Introduction to measures of inequality and concentration in tourism

! ! ! ! ! ! ! ! INTRODUCTION TO MEASURES OF INEQUALITY AND CONCENTRATION IN TOURISM ANTONIO FERNÁNDEZ MORALES UNIVERSITY OF MÁLAGA, 2016 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 1! INTRODUCTION) ! There!are!many!statistical!measures!available!in!the!literature!for!the!analysis!of!inequality!and! concentration.! ! The!most!common!measure!in!this!field!is!the!Gini!index!(Roselló!Nadal!et#al.,!2004),!which! is! associated!to!the!Lorenz!curve.!Both!were!developed!at!the!beginning!of!the!20th!century:!the! Lorenz!curve!in!1905!and!the!Gini!index!in!1914!(Giorgi,!1993).! ! A!revision!of!recent!applications!of!these!measures!in!the!field!of!tourism!can!be!found!in!CisnerosO Martín!and!FernándezOMorales!(2015)!and!Duro!(2016).! ! APPLICATIONS) ! Inequality! measures! have! many! different! applications! in! the! field! of! tourism,! either! specific! or! derived!from!applications!in!the!field!of!Economics.!Some!examples:! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! Thus,!the!Gini!index,!and!the!associated!Lorenz!curve,!is!a!useful!measure!with!different!applications! in!tourism!research,!to!get!a!better!understanding!of!the!distribution!of!wages!(Marcouiller!and!Xia,! 2008),!the!seasonal!concentration!of!the!demand!(CisnerosOMartínez!and!FernándezOMorales,!2015;! FernándezOMorales!and!CisnerosOMartínez,!2015)!or!the!supply!(Lopez!Bonilla!and!López!Bonilla,! 2006),!of!the!seasonality!of!air!passengers!traffic!(CisnerosOMartínez!and!FernándezOMorales,!2016),! the!seasonality!of!cruise!tourism!(FernándezOMorales!and!MartínOCarrasco,!Y.,!2014),!etc.!! ! ! ! INCOME)DISTRIBUTION) INEQUALITY It! was! the! first! application! of! inequality! measures,! and! still! the! most!important. WAGES)DISTRIBUTION)INEQUALITY Closely! related! with! the! previous! one.! TOURIST)DEMAND) CONCENTRATION)UN)CERTAIN) MONTHS Gini!index!can!measure!the!degree!of! annual! concentration! of! tourist! demand. TERRITRIAL)CONCENTRATION)OF) TOURISM)DEMAND/SUPPLY The! level! of! concentration! in! territorial! areas! can! be! used! to! measure!saturation!levels.! ! 2! CASE)STUDY)1) ! The!first!case!study!consists!of!a!very!simple!situation,!in!order!to!facilitate!the!understanding!of!the! Gini!index!and!the!Lorenz!curve.! ! ! ! ! ! ! ! ! ! ! If!the!salaries!in!the!travel!agency!change!to!the!distributions!shown!in!the!figures!below,!is!there! a!change!in!inequality?! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! Now,!we!analyse!some!numeric!features!of!the!salaries!distribution:! ! Xi! ni! xi=ni! pi! qi! Pi! Qi! 10000! 2! 20000! 50! 20! 50! 20! 30000! 1! 30000! 25! 30! 75! 50! 50000! 1! 50000! 25! 50! 100! 100! ! 4! 100000! 100! 100! ! ! ! A)!What!is!the!total!amount!of!salaries!received!by!the!travel!agency’s!workers?!_________€! B)!What!percentage!of!workers!are!paid!10000!€?!_____%! C)!What!percentage!of!the!total!amount!of!salaries!is!earned!by!the!worker!who!is!paid!50000!€?! ____%! D)!What!percentage!of!the!total!amount!of!salaries!is!earned!by!the!50%!of!lessOpaid!workers?! ______%! ! ! There! are! four! workers! in! a! small! travel! agency! with! annual! salaries! (see!figure):! 50000 €,! 30000€,! 10000€ ! and! 10000€ 50000€! 30000€! 10000€! 10000€! YES NO 100000€! 0€! 0€! 0€! 25000€! 25000€! 25000€! 25000€! YES  NO ! 3! 0" 25" 50" 75" 100" 0" 25" 50" 75" 100" Qi# Pi# Lorenz"curve" Equidistribu7on" THE)LORENZ)CURVE) ! The!Lorenz!curve!is!the!graphical!representation!of!the!points!(Pi,!Qi),!joined!by!straight!segments.! (Fernández!Morales!and!Lacomba!Arias,!2003)! ! Where:! Pi!:!!cumulative!percentages!of!frequencies! Qi!:!!cumulative!percentages!of!the!total!amount!of!the!variable!(cumulative!qi)! ! PROPERTIES! •" It!starts!at!(0,0)! •" It!ends!at!(100,100)! •" It!is!increasing!! •" The!greater!the!distance!between!the!Lorenz!curve!and!the!diagonal!starting!at!(0,0),!the! greater!concentration!of!the!variable! •" The!closer!to!the!diagonal,!the!less!concentration!in!the!distribution.!! ! EXTREME!CASES! •" PERFECT!EQUALITY!(EQUIDISTRIBUTION):!The!Lorenz!curve!coincides!with!the!diagonal.!This! means!that!there!is!no!inequality!in!the!distribution!of!the!variable,!since!all!the!observed! values!are!equal!to!the!arithmetic!mean!!(Gini!index!=!0)! •" MAXIMUM! INEQUALITY:! The! Lorenz! curve! connects! points! (0,0),! (100! (NO1)/N,0)! and! (100,100).!This!means!that!all!the!observations!but!one!equal!0,!so!that!the!concentration! level!is!maximum.!(Gini!index!approaches!1).!! ! ! RELATION!BETWEEN!THE!LORENZ!CURVE!AND!THE!GINI!INDEX! ! The!area!between!the!Lorenz!curve!and!the!line!of!perfect!equality!divided!by!the!area!of!the!triangle! determined!by!the!points!(0,0),!(100,0),!(100,100),!corresponds!exactly!with!the!Gini!index.!! ! SMOOTHNESS!OF!THE!CURVE! ! The!Lorenz!curve!polygon!becomes!smoother!as!the!number!of!data!(or!intervals)!increases.!! ! ! ! ! ! ! ! ! ! ! qi=100 xini xini i=1 k ∑ Gini!index!equals!1! only!with!infinite! observations.! 0%# 10%# 20%# 30%# 40%# 50%# 60%# 70%# 80%# 90%# 100%# 0%# 10%# 20%# 30%# 40%# 50%# 60%# 70%# 80%# 90%# 100%# Centro# Litoral#Este# 4!DATA! POINTS! (Case!study!1)! MORE!THAN!500!DATA!POINTS! Income!distribution!in!two! areas!of!Málaga! Source:!Martín!Reyes!et#al.!(2012)! Gini!index:!Centro!0.3214,!! !Litoral!Este!0.266! ! 4! 0" 25" 50" 75" 100" 0" 25" 50" 75" 100" Qi# Pi# Lorenz"curve" Equidistribu7on" 0" 25" 50" 75" 100" 0" 25" 50" 75" 100" Qi# Pi# Lorenz"curve" Equidistribu7on" 0" 25" 50" 75" 100" 0" 25" 50" 75" 100" Qi# Pi# Lorenz"curve" Equidistribu7on" CASE)STUDY)1) ! ! LORENZ!CURVE! Xi! ni! xi=ni! pi! qi! Pi! Qi! ! ! ! ! ! 0! 0! 10000! 2! 20000! 50! 20! 50! 20! 30000! 1! 30000! 25! 30! 75! 50! 50000! 1! 50000! 25! 50! 100! 100! ! 4! 100000! 100! 100! ! ! ! ! ! E)!What!percentage!of!the!total!amount!of!salaries!is! earned!by!the!50%!of!lessOpaid!workers?______%! ! F)!What!percentage!of!the!total!amount!of!salaries!is! received! by! the! 25%! of! highestOpaid! workers?______%! ! ! G)!Match!each!distribution!of!salaries!with!its!Lorenz!curve! ! ! ! ! ! ! ! ! ! ! ! !!!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 50000€! 30000€! 30000€! 30000€! 100000€! 0€! 0€! 0€! 25000€! 25000€! 25000€! 25000€! A B C A B  C  A B  C A B C 0" 25" 50" 75" 100" 0" 25" 50" 75" 100" Qi# Pi# Lorenz"curve" Equidistribu7on" ! 5! GINI)INDEX) ! ! The! Gini! index! is! the! most! commonly! used! statistical! indicator! for! measuring! inequality! in! the! frequency!distribution!of!a!variable.! Range:! ! ! ! ! ! ! ! ! ! ! ! ! ! ! CALCULATION! There!are!many!formulas!available.!We!will!use:!! ! Where:! Pi!:!cumulative!percentages!of!frequencies,!! Qi!:!cumulative!percentages!of!the!total!amount!of!the!variable!(cumulative!qi)! ! ! PROPERTIES! ! •" It!is!not!affected!by!scales!changes!(scale!independence)! •" Very!sensitive!to!outliers! ! ! ! ! ! ! ! ! ! ! ! ! 0 1 LOW INEQUALITY HIGH INEQUALITY EXTREME!CASE )EQUIDISTRIBUTION (PERFECT)EQUALITY) EXTREME!CASE )MAXIMUM) INEQUALITY)LIMIT G= P iQi+1− i=1 k−1 ∑P i+1Qi i=1 k−1 ∑ 10000 qi=100 xini xini i=1 k ∑ ! 6! 0" 25" 50" 75" 100" 0" 25" 50" 75" 100" Qi# Pi# Lorenz"curve" Equidistribu7on" 0" 25" 50" 75" 100" 0" 25" 50" 75" 100" Qi# Pi# Lorenz"curve" Equidistribu7on" 0" 25" 50" 75" 100" 0" 25" 50" 75" 100" Qi# Pi# Lorenz"curve" Equidistribu7on" CASE)STUDY)1) ! ! Gini!index! Xi! ni! xi=ni! pi! qi! Pi! Qi! Pi)=)Qi+1! Pi)+1=)Qi! 10000! 2! 20000! 50! 20! 50! 20! 2500! 1500! 30000! 1! 30000! 25! 30! 75! 50! 7500! 5000! 50000! 1! 50000! 25! 50! 100! 100! ! ! ! 4! 100000! 100! 100! ! ! 10000! 6500! ! G=10000 −6500 10000 =0.35 ! ! H)!Match!each!Lorenz!curve!with!its!Gini!index! ! ! ! ! ! ! ! ! ! ! ! ! ! CASE)STUDY)2) ! In! a! recent! study! about! salaries! inequality! in! tourism! related! sectors! in! Wisconsin,! U.S.A.! (Marcouiller!&!Xia,!2008),!the!following!Gini!indexes!have!been!estimated:! Subsector! ! ! ! ! ! Gini!index! A:!Performing#Arts#and#Spectator#Sports!!!! ! !0.2334! B:!Museums,#Parks#and#Historical#Sites! ! !0.2182! C:!Food#Services#and#Drinking#Places!! ! !0.0418! ! Which!subsector(s)!fit(s)!these!descriptions?! A!!!B!!C! !!‘presence!of!a!small!number!of!highly!paid!employees!and!a!relatively!large!number!of! lower!paid!employees’! ! ‘generally! common! and! persistent! problem! of! low! wages! experienced! across! this! occupational!category’! G=0 G=0.35  G=0.75 G=0 G=0.35  G=0.75 G=0 G=0.35 G=0.75 ! 7! CASE)STUDY)3! ! The!following!table!shows!the!distribution!of!women’s!monthly!salaries!in!the!hospitality!sector!in! Spain!in!2010,!according!to!the!survey!Encuesta#de#estructura#salarial#2010!by!the!National!Statistics! Institute!(the!last!interval!has!been!adjusted).!! ! Monthly)salary) )(103€))) Xi) ni) xi=ni) pi) qi) Pi) Qi) Pi=Qi+1) Pi+1=Qi) 0.0!O!13,5! 6.75! 43650! 294637.5! 25! 9.64! 25.00! 9.64! !!!!! ! 13.5!O!16,8! 15.15! 43650! 661297.5! 25! 21.64! 50.00! 31.29! ! ! 16.8!O!19,7! 18.25! 43650! 796612.5! 25! 26.07! 75.00! 57.36! ! ! 19.7!and!more! 29.85! 43650! 1302952.5! 25! 42.64! 100.00! 100.00! !! !! ! ! 174600! 3055500.0! 100! 100! ! ! ! ! ! A)!Complete!the!table,!and!calculate!the!average!salary!and!the!Gini!index!for!women.! Average!salary:!________! ! Gini!index:!____________! B)!The!men’s!average!salary!(from!the!same!survey)!is!22.6!(103€)!and!the!estimated!Gini!index!is! 0.30.!!! ! ! ! ! ! ! ! ! ! ! Identify!the!Lorenz!curves!in!the!figure! Women!!!!!!A!!B! Men!!!! ! !A!!B! ! C)!What!is!the!highest!salary!earned!by!the!25%!lessOpaid!workers?! Women:!___________! ! ! Men:!____________! ! D)!What!is!the!lowest!salary!earned!by!the!highestOpaid!workers?! Women:!___________! ! ! Men:!____________! ! E)!Do!the!25%!highestOpaid!women!earn!more!than!40%!of!the!total!amount!of!women’s!salaries?! !!Yes!!! !No! ! F)!Do!the!25%!lessOpaid!women!earn!less!than!10%!of!the!total!amount!of!women’s!salaries?! !! !Yes!!! !No! ! ! ! ! ! 0" 25" 50" 75" 100" 0" 25" 50" 75" 100" Qi# Pi# A" B" Monthly)salary) (103€)) )MEN) ni) 0.0!–!16.3! 31950! 16.3!–!19.5! 31950! 19.5!–!24.7! 31950! 24.7!and!more! 31950! ! 127800! ! ! 8! CASE)STUDY)4) ! The! table! below! shows! the! monthly! distribution! of! visitors! (thousands)! to! the! Palazzo! Ducale,! Venice,!in!2012!(source:!Panciera,!2013).!The!distribution!is!mainly!concentrated!between!April!and! October,!with!a!Gini!index!=!0.1932.!! A)!Complete!the!table.! Month) Xi) xi=ni) pi) qi) Pi) Qi) Pi=Qi+1) Pi+1=Qi) ENE! 48! 48! 8.3333! 3.670! 8.333! 3.670! 61.80! 61.17! DIC! 49! 49! 8.3333! 3.746! 16.667! 7.416! 205.15! 185.40! NOV! 64! 64! 8.3333! 4.893! 25.000! 12.309! 441.52! 410.30! FEB! 70! 70! 8.3333! 5.352! 33.333! 17.661! 835.90! 735.87! MAR! 97! 97! 8.3333! 7.416! 41.667! 25.077! 1462.16! 1253.84! OCT! 131! 131! 8.3333! 10.015! 50.000! 35.092! 2266.84! 2047.03! JUN! 134! 134! 8.3333! 10.245! 58.333! 45.337! 3255.63! 3022.45! SEP! 137! 137! 8.3333! 10.474! 66.666! 55.811! 4429.18! 4185.81! AGO! 139! 139! 8.3333! 10.627! 75.000! 66.438! 5785.55! 5536.48! JUL! 140! 140! 8.3333! 10.703! 83.333! 77.141! 7345.80! 7071.23! MAY! 144! 144! 8.3333! ! ! ! ! ! ABR! 155! 155! 8.3333! 11.850! 100.000! 100.000! !! !! ! 1308! 1308! 100.000! 100.000! ! ! ! ! B)!What!curve!corresponds!to!the!table!data?!!A!!B!!C! C)!Simulate!the!distribution!resulting!from!a!10%!increment!in!each!month.!Compare!the!new!Lorenz! curve!and!Gini!index!with!the!original!ones.! ENE! FEB! MAR! ABR! MAY! JUN! JUL! AGO! SEP! OCT! NOV! DIC! 53! 77! 107! 171! 158! 147! 154! 153! 151! 144! 70! 54! Does!the!10%!monthly!increment!cause!a!change!in!the!Gini!index?!! !Yes!!No! Why?! ! ! ! ! ! SOLUTIONS) ! CASE!STUDY!1! A)!100000!€! B)!50%! C)!50%! D)!20%! E)!20%! F)!50%! G)!C,!A,!B! H)!0,!0.35;!0.75! ! CASE!STUDY!2! A,!B! C! ! CASE!STUDY!3! A)!Average:!17.5;!G=0.26! 0" 25" 50" 75" 100" 0" 25" 50" 75" 100" Qi# Pi# A" B" C" Continue!this!activity!in!the! Seasonal!Concentration!Lab! http://goo.gl/xEcl97! (FernándezOMorales,!2015)