribution (Atkinson etal., 2011; Jenkins, 2017; Flores, 2018; Hlasny and Verme, 2018)3 and, leading to biasesin the calculation of the Gini coefficient. As Burkhauser et al. (2017) show, estimatinginequality without correcting for such bias may result in an inaccurate analysis of historical1The Gin coefficient is not the only inequality index used. For a discussion about the comparison oftwo income distributions and inequality indexes see Atkinson (1970).2For a broader discussion about the limitations of the Gini index, see Cowell and Flachaire (2018),Alvaredo et al. (2017), and Osberg (2017). For elegant usages of this index, see Corvala´n (2014) andModalsli (2017), and for an introduction to the Gini coefficient, see Ceriani and Verme (2012).3Undercoverage not only exists at the top of the distribution: Higgins, Lustig, and Vigorito (2018)and Bollinger et al. (2018) show that the entire distribution faces undercoverage, and Ceriani and Verme(2019) show “inequality almost invariably increases by adding observations on the tails of an incomedistribution but that missing a few observations at the top is much more relevant than missing manyobservations at the bottom”.97inequality changes.We study effects of two types of undercoverages at the top of the income dis-tribution: underreporting (i.e., missing income) and nonresponse (i.e., missing people).Underreporting occurs when individuals in a population report less income or wealth thanthey earn (e.g., tax evasion, top coding, information omissions in household surveys).4 Onthe other hand, nonresponse occurs when individuals in a population are unrecorded inthe data source (i.e., truncated data; e.g., people not submitting their household surveysor not declaring taxes). Bourguignon (2018), Lustig (2018), Blanchet, Flores and Morgan(2018) recently studied these two missing-information types. Their works discuss how ad-justments for these biases affect income-inequality measures. In particular, Lustig (2018)develops a taxonomy to differentiate the different types of undercoverage at the upper tail.Bourguignon (2018) shows, in a didactic manner, how different adjustments in the uppertail affect the income distribution. In particular, he argue that the adjustments of theoriginal data relies on three key parameters: i) How much is to be allocated to the top ofthe distribution; ii) how broad should the top b; iii) what share of the population shouldbe added to the top. Finally, Blanchet, Flores and Morgan develops a novel methodologyto find the point where tax data describes better the income distribution than surveydata. Their method can be used to correct for underreporting and nonresponse at thetop.In this paper, we depart from Bourginon (2018) and Blanchet, Flores and Morgan,instead of studing the whole income distribution, we only study the effects underreportingand nonresponse in the Gini coefficient. Our first contribution is that we demonstratethat not correcting for underreporting and nonresponse at the top does not necessarilyresult in an underestimated Gini coefficient.5To correct the Gini coefficient for undercoverage at the top, Atkinson (2007)proposes a simple and pragmatic approximation. He uses household-survey informationand tax data, and he approximates the Gini index as G = G1−p(1−Sp) +Sp, where G1−p4Flores (2018) also shows that household surveys do not adequately account for capital income. Thisproblem is also considered underreporting.5Regarding the effect of under coverage on the Gini coeffcient, Ceriani and Verme (2019) shows thatadding unit at the bottom or the top not necessary increase the Gini coefficient. Thus, our contributionextends and complements their results.98is the Gini coefficient computed from a household survey representative of a population’spoorest 1 − p percent, and where Sp is the income share owned by the population’s topp percent (e.g., the share of the top 1%) and computed from income-tax data (the sizeof the top). Alvaredo (2011) further develops this procedure and analytically derivesand extends his formula, proposing an exact Gini decomposition to be used when a pproportion of the population is not well measured in a data source but is better measuredin another source.6 However, Alvaredo’s exact decomposition requires to know: (i) howbroad the size of p is and (ii) the income distribution within the top p population. Aswas discussed by Cowell and Flachaire (2015) and Higgins, Lustig and Vigorito (2018)estimating this p is a major challenge.Some scenarios lack information on either of these elements (e.g., measuring eitherincome inequality adding undistributed profits or tax-haven wealth7). that is, we onlyknow i) how much should be allocated to the top of the distribution. In this context, andwithout knowing p. A modified version of the Atkinson approximation can be used undersuch information scarcity and thereby can correct the Gini coefficient.Thus, this paper’s second contribution is that it proposes a simple approximationof the Gini coefficient in the case of underreporting at the top which is a slightly modifiedversion of the traditional Atkinson approximation without the necessity of knowing thesize of the top p. In addition, the approximation’s analytical bias is computed. In addition,we show that the bias is higher when the traditional Atkinson approximation is used forsolving nonresponse and underreporting instead of the new adjusted formula to correctunderreporting. It shows, numerically, that the proposed approximation is near exactwhen used to correct the Gini coefficient for underreporting but may be heavily upwardbiased for correcting the Gini coefficient for nonresponse. That is, in order to use theunderreporting methodology we need to first correct for missing people at the top.Thus, this paper’s third contribution is to propose and apply a methodologyfor estimating the missing proportion at the upper tail. Thus, we can estimate an un-6Diaz-Bazan (2015) present another exact interpolation which is more exact than the Atkinson ap-proximation, but requires additional data.7Issues that are tremendously relevant for inequality measurement, see for instance, Alstadsæter etal. (2017, 2018).99derreporting and nonresponse corrected Gini coefficient by estimating the proportion ofnonrespondants and then apply the underreporting correction. It applies this method-ology to two countries: Chile and Canada, and corrects the income Gini coefficient byadding undistributed business profits, a source of capital income underreported in house-hold surveys, and administrative-tax-declaration data. Indeed, as Smith, Yagan, Zidarand Zwick (2019) argue “a primary source of top income is private “pass-through” busi-ness profit, which can include entrepreneurial labour income for tax reasons”, thus somepart of labour income is transformed into capital income and left inside the firm. Indeed,some tax reforms induces to keep business income inside the firm whereas others generateincentives to take out profits as dividends (see for instance the 2003 US dividend tax re-form). Thus, not accounting for undistributed business income when we measure incomeinequality could lead to artificial changes that bias levels and trends of income inequalityestimates. Thus, the methodology developed here could be used to estimate level andtrends of income inequality that are robust to tax changes and tax avoidance behaviour.The next section discusses the Gini coefficient in an undercoverage context. Sec-tion 3 proposes a Gini approximation when used for solving undercoverage at the top.Section 4 proposes a methodology to correct the Atkinson non nonresponse approxima-tion. Section 5 presents empirical aplications to test the methodology developed here,and Section 6 concludes the paper.3.2 Gini coefficient and undercoverage at the topThis chapter studies the Gini coefficient in the case of underreporting and non-responseat the top.8 The underreporting and non-response problems studied are discrete andonly affect the upper tail of the income distribution. The intuition to do such analysisis that, in some cases, there are some tax avoidance technologies that are costly andonly available for individuals at the top of the income distribution. Thus, only those8There are other biases that may affect measurement of top income inequality. For instance smallsample bias, that is, incomes at the very top span very large dollar ranges - the difference between the topand bottom of the top 0.01% is measured in multiple millions. But only a few hundred dollars separatepeople whose incomes are near the median - e.g. within the 51st percentile. When a random sample isdrawn, sampling variability can matter at the top end even if it is inconsequential for middle incomes.100who can afford this technology can underreport their income or, in the most extremecases, disappear completely from the income distribution. Another reason to assume adiscrete undercoverage is that in general income taxes are progressives with an importantproportion of the population that are exempt from income taxes, and those that areexempt from paying taxes do not have any incentives to under declare their income tothe tax authority. Finally, and this is relevant for capital income, in order that peoplecan start saving money (and receiving benefits from capital income), one needs first toget enough to consume the basics or to get certain income such that liquidity constraintsare not binding. After that people can satisfy their basics needs, people can start buyingassets to receive capital income. Thus, only after certain income level, people have somecapital income to underreport.3.2.1 Underreporting at the topSuppose the top p proportion of a population underreports their income, due to survey-measurement error, tax evasion, or top coding. Let yi be the true income of each individualor household unit i, and y∗i the observed income for this unit i. Thus,yi = y∗i + zi, (3.1)with zi being the amount underreported for each unit i and zi ≥ 0 ∀i. In addition,z ≡∑ni=1 zi∑ni=1 yiis the proportion of the total underreported income over the total true in-come. Figure 3.1 represents this type of under-reporting graphically. The dashed Lorenzcurve describes the observed information y∗i , and the bold Lorenz curve represents thedistribution of true income, yi. Also, φ ≡∑ni=n·p+1 yi∑ni=1 yiis the share of the total income heldby the top p proportion of the population.Note that the distribution for people whose income is lower than the 1 − p per-centile is the same with and without underreporting. In this context, the true incomedistribution is not necessarily more unequal than the observed one. Proposition 1 estab-lished the circumstances under which this type of outcome occurs.Proposition 4. Given nonnegative underreporting at the top, the Gini coefficient (mea-101Figure 3.1: Relative Lorenz curves with underreported income at the topbbbbbbbbbbbbbbbbbbbbSp1− p1− z1od1d2abThe dashed relative Lorenz curve describes the observed income distribution and the bold Lorenz curvedescribes the true income distribution, a, b, d1 and d2 are areas; and 1− z is the proportion of the totalobserved income divided by the total true incomesured using the observed income distribution) is not necessarily lower than the Gini coeffi-cient measured using the true income distribution. The true Gini coefficient is lower thanthe observed Gini coefficient when underreporting sufficiently reduces inequality inside theunderreported group.Proof. Suppose that a fraction z of income is underreported, that S1−p is the incomeshare of the group reporting their income accurately, and, Sp is the income share of thegroup underreporting their income. Total reported income is the denominator for bothshares. S¯1−p and S¯p are the income shares of the same groups when underreported incomeis included. Then, the following relationships hold:S¯1−p = S1−p (1− z)S¯p = Sp (1− z) + zFrom Alvaredo (2011), the Gini coefficient G∗ (uncorrected for underreporting)isG∗ = (1− p)S1−pG1−p + pSpGp + Sp − p (3.2)102where G1−p is the Gini coefficient of those who report accurately; Gp is the Gini coefficientof reported income for underreporters (i.e., this income is lower than their true income);and p is the fraction of underreporters. The Gini coefficient for true income G isG = (1− p)S1−p (1− z)G1−p + p (Sp (1− z) + z) G¯p + Sp (1− z) + z − p (3.3)with G¯p as the income underreporters’ Gini coefficient, including their underreportedincome. Thus,G∗ −G = zS1−p (1− p)G1−p − p[Sp(G¯p −Gp)+ G¯pS1−pz] − zS1−p. (3.4)And, G∗ −G > 0 if:G1−p >p[Sp(G¯p −Gp)+ G¯pS1−pz]+ zS1−pzS1−p (1− p) . (3.5)Thus, G1−p can only be less than 1 only if(G¯p −Gp)< 0.As an example, assume the observed income distribution is (1, 20, 50) and the trueincome distribution is (1, 120, 150). In the first distribution, the Gini coefficient is 0.460,and in the second distribution, it is 0.367; however, if underreporting increases inequalitybetween the underreporters, then the Gini coefficient of the true income distribution willbe higher than the observed distribution Gini coefficient.3.2.2 Nonresponse at the topSuppose that the richest p proportion of the population does not report any information(e.g., if p equals 0.01, the income information for the top 1% is truncated), that thisnonreported population holds a proportion φ of the total income, and that the trueincome yi for the other 1−p fraction of the population is observed. Figure 2 characterizes103this situation using Lorenz curves.Figure 3.2: Lorenz curve in a context of nonresponse at the topbbbbbbbbbbbbbbbbbbbb1− p1− φo 1bb1Spp is the proportion missing, φ is the proportion of the total income held by missing population, the boldblack line is the equality line for the entire population, and the dashed black line is the equality line forthe population observedSince the shape of the missing Lorenz curve segment from 1−p to 1 is unknown, wecannot say anything a priori about the Gini of the reporters versus the true distribution,as described in proposition 2, not necessarily the true income distribution is more unequalthan than the observed income distribution.Proposition 5. Given nonresponse at the top, the Gini coefficient (measured using ob-served information) can be greater than the Gini coefficient of the true-income distribution(including nonrespondents’ information).Proof. The true Gini coefficient can be written asG = (1− p)S1−pG1−p + pSpGp + Sp − p, (3.6)where G1−p is the observed Gini coefficient, 1 − p is the proportion of peopleobserved, S1−p is the share of the total income (observed and unobserved) received by theobserved people, and Gp is the Gini coefficient of the unobserved people. Thus,104G1−p −G = pG1−pS1−p − pSpGp − Sp + p. (3.7)And, G1−p −G > 0 if:G1−p >Sp (Gpp+ 1)− ppS1−p(3.8)For example, if the truncated distribution is (0, 1) and the true distribution is(0, 1, 1), the Gini coefficient for the truncated is 0.5, while for the observed is 0.33. How-ever, if the observed Gini coefficient is low enough, the Gini coefficient, computed afterincluding nonrespondents, will be higher than the observed Gini coefficient.9 Ceriani andVerme (2019) prove a similar version of Proposition 2: they show that including additionalpeople in the upper part of the income distribution, each of them earning the maximumobserved income does not necessarily increase the Gini coefficient.3.2.3 Nonnegative underreporting and nonresponse: the jointcaseIn real-world applications underreporting and nonresponse at the upper tail are oftenboth presented. Figure 3.3 uses Lorenz curves to describe this situation.Proposition 6. Given underreporting and nonresponse at the top, the Gini coefficient(measured using the observed-income distribution) can be greater than the Gini coefficientof true-income distribution.Proof. Follows directly from propositions 1 and 2.From Propositions 1, 2, and 3, it can be concluded that the existence of un-dercoverage at the top does not imply an over or underestimated Gini coefficient if this9Bollinger et al. (2018) present evidence that the income distribution, including nonrespondents, ismore unequal than the observed distribution.105Figure 3.3: Relative Lorenz curves in a context of underreported income and nonresponseat the topbbbbbbbbbbbbbbbbbbbbSp1− puc1o 1− pnr1− φ1− φ− z1The dashed relative Lorenz curve describes observed-income distribution; the bold Lorenz curve describestrue income; 1 − φ − z is the proportion of total income observed in a dataset; z is the proportion oftotal income underreported in that data set; φ is the propotion of income belonging to nonrespondants;1−puc is the proportion of people reporting their total income in the dataset; puc−pnr is the proportionof people who appear in the dataset but underreport their income; and pnr is the proportion of peoplenot appearing in the dataset.undercoverage is uncorrected.3.3 A Gini approximation for undercoverage at thetop3.3.1 An approximation as a solution for underreporting at thetopOne way to correct for underreporting is to use the Atkinson approximation (Atkinson,2007; Alvaredo, 2011); Gatk, defined as:Gatk = G1−p · (1− Sp) + Sp, (3.9)where, G1−p is the Gini coefficient for the bottom 1 − p proportion of the population,and Sp is the share of the total income held by the top p proportion. Alvaredo (2011)shows that if p is infinitesimal, then the true Gini coefficient G is identical to the Atkinson106approximation.Equation (3.9) can be modified slightly to estimate the Gini coefficient whenunderreporting is presented using the observed (and underreported) income, y∗i , and theproportion of total income that is underreported, z. We simply replace G1−p with G∗, theGini coefficient of y∗i , and Sp with z. Denote this modified Atkinson approximation byGur :Gur = G∗ (1− z) + z. (3.10)Equation (3.10) can be shown to be an upper bound of the true Gini coefficient G.Proposition 7. Gur is an upper bound of the true Gini coefficient G.Proof. Using Figure 3.1,a + b+ d1 + d212= G+ 2 · (d1 + d2) , (3.11)where G is the true Gini coefficient. Notice that(a+ d1)12=(a + d1)(1−z)2· (1− z) = G∗ · (1− z) , (3.12)where G∗ is the Gini coefficient of the observed data. In addition,b+ d212= z (3.13)Thus,Gur −G = 2 · (d1 + d2) .In Figure 3.1, the bias generated by (3.10) is equal to 2 (d1 + d2). This biasstems from the Atkinson approximation’s assumption that one individual owns all theunderreported income. Proposition 5 computes this bias analytically.Proposition 8. Biasur = Gur −G is given by:107p [z·( 1−Gp )−θ(z) (Sp (1− z) + z)] , (3.14)where p is the proportion of people underreporting their income; z is the proportionof total income that is underreported; Gp is the Gini coefficient of the underreporters’observed income; θ(z) is the difference between the true Gini coefficient of underreportersG¯p, and the observed Gini coefficient of the underreporters Gp; and Sp is the observed-income share of the p proportion that underreports their income.Proof. From Proposition 1, the true Gini coefficient can be written asG = (1− p)S1−p (1− z)G1−p + p (Sp (1− z) + z) G¯p + Sp (1− z) + z − p, (3.15)where S1−p and Sp are the observed income shares of the people who, respectively, doand do not underreport their income. Those shares use total reported income as thedenominator. Also, G¯p = Gp+θ(z); that is, the true Gini coefficient of the underreportersG¯p is the Gini coefficient computed using those underreporters’ reported income Gp plusthe change in the underreporters’ Gini coefficient after including their underreportedincome θ(z). Thus,G = (1− z) [(1− p)S1−pG1−p + pSpGp + Sp − p]+pθ(z) (S1−p (1− z) + z)+pG1−pz+z−zp.(3.16)Given that the observed Gini coefficient is equal toG∗ = [(1− p)S1−pG1−p + pSpGp + Sp − p]and that Gur = G∗ (1− z) + z,G = Gur + p [z (Gp − 1) + θ(z) (Sp (1− z) + z)] , (3.17)and,Gur −G = p [z( 1−Gp )−θ(z) (Sp (1− z) + z)] . (3.18)108In Proposition 5, the underreporting approximation bias increases with p butdecreases with θ(z) and Sp; however, an increase in z does not necessarily increase thebias.3.3.2 A Gini approximation as a solution for nonresponse at thetopThe Atkinson approximation can be directly used to correct the Gini coefficient for non-response. Assuming the nonrespondents’ income share is φ, the Atkinson approximationfor the full population’s Gini coefficient in the case of nonresponse at the top is:Gnr = G1−p (1− φ) + φ, (3.19)where, G1−p is the observed population’s Gini coefficient. This is an upper boundof the Gini coefficient.Proposition 9. Assuming nonrespose at the top, Gnr is greater than the true Gini coef-ficient, and the bias of this approximation is given by:Biasnr ≡ Gnr −G = p [G1−p (1− φ) + 1− φGp] , (3.20)where p is the proportion of people underreporting their income; φ is the proportion oftotal income that is nonreported; G1−p is the Gini coefficient of those with known income;and Gp is the Gini coefficient of the nonrespondents.Proof. Using the previously applied Gini decomposition, the true Gini coefficient isG = (1− p)S1−pG1−p + pSpGp + Sp − p, (3.21)109and the Atkinson approximation for solving nonresponse isGnr = G1−p (1− Sp) + Sp. (3.22)Thus,Gnr −G = p [G1−p (1− Sp) + 1−GpSp] .Recall Sp = φ. Also, because φ < 1 this is always greater than 0. Thus, Gnr > G.An increase in p increases the nonresponse bias, but an increase in the share ofnonrespondents reduces this bias. Greater inequality among nonrespondents also reducesthe bias.3.3.3 A Gini approximation for underreporting and nonresponseat the topIn Figure 3.3, underreporting and nonresponse are both presented. Using the Gini ap-proximation proposed to correct underreporting without correcting for nonresponse, orcorrecting only for nonresponse without correcting for underreporting, could lead to Ginicoefficients lower or higher than the true one.Proposition 10. When underreporting and nonresponse both exist in the top of the in-come distribution, estimating Gur to correct only for underreporting without correctingnonresponse does not necessarily establish an upper bound of the true Gini coefficient.Proof. Follows directly from proposition 2 and proposition 4.Proposition 11. When underreporting and nonresponse jointly exist in the top of theincome distribution, use of the Atkinson approximation to correct only for nonresponsewithout correcting for underreporting does not necessarily establish an upper bound of thetrue Gini coefficient.Proof. Follows directly from proposition 1 and proposition 6.110Another way to correct for both underreporting and nonresponse is to use theAtkinson approximation to consider underreporting and nonresponse both as a form ofnonresponse. This procedure generates a Gini coefficient that is an upper bound of thetrue Gini coefficient.Proposition 12. When underreporting and nonresponse jointly exist in the top of theincome distribution, use of the nonresponse Atkinson approximation to correct for non-response and underreporting generates an upper bound of the Gini coeffient. The bias isgiven byBiasnr + (1− φ)Biasur1−pnr ,where Biasnr is the bias for non-response Atkinson approximation and Biasur1−pnr is theGini approximation for correcting underreporting in the proportion of the population thatunderreports, pu.Proof.(1− z − φ)Gpu + φ+ z(1− φ)((1− φ− z)1− φ G∗pu+z1− φ)+ φ(1− φ) (Gpu +Biasur1−pnr)+ φG+Biasnr + (1− φ)Biasur1−pnr3.3.4 The underreporting vs the nonresponse approximationAs established in propositions 4, 6 and 9, the approximations for correcting underreportingGur and/or nonresponse Gnr are upper bounds of the Gini coefficient. There are twoways to correct for underreporting, one is to correct using the whole population evenif that population is not well measured and the other is to eliminate the proportionthat is not well measured and correct the gini coefficient using the straight Atkinsonapproximation. Assuming 1) φ and z are known, 2) the true income distribution until the1111− p percentile is known, and 3) an income level y∗i lower than the true income is knownfor the top p proportion of the population, then either Gnr or Gur can be used to correctthe Gini coefficient for undercoverage. Gnr can be estimated using the Gini coefficientof those accurately reporting their income and φ. Gur is estimated by computing theGini coefficient of the observed distribution and z.10 Proposition 10 establishes that thenonresponse approach Gnr (i.e., dropping underreporters’ information) generates a biggerbias than using the underreporting approach Gur. Figure 3.4 illustrates these results.Figure 3.4: Relative Lorenz curves of both underreporting and nonresponse problemsbbbbbbbbbbbbbbbbbbbbSp1− p1− φ1− z1ae1od1e2e3c1fbd2c2a, b, c1, c2, d1, d2, e1, e2, e3 and f are areas; 1 − p is the proportion of the population that is wellmeasured, 1 − φ is the proportion of the total income held by the first 1 − p percentiles; 1 − z is theproportion of the total income observed; the bold line is the equality line for the whole population; theblack line is the equality line for the reported population; and the dashed line is the equality line for thep richest percentilesProposition 13. Gnr is higher than Gur.Proof. From Figure 3.4, given Gp ≡ a(1−φ)·p2. Thus,a+ b+ c1 + c2 + d1 + d2 + e1 + f12= Gp (p · (1− φ))+φ−(1− p)+φ (1− p) ≤ Gp·(1− φ)+φ.(3.23)10φ is greater than z because φ also includes the proportion of underreporters captured by the house-hold survey. One way to know φ and z is by having two harmonized income-distribution informationsources: a household survey and income-tax data. Harmonized sources are essential, but in practice, suchharmonized sources may not exist —e.g., Atkinson, Piketty, and Saez (2011) used nonharmonized data,but Jenkins (2017), Burkhauser et al. (2018), and Piketty, Saez, and Zucman (2018) did.112And,Gur + 2 · e1 = G+ 2 · (d1 + d2 + e1) ≤ Gnr. (3.24)For nonresponse, the bias is greater than 2 (d1 + d2 + e1). This bias is higherthan the underreporting case 2 (d1 + d2) because the proportion of income assumed tobe held by one individual is larger when using the Atkinson approximation to correctfor nonresponse instead of the approximation proposed for underreporting (φ is greaterthan z). This shows that an imperfect measure of the income held by the top part of thedistribution is preferable to not having any measure of that same income. For instance,it is better to have the upper part with top coding, even if the top coded value is far fromthe real income. Thus, before applying the corrected underreporting approximation, weneed to correct for nonresponse first.3.3.5 Montecarlo simulationTo evaluate the magnitude of the benefits from using the underreporting approxima-tion over the nonresponse approximation we performed Montecarlo simulations. In thesesimulations, we assumed that (i) the bottom 1 − p of the population is drawn from anexponential distribution;11 that (ii) the true income for the richest p of the population isdrawn from a Pareto type I distribution (with αr as the Pareto parameter and w¯ as thethreshold income); that (iii)the observed income for the top p proportion is drawn froma Pareto type I distribution with αs parameter and w¯ income threshold with αs > αr(i.e., the income distribution for the observed top tail is less unequal than the real incomedistribution).12 In addition, this modified Atkinson approximation can be implementedin a top-coded environment (assuming top coding at w¯). We denote to this estimated11Results are presented via a single parametric distribution; however, the bottom part of the distri-bution could also be approximated with Singh-Maddala, Dagum, or GB2 distributions, which producesimilar results.12For simplicity, results are presented using a Type I Pareto distribution; however, as Atkinson (2017)shows, Pareto type I is not a perfect tool for studying top income shares and is rather “at best a convenientfirst summary of the extent of the income concentration.” In addition, Jenkins (2017) showed (for theUnited Kingdom) that a Pareto type II is preferable to the Pareto type I typically used at the thresholds.Blanchet et al. (2018) also used a Pareto type II to fit income inequality at the top.113Gini coefficient assuming top coding by Gtop.Table 3.1: Montecarlo simulations1− p Gur Gtop Gnrαr = 1.10.9 0.0057 0.0082 0.07520.95 0.0023 0.0033 0.05010.99 0.0002 0.0003 0.0156αr = 1.50.9 0.0042 0.0085 0.12440.95 0.0015 0.0030 0.07330.99 0.0001 0.0002 0.018αr = 20.9 0.0017 0.0068 0.14570.95 0.0006 0.0023 0.08180.99 0.0001 0.0002 0.0188αr = 2.40.9 0.0003 0.0057 0.150.95 0.0001 0.0019 0.0850.99 0.0001 0.0001 0.016Mean Square Error of the difference between the real Gini coefficient (G), Gur , Gtop and Gnr. Averageover 1000 simulations with n = 100, 000, λ = 140,000(exponentinal parameter), αr (real Pareto parameter)and αs = 2.5 (survey Pareto parameter).Table 3.1 shows that a higher p generates a larger bias; however, the bias causedby underreporting or top coding is quite small, which implies that knowing an income dis-tribution for underreporters is unnecessary and only the proportion of total underreportedincome is needed to correct the Gini coefficient.13In contrast, the bias in a nonresponse context can approach 15 percentage pointsof the Gini coefficient. This bias depends on the proportion of nonrespondents, p, which,in practice, is difficult to determine. The next section proposes a methodology to estimatethe missing population at the top p when only φ, µ1−p (the average income of truthfullreporters), a top-coded value χ, and an estimate for the Gini coefficient of nonrespondentGp are known.13In this theoretical exercise we assume that we know 1 − p but in real world applications p is veryhard to estimate. The recent literature (Atkinson, 2017; Jenkins, 2017; Bourguignon, 2018) suggest totry with p=0.01, p=0.05 and p=0.1.1143.4 An extension of the Atkinson approximation inthe case of nonresponseThis section proposes a methodology to estimate the proportion of the missing popula-tion p.14 To understand the estimator’s construction, assume that p (the proportion ofnonrespondents), φ (the proportion of income belonging to nonrespondents), and Gp (theGini coefficient of nonrespondents) are all known. Alvaredo (2011) shows the true Ginicoefficient G is equal toG = (1− p)(1− φ)G1−p + pφGp + φ− p, (3.25)where G1−p is the observed population’s Gini coefficient. Also, the observations of re-spondents are given by (x1, ..., xn), and those of nonrespondents are (xn+1, ..., xN), withxi ≤ xi+1 ∀i. We begin by constructing a synthetic distribution Ω for the income distri-bution such that the new distribution is composed byΩ ≡ ((x1, ..., xn), (x∗, ..., x∗)) ,where x∗ is a nonnegative value such that xn ≤ x∗ ≤ µp and µp is the average nonre-spondent income. Notice that φ is divisible into two proportions, δ and λ, and δ can bedefined asδ ≡ px∗(1− p)µ1−p + pµp ,with µ1−p being the average income of the respondents or average observed income. Also,define χ ≡ xn, the maximum value from the observed 1− p proportion of the population.Then, for instance, if δ = φ, then δ is formed by attributing the average nonrespondentincome to each nonrespondent, or δ could be formed by assuming each nonrespondentearns the highest reported value χ. Consequently, λ is defined as λ ≡ φ− δ. Thus, (3.25)14Blanchet, Flores, and Morgan (2018) propose a new methodology to find this proportion, what theycall a merging point—the point at which tax data becomes more representative than household data.115can be written asG = (1− λ) [(1− p)S∗1−pG1−p + S∗p − p]+ pSpGp + λ− λp, (3.26)where S∗1−p (1− λ) = 1−φ, S∗p (1− λ) = δ. The Gini coefficient of the Ω synthetic incomedistribution is[(1− p)S∗1−pG1−p + S∗p − p].15 Then, (3.26) can be written as:G = (1− λ) [G∗∗] + λ+ p (φGp − λ) , (3.27)where G∗∗ is the Gini coefficient of the income distribution that includes the observedpeople and the synthetic nonrespondents. Also, G∗∗ur ≡ (1− λ) [G∗∗] + λ is the Giniapproximation for correcting underreporting in this new synthetic income distribution.Thus,G = G∗∗ur + p [φGp − λ] . (3.28)This means the Gini coefficient can be written as the sum of two terms: an underreportingGini approximation and p [φGp − λ]. If λ = φGp, then G = G∗∗nr. Choosing λ (i.e., themissing-income proportion left as underreporting in the synthetic income distribution Ω)transforms the true Gini coefficient into an underreporting Gini approximation.Though, in real world applications, neither p norGp are known, p can be estimatedvia an estimated value for Gp (this estimator is Gˆp). To estimate p assume that δ iscomposed of nonrespondents, each with income equal to χ, chose λˆ = φGˆp, and use thefact that S∗p(1− λˆ)= δˆ . Moreover, S∗p =pχ(1−p)µ1−p+pχand δˆ = φ− λˆ. Thus,pˆ =φ(1− Gˆp)µ1−pχ1− φ+ φµ1−pχ(1− Gˆp) . (3.29)Notice that if Gˆp = 1, then pˆ = 0, the empirical counterpart to (3.27) returns to theAtkinson approximation for nonresponse. In addition, the bias of this approximation can15Note that the Gini coefficient of (x∗, ..., x∗) is 0.116be written asGˆ∗∗nr −G = −pφ(Gp − Gˆp)− (1− λ) (pˆ− p) (1 + (1− φ)Gp) . (3.30)Let assume that adding nonrespodants did not change the Gini coefficient for the top ofthe distribution. In particular, let Gˆ∗1% the estimated Gini coefficient of the top 1% usingobserved data. Assume that the Gini coefficient of the true upper tail Gˆp (the upper tailthat includes nonrespondants) is the same that Gˆ∗1.16 An algorithm can thus be writtento generate upper and lower bounds for the Gini coefficient under nonresponse.Algorithm 1: Correcting for nonresponse If the distribution for the bottomp fraction is known, then the following are also known: the average of the known distri-bution µ1−p, the Gini coefficient G1−p, and the maximum observed value in the observedincome distribution, χ.i Compute Gˆ1−p using observed dataii Knowing φ, χ and assuming Gp = Gˆ∗1%, λˆ and pˆ can be obtained from (3.28) and(3.29) respectively.iii Having λˆ, µ and pˆ means δˆ can be computed from δˆ = φ− λˆ.iv Having δˆ means G∗∗ can be computed as G∗∗ = (1− pˆ)(1− δˆ)G1−p + δˆ − pˆ. Theapproximation for the Gini coefficient G can be estimated as G∗∗nr =(1− λˆ)G∗∗+λ.Notice that surveys with high maximum values (χ) related to the average (µ) will estimatevery low missing proportions, this is because an additional synthetic individual will covera higher proportion of the missing income. The following section tests this methodologyon household surveys.16The Gini coefficient of the observed top 1% and the nonrespondants is G1%+p =S1%S1%+Sp0.010.01+pG∗1 +SpS1%+Spp0.01+pGp +SpS1%+Sp− p0.01+p.1173.5 Empirical ApplicationsThis section presents an empirical application, using data from two countries, Canadaand Chile. The applications use the Atkinson approximation to solve underreporting andnonresponse at the top. The application is meant to test the methodology developed inthe previous section.3.5.1 Application. Income inequality and undistributed busi-ness profits.It is known that the line between labour and capital is inherently imprecise for the smallbusiness sector, and it is certainly possible that tax accounting differs from the common-language way of separating labour from capital (Kopczuk, 2016). In some cases, partof the labour income is left inside the firm as undistributed business profits and notaccounted in the year that income was generated. Indeed, undistributed business profitsare typically an underreported component of income. It is not declared in personal taxrecords nor requested in surveys. Recent works (Adalæster et al., 2017; Flores et al.,2019; Fairfield and Jorrat, 2016; Gutierrez et al., 2015; Smith, Yagan Zidar and Zwick,2019; Wolfson et al., 2016) show that including undistributed business profits in inequalitymeasurements increases income inequality measures.Moreover, the level of undistributed business profits is closely related to taxchanges, as Smith, Yagan, Zidar and Zwick (2019) argue, an important proportion oftop income is private “pass-through” business profit for tax proposes. Indeed, some taxreforms induces to keep business income inside the firm whereas others generate incentivesto take out profits as dividends (see for instance the 2003 US dividend tax reform). Thus,not accounting for undistributed business income could lead to artificial changes thatbias levels and trends of measured income inequality. Thus, the methodology developedhere could be applied to estimate level trends of income inequality that are robust to taxchanges and tax avoidance behaviour.The Canadian data we used in this application comes from the Survey of Finan-118cial Security, while Chilean data comes from Encuesta de Caracterizacio´n Socioecono´micaNacional. The Survey of Financial Security, which has four available waves (1999, 2005,2012, and 2016), provides a record of Canadian residents’ market incomes, assets, anddebts. The Encuesta de Caracterizacio´n Socioecono´mica Nacional survey is a household-level survey that measures different types of income in Chile. We used household marketincome to compute each country’s Gini coefficient. Canadian undistributed business prof-its data comes from CANSIM Table 36-10-0117-01 and Chilean application data comesfrom Flores et al. (2019).17 We use Chile and Canada because both countries have in-tegrated tax systems where the corporate tax paid by firms could be partially used as acredit for the dividends received by individuals. This creates incentives for tax planning.18With the above dataGur can be computed by estimating the Gini coefficient of thehousehold survey G∗, and the proportion of total income corresponding to undistributedbusiness profits z can be used to adjust the Gini coefficient. Here we are not correcting fornonresponse by assuming that undistributed business profits are pure underreporting Gur.Otherwise, we can correct for nonresponse at the top and then correct for underreportingG∗∗nr. For the latter estimation we need to estimate the proportion of nonrespondant, to dothat, we will use the methodology developed in the previous section. Then, we compareGur, G∗∗nr with the uncorrected Gini coefficient G∗.ResultsFigure 3.5 shows the empirical application’s Chilean results. The Gini coefficient decreasedthroughout 2003 to 2013, then increases from 2013 to 2015 to return to 2013 levels in 2017.This can be caused by retained earnings. In particular, retained earnings (as a proportionof total income) increased between 2003 and 2015. In this last year, retained earningswere 16 percent of the total income. However,retained earnings fell to 12 percent in 2017.This can be explained by the 2014 tax reform that reduced the integration rate betweencorporate and personal taxes along with others tax changes that reduces the incentivesto retain earnings inside the firms.17Canadian retained earnings data comes mostly from financial corporations.18Tax planning using small businesses in Canada and Chile was broadly studied in the literature. Seefor instance Fairfield and Jorrat (2016); Lopez et al. (2016); Wolfson et al. (2016).119Figure 3.5: Corrected vs uncorrected Gini coefficient for Chile.5.55.6.65.72003 2006 2009 2011 2013 2015 2017Nonresponse and underreporting UnderreportingUncorrectedThe Chilean Gini coefficient for income using Encuesta de Caracterizacio´n Socioecono´mica Nacionalinformation and Flores et al. (2016)This result is robust for Gur and G∗∗nr. The difference between the adjusted Ginicoefficient and the observed Gini coefficient increases during the analyzed period. Inaddition, the difference between Gur and G∗∗nr are tiny, this is because the maximumincome reported in CASEN survey is very high compared to the average income.Figure 3.6 shows Canadian results. The adjusted Gini coefficient increases be-tween 1999 and 2005 and decreases between 2005 and 2016. However, the unadjustedGini coefficient always increases between 1999 and 2016. One possible explanation for thehuge jump in the Gini coefficient in 2005 is because retained earnings in 2005 where muchmore bigger than they were in other years. In particular, 2005 retained earnings weremore than 10 percent of total income, whereas for other years, retained earnings were atmost 5 percent. Retained earnings could be bigger in 2005 because it was that it was ayear before the 2006 dividend tax reform that increases the dividend tax credits receivedfrom publicly traded firms and large CCPC. This reform creates incentives to pay moredividends instead of retain earnings inside firms.120Figure 3.6: Corrected vs uncorrected Gini coefficient for Canada.45.5.55.61999 2005 2012 2016Nonresponse and underreporting UnderreportingUncorrectedThe Canadian Gini coefficient for income using SFS information and CANSIM Table 36-10-0117-01Both examples shows that trends of the nonresponse and underreporting Ginicoefficients for both countries are different than the observed Gini coefficient. In addition,given the survey structure and maximum values, we can conclude that retained earningsare more an underreporting issue than a nonresponse one.3.6 ConclusionThis work studied the Gini coefficient for underreporting and nonresponse at the top twoissues that recently attract the attention of several scholars that estimate incore inequality.The first contribution is that this work proves that underreporting or nonresponse doesnot necessarily result in a true Gini coefficient that is higher than the estimated Gini.That is, we are not necessarily estimating underestimated Gini coefficients when we usea household survey.In addition, a correction of the Atkinson approximation approximates the Gini121coefficient for correcting underreporting or top coding well —this approximation can beused to correct inequality measurements of income sources (e.g., undistributed businessprofits) concentrated at the top but unreported in household surveys. A key feature ofthis underreporting approximation is that it is not necessary to know the proportion ofunderreporters and sometimes it is also not necessarily to know the missing populationeither. Thus, despite not being estimating the whole income distribution, contribute toBourginon (2018) and Blanchet, Flores and Morgan (2018) by developing an adjustmentwhere it is not necesseraly to find the size of the top neither the size of the missing peopleto obtain results close to the true ones.Moreover, we developed a simple adjustment that combines top coding and theunderreporting approximation to construct an estimation of the Gini coefficient in thepresence of nonresponse and underreporting at the top. We estimate the missing popula-tion and then correcting for underreporting income. We applied this methodology to twocountries, Chile and Canada where we show that retained earnings is more an underre-porting issue than a nonresponse one. Our methodology can be easily replicated for othercountries and additional undercoverage examples.122ConclusionIn this thesis, I present new evidence on economic inequality and intergenerational mo-bility in the Canadian and Chilean context. Chapter 1 first study that estimate intergen-erational mobility in Chile using administrative records. We build a data set that linksparental and child earnings using information from the formal labour sector and the placeof residence of children during their adolescence. Our analysis reveals that intergener-ational mobility at the national level is significantly lower than what was estimated inprevious research. However, intergenerational mobility is extremely non-linear. We foundthat mobility is very high for the bottom 80 percent of the earnings distribution but isvery persistent at the upper tail of the parental and child distributions.In addition, Chile is a highly heterogeneous country in its intergenerational mo-bility measures at the regional level. For instance, Antofagasta, which is a mining region,has a probability of rags to riches higher than 0.3. This result is in line with what Conollyet al. (2018) founds for the US and Canada. Meanwhile, regions like Araucan´ıa or ElMaule have a circle of poverty probability higher than 0.3. It is worth digging a littledeeper in future research to understand why those regions are so persistent in poverty.We also find heterogeneity within the Metropolitan region, with municipalitieshaving a circle of privilege probability higher than 0.7, and other municipalities with acircle of poverty probability closer to 0.3. We also learn that the variance of persistenceat the top is higher than the variance of upward mobility. This means that the place ofresidence affects children of upper-earnings parents more than middle- or poor-class par-ents. Future research should focus on understanding the causes behind these differences.Although our work is descriptive in nature, it sheds lights on intergenerational mobility123in a highly unequal country that does not belong to the advanced economies.Moreover, we make a some methodological contributions. We use RIF regressionsand Kernel conditional densities to study intergenerational mobility at the top. Thosetools help usus to show that intergenerational mobility is very persistent at the top inChile. In addition, we differentiate the Gatsby curve for Chile and Santiago using twomeasures of intergenerational mobility: absolute intergenerational mobility and relativeintergenerational mobility. We show that the Gatsby curve is valid for persistence andupward mobility for Chile but only for persistence for la Regio´n Metropolitana. Thishelp us to differentiate different mechanisms that may affect intergenerational mobilityfor Chile.This work builds on previous national literature and brings the state of researchup to the robustness of analysis seen among works in developed economies. As such, notonly does it provide more useful information for academics; it also provides an importantcounterpoint to similar works from developed economies by analyzing intergenerationalearnings mobility in a non-developed [o developing] economy in a way that can be con-trasted with the results of that literature. We believe that, by providing a clearer pictureof how intergenerational earnings mobility occurs in Chile at a regional level, this work canboth inspire further research on the matter both in Chile and other developing economies.These results can also help Chilean authorities better understand how and where to ap-ply certain related social/economic programs in order to improve their impact, as well asprovide input for drawing up and discussing proposed bills affected by this study’s results.Chapter 2 argues in favour of using retained earnings as a measure of accruedcapital gains, clarifies the difference between left money inside the firm for closely held cor-porations and for publicly traded firms. In particular, we make a conceptual contributionby showing that for closely held corporations retained earnings could have a consumptionvalue in addition to the change of wealth value that the literature acknowledged. Theintuition is that goods can be bought inside the firm to satisfy personal consumption.In addition, we propose a methodology to impute corporate retained earningsto households. The convenience of this methodology is that it only needs market income124information from a household survey and aggregated retained earnings information from ahousehold survey. We apply this methodology to Canada. We show that including accruedcapital gains increases measured income inequality and, more importantly, changes theobserved trend of inequality which is similar to what Davies, Fortin and Lemieux (2017)found for Canadian wealth.We compare our imputation method with the broadly used capitalization impu-tation method. We obtain similar results by using the capitalization method to imputeretained earnings. Even though the parametric imputation procedure developed here isnot perfect, it gives results that are close enough to the capitalization method. This factsuggests that in the context of scarcity of data, for example, developing countries wherewe cannot access easily to administrative data, and this procedure could be useful forimputing capital income in such contexts.Another application of the imputation methodology developed here can be usedin the context of undercoverage at the upper tail. Recent literature, such as Flores (2018),show that because of nonresponse and underreporting, household surveys only accountfor 20 percent of the capital income that appears in national account data. Thus, becausethe parametric imputation procedure developed here only need a household survey andnational account data, could be used to correct for underreporting of capital income inhousehold surveys. Future research is needed to study the imputation procedure in suchcontexts.Chapter 3 studied the Gini coefficient for underreporting and nonresponse atthe top two issues that recently attract the attention of several scholars that estimateincore inequality. The first contribution is that this work proves that underreporting ornonresponse does not necessarily result in a true Gini coefficient that is higher than theestimated Gini. That is, we are not necessarily estimating underestimated Gini coefficientswhen we use a household survey.In addition, a correction of the Atkinson approximation approximates the Ginicoefficient for correcting underreporting or top coding well —this approximation can beused to correct inequality measurements of income sources (e.g., undistributed business125profits) concentrated at the top but unreported in household surveys. A key feature ofthis underreporting approximation is that it is not necessary to know the proportion ofunderreporters and sometimes it is also not necessarily to know the missing populationeither. Thus, despite not being estimating the whole income distribution, contribute toBourginon (2018) and Blanchet, Flores and Morgan (2018) by developing an adjustmentwhere it is not necesseraly to find the size of the top neither the size of the missing peopleto obtain results close to the true ones.Moreover, we developed a simple adjustment that combines top coding and theunderreporting approximation to construct an estimation of the Gini coefficient in thepresence of nonresponse and underreporting at the top. We estimate the missing popula-tion and then correcting for underreporting income. We applied this methodology to twocountries, Chile and Canada where we show that retained earnings is more an underre-porting issue than a nonresponse one. 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Elite colleges and upward mobility to top jobs and topincomes. American Economic Review, 109(1), 1-47.140Appendix AAppendix for Chapter 1A.1 Data appendixThis appendix discussed the linkage and cleaning process of the data used in this disser-tation. Fist we deliver to the Social Registry Services with 10,000,000 of Tax Numbers(RUT) so that they can create the matching between parents and Children. Then, wework with the unemployment insurance program dataset and we use the RUT as linkagekey. Finally, we use information from the education ministry to obtain children address,where we use RUT as linkage key.Table A.1: Linkage unitsSubset NTotal children cohort 2,817,428Children that worked more than 6 months 1,889,107Total parental cohort 3,021,785Parents that works more than 6 months 1,275,664Total number of linkages (excluding duplicates) 505,524A.1 shows the total number of children that are in the cohort studied. Then weshow the total amount of children that works more than 5 months and who earned morethan half the minimum wage in average. In addition, we included the total number of141potential parents and the number of parents that works more than 5 months receivingmore than half the minimum wage. Finally, this table shows the number of total linkagesused for the estimates. Comparing with Corak and Heisz (1999) we have more than100,000 linkage for a country with half the size of the Canadian population.Table A.2: Educational linkagesGrade included Linkage rate12th grade 76.17%Including also 11th grade 84.41%Including also 10th grade 87.77%Including also 9th grade 91.96%Including also 8th grade 92.87%Including also 7th grade 93.37%A.2 shows the proportion of the children that worked more than 5 months withwages in average greater than half the minimum wage. When we include only 12 grade,the linkage rate is only 76.17%. When we include 11th grade information, the linkagerate is 84.41% and when we include 12th to 7th grade, the linkage rate is 93.37%. Thisnumber is close to Chetty et al. (2014) and Acarci et al. (2020) linkage rates.Table A.3: Age distributionAge CASEN UID28 297,739 313,69629 257,271 301,83930 292,403 282,43031 205,791 254,60232 233,878 234,76233 246,575 211,548Total 1,533,657 1,598,877In addition, we can see from Table A.3 that the numbers per age are somewhat142similar between CASEN survey, which is a national representative survey and UID infor-mation.Table A.4: Sex distributionSource Female percentageUID 47.32%CASEN 2017 51.41%Finally, from Table A.4 we can see that the female proportion is lower in the UIDdatabase than the national proportion given that the female labour participation in theformal private sector is lower than the female proportion.143A.2 Additional regressionsTable A.5: Period robustness checks for IGE(1) (2) (3) (4) (5) (6)yp 0.288*** 0.297*** 0.311*** 0.323*** 0.333*** 0.354***(0.001) (0.001) (0.002) (0.002) (0.003) (0.005)Constant 9.506*** 9.426*** 9.298*** 9.193*** 9.135*** 8.942***(0.016) (0.018) (0.021) (0.027) (0.039) (0.058)Observations 505,524 416,818 282,979 173,683 83,668 39,160R-squared 0.091 0.098 0.108 0.117 0.124 0.134Standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1Earnings are measured as average earnings over the months where a children-parents pair report positiveearnings over the studied 5-year period. We keep individuals that appear at least 6 times with positiveearnings in the dataset with average earnings greater than half of the corresponding minimum wage.Columns (1) to (4) report results for male and female children. (1) considers individuals with at least 6months of positive earnings, (2) considers individuals with at least 12 months of positive earnings, (3)considers individuals with at least 24 months of positive earnings, (4)considers individuals with at least36 months of positive earnings, (5) considers individuals with at least 48 months of earnings and (6)considers individuals with at least 54 months of earnings.144Table A.6: Age robustness checks for IGE(1) (2) (3) (4) (5) (6)yp 0.289*** 0.299*** 0.313*** 0.326*** 0.335*** 0.353***(0.001) (0.001) (0.002) (0.002) (0.003) (0.005)Constant 9.496*** 9.411*** 9.278*** 9.167*** 9.116*** 8.962***(0.017) (0.019) (0.023) (0.029) (0.042) (0.062)Observations 440,222 363,667 247,419 152,032 73,177 34,214R-squared 0.093 0.101 0.112 0.121 0.128 0.137Standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1Earnings are measured as average earnings over the months where a children-parents pair report positiveearnings over the studied 5-year period. We keep individuals that appear at least 6 times with positiveearnings in the dataset with average earnings greater than half of the corresponding minimum wage.Columns (1) to (4) report results for male and female children. (1) considers individuals with at least 6months of positive earnings, (2) considers individuals with at least 12 months of positive earnings, (3)considers individuals with at least 24 months of positive earnings, (4)considers individuals with at least36 months of positive earnings, (5) considers individuals with at least 48 months of earnings and (6)considers individuals with at least 54 months of earnings.A.3 Penn paradeTo evaluate the validity of the earnings distribution, we will plot the Pen parade -therelationship between earnings percentiles and the percentiles, for the UID dataset and theCASEN dataset from 2003 (after correcting for inflation).145Table A.7: Period robustness checks for rank-rank correlation(1) (2) (3) (4) (5) (6)rp 0.257*** 0.264*** 0.273*** 0.279*** 0.281*** 0.293***(0.001) (0.002) (0.002) (0.002) (0.003) (0.005)Constant 37.833*** 39.075*** 41.208*** 43.635*** 46.905*** 49.420***(0.087) (0.096) (0.119) (0.152) (0.215) (0.312)Observations 440,222 363,667 247,419 152,032 73,177 34,214R-squared 0.065 0.069 0.075 0.080 0.086 0.096Standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1Earnings are measured as average earnings over the months where a children-parents pair report positiveearnings over the studied 5-year period. We keep individuals that appear at least 6 times with positiveearnings in the dataset with average earnings greater than half of the corresponding minimum wage.Columns (1) to (4) report results for male and female children. (1) considers individuals with at least 6months of positive earnings, (2) considers individuals with at least 12 months of positive earnings, (3)considers individuals with at least 24 months of positive earnings, (4)considers individuals with at least36 months of positive earnings, (5) considers individuals with at least 48 months of earnings and (6)considers individuals with at least 54 months of earnings.Table A.8: Age robustness checks for rank-rank correlation(1) (2) (3) (4) (5) (6)rp 0.257*** 0.264*** 0.273*** 0.279*** 0.281*** 0.293***(0.001) (0.002) (0.002) (0.002) (0.003) (0.005)Constant 37.833*** 39.075*** 41.208*** 43.635*** 46.905*** 49.420***(0.087) (0.096) (0.119) (0.152) (0.215) (0.312)Observations 440,222 363,667 247,419 152,032 73,177 34,214R-squared 0.065 0.069 0.075 0.080 0.086 0.096Standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1Earnings are measured as average earnings over the months where a children-parents pair report positiveearnings over the studied 5-year period. We keep individuals that appear at least 6 times with positiveearnings in the dataset with average earnings greater than half of the corresponding minimum wage.Columns (1) to (4) report results for male and female children. (1) considers individuals with at least 6months of positive earnings, (2) considers individuals with at least 12 months of positive earnings, (3)considers individuals with at least 24 months of positive earnings, (4)considers individuals with at least36 months of positive earnings, (5) considers individuals with at least 48 months of earnings and (6)considers individuals with at least 54 months of earnings.146Figure A.1: Pen parade for parental earnings010002000300040005000Earnings percentile0 20 40 60 80 100Parent rankingCasen 2003 Pen parade UID Pen paradePen parade for parental earningsThis figure shows the parental Pen parade using the Casen 2003 survey and the UID information.Figure A.2: Pen parade for child earnings010002000300040005000Earnings percentile0 20 40 60 80 100Child rankingESI 2019 Pen parade UID Pen paradePen parade for child earningsThis figure shows the children Pen parade using the ESI 2019 survey and the UID information.As we can see, both parental and children earnings are close to other informationsources that includes all workers.147A.4 Additional TablesTable A.9: Descriptive statistics of the UIP databaseYear N Min Mean Max P1 P5 Median P95 P992003 1,349,584 2 384,908 12,379,695 18,804 67,062 267,458 1,105,923 2,475,9242004 1,849,529 2 428,928 12,432,321 23,508 76,579 289,927 1,316,578 2,486,4672005 2,337,830 2 456,767 11,231,467 24,410 86,533 306,313 1,429,162 2,477,6612006 2,701,345 2 491,342 7,432,997 27,325 99,207 327,247 1,576,341 2,477,6662007 3,103,157 1 526,906 12,848,755 30,519 101,730 350,557 1,663,517 2,495,2742008 3,309,297 1 538,045 34,705,504 26,991 97,940 353,730 1,720,412 2,479,4602009 3,419,851 1 581,561 21,906,338 28,923 108,215 382,093 1,890,408 2,472,3602010 3,742,474 1 622,926 8,101,156 30,870 110,580 407,878 2,038,509 2,678,7782011 4,052,453 1 655,923 7,992,452 33,467 112,763 434,478 2,158,657 2,734,9462012 4,286,460 1 698,879 10,768,408 38,684 118,068 468,135 2,296,852 2,777,1162013 4,404,045 1 745,498 14,414,099 40,927 124,889 507,729 2,426,510 2,913,6422014 4,502,329 1 758,235 16,735,737 38,477 125,925 518,157 2,476,340 2,990,9062015 4,611,049 1 772,466 16,963,680 34,989 129,608 532,841 2,497,008 3,026,7802016 4,695,182 1 790,532 19,047,738 35,946 134,800 546,676 2,528,353 3,073,0802017 4,861,557 1 813,370 44,150,312 40,455 138,969 565,972 2,574,900 3,131,0132018 5,010,358 1 835,629 38,659,216 40,000 144,000 580,000 2,623,760 3,238,980148Figure A.3: Regions of ChileRegions Chile149Table A.10: Regional informationRegion Size KM2 Population Region Number Poverty rate % GDP ChileRegio´n de Arica y Parinacota. 16,873 226,068 15 8.4% 0.8%Regio´n de Tarapaca´. 42,226 330,558 1 6.4% 2.5%Regio´n de Antofagasta. 126,049 607,534 2 5.1% 10.1%Regio´n de Atacama. 75,176 286,168 3 7.9% 2.6%Regio´n de Coquimbo. 40,580 757,586 4 11.9% 3.1%Regio´n de Valpara´ıso. 16,396 1,815,902 5 7.1% 9.1%Regio´n Metropolitana de Santiago. 15,403 7,112,808 13 5.4% 46.4%Regio´n del Libertador General Bernardo O’Higgins. 16,387 914,555 6 10.1% 4.8%Regio´n del Maule. 30,296 1,044,950 7 12.7% 3.4%Regio´n de N˜uble 13,179 480,609 16 16.1% –Regio´n del Biob´ıo. 23,890 1,556,805 8 12.3% 7.9%Regio´n de La Araucan´ıa. 31,842 957,224 9 17.2% 2.8%Regio´n de Los Rı´os. 18,430 384,837 14 12.1% 1.4%Regio´n de Los Lagos. 48,584 828,708 10 11.7% 3.3%Regio´n de Ayse´n del General Carlos Iba´n˜ez del Campo. 108,494 103,158 11 4.6% 0.6%Regio´n de Magallanes y la Anta´rtica Chilena 132,291 166,533 12 2.1% 1.2%150Table A.11: Intergenerational mobility indicators by municipality in the Metropolitanregion. “Santiago” refers to the municipality, not the city.Region N βr αr p15 p11 p55 rabs rperSantiago 4711 0.212 43.360 0.183 0.207 0.388 48.654 63.474Cerrillos 2207 0.175 43.128 0.165 0.184 0.291 47.514 59.792Cerro Navia 4743 0.140 42.220 0.096 0.216 0.240 45.720 55.520Conchal´ı 3876 0.131 46.015 0.159 0.163 0.314 49.286 58.444El Bosque 5855 0.171 40.689 0.132 0.248 0.258 44.954 56.893Estacio´n Central 3277 0.207 42.257 0.154 0.203 0.359 47.434 61.931Huechuraba 2523 0.269 36.537 0.123 0.226 0.366 43.263 62.098Independencia 1664 0.197 43.278 0.170 0.175 0.368 48.207 62.009La Cisterna 2259 0.197 44.585 0.236 0.189 0.379 49.506 63.283La Florida 11614 0.226 42.181 0.154 0.192 0.388 47.819 63.607La Granja 4257 0.162 43.585 0.158 0.208 0.300 47.646 59.015La Pintana 6475 0.141 38.496 0.090 0.264 0.173 42.021 51.889La Reina 2460 0.313 43.944 0.268 0.158 0.582 51.778 73.713Las Condes 5619 0.409 41.259 0.235 0.166 0.676 51.489 80.132Lo Barnechea 2353 0.551 29.397 0.149 0.222 0.771 43.172 81.741Lo Espejo 3789 0.144 41.046 0.091 0.237 0.211 44.641 54.709Lo Prado 3061 0.180 41.495 0.135 0.188 0.277 45.991 58.578Macul 3074 0.247 39.830 0.145 0.215 0.372 46.008 63.307Maipu´ 17481 0.186 45.625 0.197 0.190 0.351 50.286 63.337N˜un˜oa 3655 0.264 44.424 0.234 0.227 0.467 51.024 69.502Pedro Aguirre Cerda 3350 0.174 41.523 0.130 0.235 0.283 45.879 58.075Pen˜alole´n 7261 0.288 34.718 0.109 0.263 0.400 41.917 62.075Providencia 1627 0.248 50.776 0.284 0.205 0.589 56.983 74.363Pudahuel 7465 0.151 43.272 0.137 0.221 0.265 47.055 57.646Quilicura 5469 0.184 43.138 0.114 0.170 0.293 47.728 60.580Quinta Normal 3050 0.169 44.525 0.153 0.195 0.302 48.760 60.618Recoleta 4729 0.162 43.364 0.126 0.214 0.267 47.416 58.763Renca 4892 0.142 44.065 0.135 0.186 0.266 47.620 57.573San Joaqu´ın 2650 0.160 43.306 0.102 0.208 0.273 47.304 58.498San Miguel 2111 0.247 42.157 0.184 0.197 0.420 48.328 65.607San Ramo´n 2890 0.153 38.985 0.098 0.250 0.179 42.799 53.478Vitacura 1570 0.271 56.960 NA NA 0.721 63.740 82.724Puente Alto 19612 0.189 41.116 0.145 0.220 0.292 45.852 59.113Pirque 510 0.262 38.296 0.103 0.191 0.509 44.857 63.227San Jose´ de Maipo 385 0.332 36.005 NA NA 0.381 44.297 67.516Colina 3551 0.223 38.426 0.098 0.227 0.350 43.990 59.571Lampa 1593 0.186 39.806 0.081 0.244 0.343 44.466 57.513Tiltil 630 0.138 46.137 0.195 0.195 0.326 49.599 59.292San Bernardo 9491 0.189 39.324 0.102 0.250 0.286 44.049 57.277Buin 2918 0.235 36.048 0.086 0.299 0.313 41.932 58.408Calera de Tango 650 0.360 32.700 0.144 0.288 0.568 41.697 66.887Paine 1970 0.184 38.631 0.092 0.235 0.311 43.235 56.125Melipilla 3228 0.182 39.959 0.110 0.266 0.286 44.498 57.207Alhue´ 172 0.244 46.174 NA NA NA 52.280 69.380Curacav´ı 783 0.134 43.136 0.133 0.235 0.216 46.489 55.876Mar´ıa Pinto 327 0.136 41.334 0.058 0.256 NA 44.740 54.277San Pedro 196 0.150 43.680 NA NA NA 47.426 57.914Talagante 2241 0.234 38.799 0.113 0.242 0.363 44.641 60.999El Monte 827 0.213 38.600 0.072 0.258 0.268 43.930 58.854Isla de Maipo 1028 0.154 39.982 0.125 0.253 0.330 43.832 54.612Padre Hurtado 1494 0.126 43.783 0.105 0.145 0.270 46.939 55.775Pen˜aflor 2381 0.178 41.758 0.144 0.187 0.319 46.214 58.689151A.5 More on non-linearitiesOne way to study non-linearities is to study the effect that a change on parental earningshas on child earnings for different quantiles of the child earnings distribution. One wayto do it is using the Unconditional quantile regressions developed by Firpo, Fortin andLemieux (2009). First, let’s assume that:yci = h(yp, ǫ) (A.1)where ǫ is the unobservable component and h(·) is stricly monotonic in ǫ. We candefine the unconditional partial effect as the small location shift in the distribution of ypon a distributional statistic v(Fy). We can write this as:α(v) =∫dE[RIF (Y c, v)|Y p = yp]dypdF (yp), (A.2)where RIF (Y c, v) is the recentered influence function. When the distributionalstatistic v is the rth quantile function, qτ = inf{q : Fyc(q) ≥ τ} and we can write theRIF (Y c, qτ ) as:RIF (Y c, qτ ) = qτ +τ − 1{yc ≤ qt}fY c(qτ ), (A.3)where fY c(qτ ) is the density function of Yc evaluated as qτ . From Firpo, Fortinand Lemieux (2009) we know that the unconditional quantile partial effect (UQPE) canbe expressed as the weighted average of the conditional quantile partial effects (CQPE):UQPE(τ) = E [ωτ (Yp) · CQPE (ξτ (yp), yp)] (A.4)where CQPE(τ, yp) = ∂Qτ [h(yp,ǫ)|Y p=yp]∂yp, and define CQPE(τ) ≡ E [CQPE(τ, yp)],ωτ (Yp) ≡ fY c|Y p(qτ |yp)fY c(qτ )and ξτ : Yp → (0, 1) is given by:ξτ(yp) = {s : Qs[Y c|Y p = yp] = qτ} = FY c|Y p(qτ |Y p = yp) (A.5)152ξτ is a “matching function” that indicates when the unconditional quantile qτfalls in the conditional distribution of Y c given Y p.A.6 Why imputation-based IGE estimates may fail:the importance of administrative data useThe main challenge in studying intergenerational mobility is to link child and parentalpermanent income. In the absence of administrative records, studies on intergenerationalmobility often rely on low-quality data, which do not allow to establish a parent-childlink with adequate income information. This limitation is detrimental for the study ofintergenerational mobility because it affects the credibility and precision of IGE estimates,which may cause to arrive at a misleading conclusion (Emran and Shilpi, 2019).Two-Sample Two-Stage Least Squares (TSTSLS) estimatorWith no access to administrative data, the most used methodology for IGE estimationis the Two-Sample Two-Stage Least Squares (TSTSLS). This estimator was originallyintroduced by Bjo¨rklund and Ja¨ntti (1997) for IGE estimation in a setting with missingparental income, and it has been used in several empirical studies (e.g., Aaronson andMazumder, 2008; Gong et al., 2012; Olivetti and Paserman, 2015; Piraino, 2015). TheTSTSLS estimator uses retrospective information on parents’ socioeconomic backgroundalong with a sample of “pseudo”-parents to impute parental income through a Mincer’sequation. Since background information of this type is more likely to be available insurvey datasets (or historical censuses), the TSTSLS methodology has allowed the IGEestimation for a significantly larger number of countries and historical periods, especiallyin developing nations (Narayan et al., 2018; Brunori et al., 2020). The standard empiricalspecification for estimating intergenerational income mobility is given by the followingequation:yci = α + βypi + ǫi, (A.6)153where yci is the logarithm of a child’s permanent individual income and ypi is the logarithmof her parent’s permanent individual income.1 The coefficient β is generally called “in-tergenerational elasticity” (IGE) and forms the basis for comparisons of intergenerationalincome mobility across countries. Among the existing IGE estimates in the literature,virtually all of those for developing countries are obtained through the TSTSLS method-ology proposed by Bjo¨rklund and Ja¨ntti (1997). This estimation procedure is based ontwo samples. The main sample contains information on individual incomes and recallsocioeconomic information about their parents. The auxiliary sample is typically derivedfrom an earlier survey of the same population where individuals (pseudo-parents) reporttheir income as well as socioeconomic information such as that recalled by respondents inthe main sample. The estimation then proceeds in two steps. First, the auxiliary sampleis used to estimate a Mincer’s equation by OLS:yspit = ωzi + vit, (A.7)where yspit is the income of pseudo-parents for child i at time t, zi is a vector of timeinvariant characteristics, and vit is the residual component of (A.7)). Then, we estimateypit as yˆpit = ω · zitp, that is, we impute the income of unseen parents. With this, we adjustthe following relationship between yc and yˆpiycit = α + βTSTSLSyˆpi + ψi (A.8)where βTSTSLS is the elasticity between the imputed parental income and childincome, while ψi is an error term. We estimate βTSTSLS as:βˆTSTSLS =ˆcov (yc, yˆp)ˆvar (yˆp)(A.9)There are many ways to create yˆp. Let us call Ω = {yˆp(j)}∞j=1 the set of predictions1Solon (1992) discuss that (1) could not be the true income model. In particular, there could beomitted variables that affect child earnings that are correlated with parental income. For instance,parents’ education. Following Chetty et al. (2014), Corak (2018) and Accarci et al. (2020), we assumethat (1) is the correct model. That is, there are not omitted variables problems. In addition, we alsoignore any measurement error of the children’s and parents’ income.154for yp. Each element yˆp(j) of Ω defines a different parameter, call it β(j)TSTSLS with itsrespective TSTSLS estimator as βˆ(j)TSTSLS. Significantly, we can stablish a relationshipbetween plimβˆ and plimβˆTSTSLS(j), withβˆ =ˆcov(yc, ypˆvar (yp)(A.10)Proposition 14. plimβˆTSTSLS(j) could be theoretically be higher or lower than plimβˆ.Proof. We know thatplimβˆ = cov(yc,ypvar(yp)= cov(yc,yp+yˆp(j)−yˆp(j)var(yp)= cov(yc,yp−yˆp(j)var(yp)+ cov(yc,yˆp(j)var(yp)Define (yp − yˆp(j)) ≡ ϕ(j)=cov(yc, ϕ(j)var (yp)+var (yˆp)var (yp)cov(yc, yˆp(j)var (yˆp)Define cov(yc,ϕ(j)var(yp)≡ η(j) and var(yˆp)var(yp)≡ κ(j) also notice that plimβˆTSTSLS(j) = cov(yc,yˆp(j)var(yˆp)Then, we have thatplimβˆTSTSLS(j) =plimβˆ − η(j)κ(j)(A.11)Now, if we assume that E(ǫi|ypi ) = 0. Then, plimβˆTSTSLS(j) = β−η(j)κ(j)Let us call κ(j) as the lack of variance bias and η(j) as the projection bias associ-ated to the projection yˆp(j). Our administrative records not only allow us to estimate βˆ,but also allow us to mimic a TSTSLS estimation setting for different yp(j) to measure themagnitudes of kappa(j) and η(j). To evaluate the importance of the use of administrativedata for IGE estimation, we proceed by simulating the followingb exercise:Simulated exersice 1:i From the main sample, we keep only children and their parents, and obtain β fromequation (A.6) by OLS.155ii We take a random subsample Σ of 50,000 parents’ and children’s information fromthe main sample. We randomly divide this subsample into two sub-subsamples Σ1and Σ2 of 25,000 observations. One sub-subsample (Σ1) is used to estimate thefollowing Mincer-type equation for pseudo-parents.yspi = γ′x′i + vi, (A.12)where x′i is composed by age, age squared, occupational sector, education type, andtype of contract. We estimate γ by OLS and we use Σ2 parental information tocompute a prediction for yˆp call it yˆp(1).iii We compute βˆTSTSLS(1) by regressing yc on yˆp(1) from Σ2iv We repeat ii)-iii) 1,000 times.Table 14 shows the distribution of this simulated procedure. As we can see,βˆTSTSLS(1) is much larger than βˆ. βˆTSTSLS(1) is closer to what Nunez and Miranda(2010, 2011) previously estimate for Chile. They find that IGE is in the ranges of 0.5-0.6.Table A.12: Results from simulated exercise 1Coefficient Min. 1st Qu. Median Mean 3rd Qu. Max.βˆ 0.287 0.287 0.287 0.287 0.287 0.287βˆTSTSLS(1) 0.448 0.479 0.488 0.488 0.498 0.538η(1) 0.142 0.155 0.159 0.159 0.162 0.180κ(1) 0.238 0.258 0.263 0.263 0.267 0.286(βˆ−η(1))κ(1)0.413 0.473 0.489 0.489 0.504 0.559As we can see, there is overestimation driven by κ(1). The lower value of var(yˆp(1))var(yp)has been deeply discussed in the statistical literature on imputation of missing data (Ru-bin and Little, 2019, Rubin, 2004; Rubin, 1996). Note that the setting of the TSTSLSestimation is a problem of missing data, where the unseen parental incomes are imputedusing regression imputation. However, even under a correctly specified model, regression156imputation does not properly reflect the uncertainty of the missing data. The issue isthat the imputed missing parental incomes from the regression model do not include anyresidual term, not providing enough uncertainty about the missing data. To overcome thisproblem, one possibility is to estimate the unseen parental incomes by using stochasticregression imputation. Specifically, to correct the lack of an error term in regression impu-tation, we can introduce error by adding a noise with zero mean and estimated regressionvariance to the regression imputation, that is, the predicted value from a regression plusa random residual value: yˆpi = γˆ′x′i + N(0, σˆ2v), where σˆ2v is the estimated variance ofthe Mincer equation (yspi =?′x′i + vi). Significantly, to estimate each missing parent’sincome, the stochastic regression imputation can be repeated several times in the spirit ofMultiple Imputation (Rubin, 1978). We now repeat exercise 1 using stochastic regressionimputation. The results are in Table 15:Table A.13: Results from simulated exercise 1 with additional varianceCoefficient Min. 1st Qu. Median Mean 3rd Qu. Max.βˆ 0.287 0.287 0.287 0.287 0.287 0.287βˆTSTSLS(1b) 0.109 0.123 0.128 0.128 0.132 0.149η(1b) 0.135 0.154 0.159 0.159 0.164 0.191κ(1b) 0.941 0.989 1 1 1.012 1.06(βˆ−η(1))κ(1)0.097 0.123 0.128 0.128 0.133 0.151As can be seen in Table A.13, κ(1) increases significantly as expected, attaining amedian value of 1. However, βˆTSTSLS(1b) is a lower bound of βˆ because the prediction biasremains. We can improve yˆp by adding an additional predictor to the Mincer’s equation:a measure of the parental earning ability. We can estimate that value because of the panelstructure on our data. To do this, we use our administrative dataset from 2003 to 2019to estimate a panel Mincer equation with fixed effects using our main parents sample:yspit = αi + ωzit + ψit (A.13)157where αi is a parental fixed effect, zit is a vector of time variant observables, andψit is a white noise. After adjusting this model by OLS, we recover the estimated fixedeffect associated to each parent to be used as measure of the parental earning ability. Withthis, we repeat our exercise by estimating yˆp(2). This prediction includes the estimatedfixed effect as predictor. We also can estimate yˆp(2b) which is yˆp(2) but after correctingfor the lack of variance.Table A.14: Results from simulated exercise 1 with additional variance and more predic-tionCoefficient Min. 1st Qu. Median Mean 3rd Qu. Max.βˆ 0.287 0.287 0.287 0.287 0.287 0.287βˆTSTSLS(2) 0.317 0.335 0.34 0.34 0.345 0.363βˆTSTSLS(2b) 0.253 0.272 0.276 0.276 0.28 0.298κ(2) 0.787 0.805 0.81 0.81 0.815 0.833κ(2b) 0.972 0.993 1 1 1.006 1.033η(2) 0 0.009 0.011 0.011 0.013 0.020η(2b) -0.003 0.008 0.011 0.011 0.014 0.023βˆ−η(2)κ(2)0.327 0.338 0.34 0.34 0.343 0.355βˆ−η(2b)κ(2b)0.262 0.273 0.276 0.276 0.279 0.293From this exercise, we understand that the reasons that the traditional βˆTSTSLSdoes not work to estimate β are insufficient prediction power for yp and the lack of vari-ance of that prediction. Typically, predictions for yp cannot be improved by informationon cross-sectional household surveys, and even in the case of having the right Mincer’sequation, TSTSLS can be still biased for the lack of variance of the imputed parentalearnings. Of course, the main challenge is to build a good model to impute yp, especiallyin developing countries where earnings/income are usually determined by unobservablecovariates such as social capital, non-cognitive skills, or neighborhood. Thus, this exerciseshows the importance of using administrative information to estimate intergenerationalmobility.22As Chetty et al. (2014) pointed out, income/earnings intergeneration elasticity is highly sensitive158to the inclusion of parents with 0 earnings. We find the same for Chile, a result that is consistent withCorak and Heisz (1999), who also found that IGE is highly non-linear.159Appendix BAppendix for Chapter 2B.1 A stochastic model for income processFollowing Dı´az et al. (2020), we assume that time is continuous and there is a continuumof agents indexed by i. Workers are heterogeneous in their total income yi. Total incomeis equal toyi ≡ wi + xi, (B.1)where wi is the part of the income that does not depend on the level of previous oractual income (the additive part, e.g., labour income or social security income), and xi isthe piece of income that depends on the level of previous income or actual income (themultiplicative part, e.g., dividends, interest payments, real estate income). Assume thatan agent needs some level of income w¯ (threshold income) to buy assets that generatesome income xi. If wi < w¯ then xi = 0.1 Suppose that the dynamics of w is given by thefollowing reduced-form model.21One might think that a retired agent has wi = 0 but x > 0. However, wi is not only labour income; itis all income that does not depend on the previous income. Assuming that a retired agent gets a constantflow of money from pension plans (such as RPP, RRSP, or social security) then this is an additive sourceof income instead of a multiplicative source. Besides, on average we can argue that a retired agent alsoneeds a minimum income to start saving and we assume that on average is w¯.2See Champernowne (1953), Aoki and Nirei (2016), Kim (2015) and Gabaix et al. (2016). Thosepapers use models with this reduced form. In particular, Gabaix et al. show that this equation could bederived from a general equilibrium model with individual optimization.160dwdt= µ+ σǫ(t), (B.2)where µ is the drift term, σ is the amplitude term and ǫ(t) is white noise.3 Because dwdoes not depend on w itself, neither µ nor σ depend on w. In the case of x, we candescribe its dynamics using the following stochastic process:dxdt= µ(x) + σ(x)ǫ(t). (B.3)Here, µ(x) and σ(x) depend on x. Thus, y has a reflecting barrier at w¯ but for incomes thatare greater than w¯.4 To solve those equations, we need to find a stationary distribution,one method to do this is using the Kolmogorov forward equation, which is given by:∂f(m, t)∂t= −µ(m)∂f(m, t)∂m+σ(m)22∂2f(m, t)∂m2, (B.4)where f(m, t) is the probability density function implicit in the stochastic process de-scribed by (B.4).In order to find a parametric solution, we solve for the stationary distribution,that is, we impose ∂f(m,t)∂t= 0. Call this pdf f¯(m). Solving equation (B.2) we have thefollowing stationary distribution:5f¯(w) = λe−λw. (B.5)That is, the stationary distribution of w is an exponential distribution with parameter λ.3Some literature estimates a reduced-form labour income process using equation (B.2). For example,Hearthcote et al. (2010) and Meghir and Pistaferri (2011).4To ensure the existence of a stationary distribution, we need to add some “stabilizing force” (Gabaix,2009). In this case, we add a reflecting barrier at w¯; that is, the income cannot be below w¯ after the levelw¯ is achieved.5In order to build the imputation method, it is necessary to work in terms of levels. However, onemay think that equation (B.2) should be in terms of logs instead of levels, but there are two argumentsto address this with this. First, a reflecting barrier around 0 could be added to address the fact that ǫ(t)could be highly negative. Second, the stationary distribution is an exponential which works only withpositive numbers.161By equation (B.3), assuming that µ(x) = µx and σ(x) = σ√x we have:f¯(x) = ηxw¯ηxxηx+1. (B.6)That is, the stationary distribution of x is a Pareto distribution with parameter ηx andw¯ threshold parameter.Now, following Gabaix (2009), if w and x are independent power law processeswith ηw and ηx as exponents.6 then, the process composed by the sum of w and x follows:ηy ≡ ηw+x = min(ηw, ηx), (B.7)where ηy is equal to the power law exponent of y. When one combines two indepen-dent power law processes, the fattest (the one with the smallest exponent) power lawdominates.7 In simple words, in the tail of the distribution, the unequal process domi-nates. This result is also derived by Clemens, Gottlieb, He´mous and Olsen (2017); theyuse a different framework to arrive at the conclusion that the wage distribution of doc-tors is dominated by the shape parameter of the distribution of practitioners which isa distribution that generates higher inequality.8 In our case, where w is distributed asan exponential, we can say that ηw = ∞ and given that x is Pareto distributed, thenηx < ∞. That is, ηw+x = ηx. Given the assumption that the income generated throughassets (multiplicative income) is more unequal to the additive income (labour income andsocial security), the distribution in the tail is dominated by the shape of the (x) process.6A power law is a relation of the type Y = kXα. A distribution that satisfies at least in the uppertail P (y > x) = k · x−η where η is the power law exponent and k is a constant.7Given that in the model x is different from 0 if w ≥ w¯ ,one may think that w and x are notindependent. However, for the application of this property is the tail of the distributions that matters.Thus, conditional on w ≥ w¯ it is not crazy to think that the return of the income generated though assetsare independent of the labour income.8In the context of Clemens et al.´s work:“suppose that there are two cities with physicians and one cityhas more inequality than the other. If we observe the aggregate income distribution, we see the shape ofthe city with the highest inequality”.162With this in mind, the cumulative distribution function (cdf) of total income is:9F (y) =1− e−λy y ≤ w¯1− e−λw¯ + e−λw¯(1−(w¯y)ηx)y > w¯. (B.8)This income distribution depends on three key parameters: λ, w¯ and ηx. The firstof those parameters is the exponential parameter which in this context gives the shape ofthe additive income process. The second parameter w¯ is the core parameter of interest; itrepresents the minimum income that an individual will require to start generating incomethrough his assets. This means that if the market income of some agent is less than w¯,this agent does not hold any retained earnings. Thus, the imputation procedure starts toassign values different from 0 only if yi ≥ w¯. Finally, ηx is the shape parameter of thePareto tail; a lower parameter means an unequal economy.B.1.1 Effect of the inclusion of retained earnings in the stochas-tic income modelIn order to have ownership over retained earnings, it is required to own a part of a firm,that is, it is necessary to hold an asset (for instance, the entire firm in the case of sole-ownership firms or some amount of stock in a publicly traded firm). Those assets couldbe bought today or have been bought in the past, that is, the income generated throughretained earnings depends on the current or previous income. Thus, the income generatedthrough retained earnings is a multiplicative part of the income. In particular, we assumethat retained earnings has the same structure as x. That is, retained earnings are onlydifferent from 0 if wi < w¯. Thus, the inclusion of retained earnings does not change w¯.Now, define zi as the income generated through retained earnings; then, we havethe following identity for the income process:y∗i ≡ wi + xi + zi, (B.9)9Notice that a normalization restriction∫∞0f(y)dy = 1 was imposed.163where y∗i is the total income including retained earnings, wi and xi are defined as before.Because xi and zi is income that comes from assets, we define hi ≡ xi + zi. Given that hiis a multiplicative part, it has the same structure as equation (B.3)dhdt= µ(h) + σ(h)ǫ(t). (B.10)As we can see, µ(h) and σ(h) depend on h. Then, using equation (B.4) in (B.10) andsolving for the stationary distribution, we see that the stationary distribution for h is aPareto process with ηh parameter and w¯ as a threshold parameter. Then, the stationarydistribution for y∗ isF (y∗) =1− e−λy y∗ ≤ w¯1− e−λw¯ + e−λw¯(1−(w¯y)ηw+h)y∗ > w¯. (B.11)If retained earnings are more concentrated than other assets (e.g., housing, bonds) it isexpected that ηh ≤ ηx, which implies that the tail of the income distribution is drivenby the shape of the distribution of retained earnings. However, it is quite difficult todetermine the exact distribution of h; we need administrative data about income andownership of firms for each individual, data which are quite difficult to obtain for longerperiods of time and different countries. For this reason, in the next section, we develop aprocedure to estimate ηh without knowing the whole distribution.B.2 Standard error estimationB.2.1 Estimation of standard errorsTo provide confidence intervals we need to estimate standard errors for the parameters.In this context, there are two possibilities, the first one is to compute the asymptoticdistribution of the estimators, and the second procedure is to use a bootstrap procedure.As Cowell and Van Kerm (2015) suggested we use the bootstrap procedure.1010Hansen (2000) in the context of Threshold autoregressive models (TAR) found that the distributionof the threshold estimate is non-standard. So the computation of an asymptotic distribution is far from164Bootstrap and threshold modelsIt is known that the standard bootstrap under mild conditions can provide an estimateof the exact variance of the estimator.11 However, following Kapetanios (2009) in thecase of the estimator of the threshold parameter, there are doubts of the consistencyof the bootstrap.12 In particular, an essential condition for bootstrap validity for anestimator is that the mapping between the joint distribution function of the sample andthe estimator is continuous. It is not clear whether this continuity assumption is satisfiedin a threshold model.13 Moreover, as Yu (2014) showed, the non-parametric bootstrapfails in discontinuous threshold regression, demonstrating the inconsistency of the non-parametric bootstrap for inference on the threshold point.14In this context, Davidson and Flachaire (2007) show that a semi-parametric modelhelps to address inference. Constructing bootstrap inference using a semi-parametricmodel enhances the precision of confidence intervals and tests. The idea is to build boot-strap samples by resampling from the survey data, that is by drawing observations withreplacement from the bottom of the sample with probability p∗ and taking observationssimulated from the Pareto distribution with probability 1 − p∗. Note that in this proce-dure, point estimates are still calculated on the basis of the full non-parametric sample(both in the full sample and in the resamples) and the semi-parametric bootstrap doesnot involve re-estimation of the parameter in each bootstrap sample.B.3 Proof of proposition 2Proof. we have:the scope of this paper.11Under the assumption of asymptotic pivotalness (independence of the asymptotic distribution fromnuisance parameters), the bootstrap estimator may converge more quickly to the true distribution com-pared to the asymptotic approximation12Diebold and Chen (1996) provide simulation evidence without theory that the parametric bootstrapworks well for structural change tests applied to AR(1) processes.13This is true either when a fixed threshold is assumed as in Chan (1993) or when a small thresholdassumption is made, such as Assumption 3, which is used by Caner and Hansen (2004).14The main reason for the non-parametric bootstrap failure is the discreteness generated from thebootstrap sampling on the local data around w¯. To break such discreteness, we can smooth the data inthe neighborhood of q = w¯. Such a procedure is termed as the smoothed bootstrap by Efron (1979)165Pr(h ≤ u) = Pr(x+ ψ(x) ≤ u)Define r(k) = h−1(k), because of assumption 1 and 2 this function is unique, then wehave:Pr(h ≤ u|x > w¯) = Pr (x ≤ r(u)|x > w¯)=∫ r(u)w¯f(x)dxThen, taking derivatives with respect to u and using the Leibniz rule we will have:fh(u) = r′(u)fx (r(u))where fh(·) is the conditional pdf of h and fx(·) is the conditional pdf of x. Now, usingassumption 3 we have:ηhw¯ηhu1+ηh= r(u)′ηxw¯ηxr(u)1+ηxUsing some algebra we get:r′(u)r(u)=ηhηxw¯ηhw¯ηxr(u)ηxu1+ηhTry the following educated guess r(u) = Auηhηx and integrating both sides, wehave:166log(r(u)) =ηhηxw¯ηhw¯ηx· A · log(u) + CGiven the educated guess we have that A = w¯ηxw¯ηhand C = A. Then we have that:r(u) =w¯ηxw¯ηhuηhηxthen r−1(x) = h is equal tohequivx+ ψ∗(x) = xηxηhw¯w¯ηxηhThenψ∗(x) = xηxηhw¯w¯ηxηh− x167Appendix CAppendix for chapter 3C.1 Tables for empirical analysisTable C.1: Data for Canada. µ in current Canadian DollarsYear µ G∗ Gini 1% z Gur p Gnr1999 41966 0.4497 0.1521 0.0241 0.4630 0.0010 0.46152005 52466 0.4753 0.1971 0.1025 0.5291 0.0025 0.52472012 65853 0.4816 0.2549 0.0488 0.5069 0.0010 0.50512016 73760 0.4846 0.3103 0.0080 0.4888 0.0002 0.4885Table C.2: Data for Chile. µ in current Chilean PesosYear µ G∗ Gini 1% z Gur p Gnr2003 301528 0.5679 0.2987 0.0395 0.5849 0.0001 0.58462006 328722 0.5524 0.2771 0.0859 0.5908 0.0003 0.58972009 419562 0.5515 0.2574 0.1108 0.6012 0.0015 0.59792011 436479 0.5391 0.2320 0.1134 0.5914 0.0009 0.58912013 414181 0.5098 0.2255 0.1086 0.5630 0.0011 0.56052015 473438 0.4970 0.2196 0.1604 0.5777 0.0006 0.57462017 526054 0.5022 0.2674 0.1281 0.5660 0.0007 0.5634168

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