Income and wealth distributions in a neoclassical growth model with σ 1

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1 Income and wealh disribuions in a neoclassical growh model wih 1 Mauro Parão March 8, 2017 Absrac The publicaion of Capial in he Tweny-Firs Cenury by Pikey helped o increase he debae abou he prospecs of he evoluion of income and wealh inequaliy in his cenury. One of he main conroversies is abou he effecs o he income and wealh inequaliies of a decrease in he growh rae g. In Pikey, 2014), i is claimed ha a decrease in g will cause an increase in he wealh inequaliy, hrough an increase in he difference r g, where r is he rae of reurn on capial. This claim was criicized by many auhors. In his paper, we presen a neoclassical growh model wih heerogeneous agens and use i o shed more ligh on his issue. Our model generalizes and improves he model inroduced in Pikey e Zucman, 2015) and he model presened in Aoki and Nirei, 2016). We also presen a resul relaing income, wealh and wage inequaliies. JEL Numbers: E13, E21, E23, E25 1 Inroducion The publicaion of Capial in he Tweny-Firs Cenury by Pikey helped o increase he debae abou he prospecs of he evoluion of income and wealh inequaliy in his cenury. One of he main conroversies is abou he Associae Professor a he Deparmen of Mahemaics, Universiy of Brasília. Campus Darcy Ribeiro. Brasília, DF, Brasil. mparao@ma.unb.br. Also a PhD suden a he Deparmen of Economics, Universiy of Brasília. 1

2 effecs o he income and wealh inequaliies of a decrease in he growh rae g. In Pikey, 2014), i is claimed ha a decrease in g will cause an increase in he wealh inequaliy, hrough an increase in he difference r g, where r is he rae of reurn on capial. This claim was criicized by many auhors. For example, in Acemoglu and Robinson, 2015), i is presened an economeric exercise suggesing a low correlaion beween he difference r g and he wealh inequaliy, while, in Sigliz, 2015), one can find a general criicism o he fac ha r is aken exogenously in he models underlying Pikey s analysis. In Pikey, 2015), afer hese criicisms, Pikey sill susain he link beween he difference r g and he wealh inequaliy, bu concedes ha a decrease in g may no necessarily led o an increase in he wealh inequaliy. In his paper, we presen a neoclassical growh model wih heerogeneous agens and use i o shed more ligh on his issue. Several models wih heerogeneous agens dealing wih income and wealh disribuions appeared in recen years, such as he models presened in Benhabib, Bisin and Zhu, 2011 and 2015) and in Hubmer, Krusell and Smih Jr, 2016). Our model generalizes and improves he model inroduced in Pikey e Zucman, 2015) and he model presened in Aoki and Nirei, 2016) in he following aspecs: 1. The rae of reurn on capial is aken endogenously hrough a producion funcion which has consan gross elasiciy of subsiuion 1. In Aoki and Nirei, 2016), he producion funcion is Cobb-Douglas = 1), while he rae of reurn on capial is aken exogenously in Pikey and Zucman, 2015). 2. The inheriance process and he populaion growh are explicily inroduced in a micro-founded way and are inerconneced, resembling he assoraive maing presened in Cowell, 1998). In Aoki and Nirei, 2016; Pikey and Zucman, 2015), agens live forever and he populaion growh is inroduced in he aggregaed level in an ad hoc way. 3. Produciviy is subjeced o idiosyncraic shocks which are known only afer he beginning of he producion process. In Aoki and Nirei, 2016), he produciviy shocks are known before he beginning of he producion process, which allows agens o avoid losses. Besides no being very realisic, he previous knowledge of he shocks may rise an inconsisence in he model, since, depending on he size of he shock, i can be raional no o engage in he producion process, which generaes 2

3 nonlineariies in he model. This poin is explained in deail in he Subsecion A.5 of he appendix. 4. The depreciaion and saving raes are also subjeced o idiosyncraic shocks. In Aoki and Nirei, 2015), hese raes are assumed o be consan, while only he saving rae is subjeced o idiosyncraic shocks in Pikey and Zucman, 2015). In our model, he shocks in he depreciaion and saving raes can be correlaed o he shocks in he produciviy. 5. The effecs of he inroducion of redisribuion hrough axes on he seady sae of he aggregaed level are considered. The main resuls for he income and wealh disribuions are he following: 1. The upper ail of he saionary disribuion of he capial sock per capia is approximaed by a Pareo disribuion whose coefficien P is a decreasing funcion of he difference r g, exacly as in Pikey and Zucman, 2015). 2. The relaion beween he Lorenz curves of he oal gross income and he capial sock disribuions are given by L Y x) = α L K x) + 1 α )x 1) and he relaion beween heir Gini indexes are given by G Y = α G K 2) where α is he capial share of he gross income see Proposiion 4.3). This resul shows why he capial sock inequaliy is much higher han he oal income inequaliy. The same resul holds replacing gross quaniies by ne quaniies. 3. The previous resuls are used o analyze how oal income and sock capial inequaliies vary wih he growh rae in he seady sae when = 1 and 2. Even in he case favored by Pikey, wih = 2, he resuls are much more ambiguous han expeced as explained in deail in he las secion. The relaions beween Lorenz curves and Gini indexes presened above follow from a more general resul which is proved in Subsecion A.6 of he appendix. 3

4 Proposiion 1.1 If he orders of capial socks and of wages coincide and if he gross rae of reurn on capial r is consan relaive o he capial sock, hen L Y x) = α L K x) + 1 α )L w x) 3) and G Y = α G K + 1 α )G w 4) where L Y, L K, L w are he Lorenz curves and G Y, G K, G w are he Gini indexes of, respecively, gross oal income, capial sock and wages and α is he capial share of he gross income. The same resul holds replacing gross quaniies by ne quaniies. The above resul is approximaely verified by empirical daa, as illusraed by he following ables, elaboraed from he ables presened in Pikey, 2014, pages ). The averages presened in he las column was calculaed using he above proposiion. Table 1: Inequaliy in Scandinavia ) Exracs Wages Capial Toal income Average α = 20%) The op 10% 20% 50% 25% 26% The op 1% 5% 20% 7% 8% The nex 9% 15% 30% 18% 18% The middle 40% 45% 40% 45% 44% The boom 50% 35% 10% 30% 30% Gini index 19% 58% 26% 27% Table 2: Inequaliy in Europe 2010) Exracs Wages Capial Toal income Average α = 30%) The op 10% 25% 60% 35% 35% The op 1% 7% 25% 10% 12% The nex 9% 18% 35% 25% 23% The middle 40% 45% 35% 40% 42% The boom 50% 30% 5% 25% 23% Gini index 26% 67% 36% 38% 4

5 Table 3: Inequaliy in he USA 2010) Exracs Wages Capial Toal income Average α = 30%) The op 10% 35% 70% 50% 46% The op 1% 12% 35% 20% 19% The nex 9% 23% 35% 30% 27% The middle 40% 40% 25% 30% 35% The boom 50% 25% 5% 20% 19% Gini index 36% 73% 49% 47% 2 Model descripion In his secion, we inroduce he main ingrediens of he model. We assume ha he basic economic unis are families which are formed as soon as heir respecive parens reire. The marriages occur beween pairs of individuals wih he same capial sock, inheried from heir respecive fahers as soon as hey reire. The capial sock of he family i a ime is a random variable K i,. As we will see laer, he dynamics of he model is such ha, if he iniial capial socks K i,0 are independen and idenically disribued variables, hen he capial socks K i, are as well independen and idenically disribued variables a each ime. Hence, if he number of families is large, he draw of all hese random variables K i, will presen a disribuion which is very close o he common disribuion underlying hem. Each family supplies one uniy of labor ime in each period, so ha he aggregae amoun of labor ime available a ime is equal o he number of families L a ime. We also assume ha here is full employmen and ha he labor marke is perfecly compeiive, so ha all families receive he same wage w a ime. All families have access o he same echnology described by a a producion funcion wih consan elasiciy of subsiuion, so ha he produc of family in he period beween and + 1 is given by Y i, = ε i, αk 1 i, ) + 1 α)a L i, ) 1 1 5) when 1, or Y i, = ε i, K α i,a L i, ) 1 α 6) when = 1, where A is he labor produciviy facor in he period beween and + 1, L i, is he amoun of labor ime hired by family i in he period 5

6 beween and + 1 e ε i, is he oal produciviy shock suffered by family i in he period beween and + 1, which is a nonnegaive random variable wih expeced value equal o one. Each family i hires an amoun of labor ime L i, so ha i maximizes he expeced value of is profis, given by, Z i, w L i,, where when 1, or Z i, = αk 1 i, ) + 1 α)a L i, ) 1 1 when = 1, subjeced o he budge consrain 7) Z i, = K α i,a L i, ) 1 α 8) w L i, 1 δ i, )K i, 9) where δ i, is he depreciaion rae suffered by family i in he period beween and + 1, which is a random variable beween zero and one wih expeced value equal o δ. The concree inspiraion o his model comes from an agrarian economy, where producers suffer, in each period, random produciviy shocks due, for example, o weaher condiions or he emergence of pess and diseases. Finally, he accumulaion or inheriance process is given by λ i, K i,+1 = 1 δ i, )K i, w L i, + s y i, Y i, + s w i,w 10) where s y i, and sw i, are he saving raes, respecively, of he produc and of he wage chosen by family i in he period beween and + 1, which are random variables beween zero and one wih expeced values equal, respecively, o s y and s w, and λ i, is a random variable relaed o he inheriance process in he following way. If he reiremen does no occur, hen we have jus he accumulaion process and hus λ i, = 1. If he reiremen occurs, hen we have he inheriance process and hus λ i, = n i,, where n 2 i, 1 is he number of children of family i a ime. We assume ha ε i,, δ i,, λ i,, s y i,, sw i,) form a sequence of random vecors which are independen and idenically disribued across families and imes and are independen of he iniial capial socks K i,0. Thus he saving raes s y i, and sw i, could be correlaed o each oher and o he produciviy shock ε i,, which is expeced o occur in concree siuaions. Besides his, since L i=1 L i, should be approximaely equal o L, i follows ha he expeced value of L i, should be equal o one, and, since L i=1 λ i, should be approximaely equal o L +1, i follows ha he expeced value of λ i, should be equal o L +1 /L. 6

7 3 Dynamics descripion In his secion, we describe he dynamics underlying he model. I is convenien o consider quaniies and heir respecive expeced values derended of he labor produciviy facor. We denoe by y he expeced value of he derended produc y i, = Y i, /A of family i in he period beween and +1, and denoe by k he expeced value of he derended capial sock k i, = K i, /A of family i in he period beween and + 1. The nex resul relaes y wih k and wih he wage w. Proposiion 3.1 The random variables K i, are independen o each oher across families and are independen o he random vecors ε i,, δ i,, λ i,, s y i,, sw i,) for each ime. If k 1, 1, and δ i, < α, hen he amoun of labor ime hired by family i a ime is given by L i, = k i, k 11) is budge consrain is sricly aained, while he derended produc of family i in he period beween and + 1 is given by y i, y = ε i, k i, k 12) he gross rae of reurn on capial of family i in he period beween and +1 is given by r i, = Y i, w L i, K i, and he derended wage is given by Furhermore, we have ha where when > 1, or when = 1. fk ) = w = ε i,y A w k A 13) = 1 α)y 1 14) A y = fk ) 15) αk 1 ) α 16) fk ) = k α 17) 7

8 Proof: The independence of he random variables K i, o each oher across families and heir independence o he random vecors ε i,, δ i,, λ i,, s y i,, sw i,) follows direcly by inducion on ime, using he equaion 130) and he independence o he iniial capial socks K i,0. If he budge consrain is sricly aained, by equaion 7), he firs order condiion of he maximizaion problem faced by family i a ime is given by 1 Z 1 i, 1 1 α)a L i, ) 1 A w = 0 18) when > 1, and, by equaion 8), i is given by ) Zi, 1 α)a w = 0 19) A L i, when = 1, so ha Zi, ) 1 1 α)a w = 0 20) A L i, when 1. Hence i follows ha ) Z i, w = = µ 21) A L i, 1 α)a On he oher hand, when > 1, by equaion 7), we have ha µ = α Ki, A L i, ) α ) 1 22) so ha K i, 1 = A L i, α µ α ) 1 = ν 23) α and, when = 1, by equaion 7), we have ha ) α Ki, µ = 24) A L i, so ha K i, = µ 1 α = ν 25) A L i, 8

9 Since he expeced value of L i, is equal o one, by equaions 23) and 25), i follows ha k = ν. Hence equaions 23) and 25) are equivalen o equaion 11), since k i, = K i, /A. Furhermore, by equaion 21) and since Y i, = ε i, Z i,, i follows ha y i, = µ ε i, L i,, so ha y = µ. In fac, we have ha L i, is independen of ε i,, since i is a funcion of K i,. Thus i follows ha he expeced value of ε i, L i, is he produc of he expeced values of ε i, and L i,, which are boh equal o one. Hence equaion 12) follows from equaions 21), 23), and 25), while equaion 13) follows from equaions 11) and 12). Furhermore equaion 14) follows from equaion 21), while equaions 16) and 17) follow from equaions 22) and 24). We now show ha he budge consrain is sricly aained. By equaion 14), we have ha w = 1 α)y 1 26) A When > 1, by equaion 16), i follows ha w = 1 α) αk 1 A If k 1, we have ha k 1 w A 1 α) αk 1 1, so ha + 1 α)k 1 ) α ) ) = 1 α)k 1 1 α)k 28) When = 1, by equaion 17), i follows ha w = 1 α)k α A 29) If k 1, we have ha w 1 α)k A 30) Hence, in boh cases, i follows ha w 1 α < 1 δ i, A k 31) This inequaliy and he equaion 11) imply ha w L i, < 1 δ i, )K i, 32) The nex resul shows ha y and k are derended per capia aggregae quaniies. 9

10 Proposiion 3.2 If k 1, 1, and δ i, < α, hen he derended per capia aggregae produc 1 L Y i, 33) A L i=1 is approximaely equal o y, while he derended per capia aggregae capial sock 1 L K i, 34) A L i=1 is approximaely equal o k. Furhermore, even wihou he represenaive agen hypohesis, he average rae of reurn on capial is approximaely equal o r = f k ) 35) which is equal o he expeced value of he rae of reurn on capial r i, of family i in he period beween and + 1. Proof: We have ha y i, and k i, are independen across families, since, by Proposiion 3.2, hey are muliple of K i,, which are independen across families. Furhermore, we have ha 1 L Y i, = 1 L A L L i=1 i=1 y i, and 1 L K i, = 1 L k i, 36) A L L i=1 i=1 so ha he firs wo claims follow direcly from he Law of Large Numbers. Now le us deermine he average rae of reurn on capial. The capial share of he produc is given by α = 1 L i=1 w L i, L i=1 Y i, 1 w L y A L 37) where denoes approximaely equal o. Hence by equaion 14), i follows ha ) α 1 1 α)y 1 1 = y 1 y 1 1 α) 38) Thus, when > 1, by equaion 16), i follows ha α α k y ) 1 39) 10

11 which is also holds when = 1. Hence, i follows ha r = α k y α k y ) 1 = f k ) 40) By equaion 13), he expeced value of he rae of reurn on capial r i, of family i in he period beween and + 1 is given by y A w k A = y ) 1 1 α)y 1 1 k = y k α k y ) 1 = f k ) 41) where we used equaion 14) in he firs equaliy and equaion 16) in he second equaliy as above. The nex resul describes he dynamics boh in he aggregae and in he disaggregae levels. From now on, we assume ha he labor produciviy facor A growhs a a consan rae a and ha he number of families L growhs a a consan rae l. Proposiion 3.3 If 1, δ i, < α, and k 1, hen he dynamics of he derended capial sock k i, of family i in he period beween and + 1 is given by 1 + a)λ i, k i,+1 = 1 δ i, w A k + s y i, ε i, y k ) k i, + s w w i, 42) A while he dynamics of he derended per capia aggregae capial sock k is given by 1 + g)k +1 = 1 δ)k + sk )fk ) 43) where he growh rae is given by and he gross saving rae is given by g = a + l + al 44) sk ) = s y 1 s w )1 α)fk ) 1 45) Proof: Dividing equaion 130) by A and using ha A +1 = 1 + a)a, i follows ha λ i, 1 + a) K i,+1 A +1 = 1 δ i, ) K i, A w L i, A + s y Y i, i, + s w i, A w A 46) 11

12 Hence, by equaions 11) and 12), i follows ha 1 + a)λ i, k i,+1 = 1 δ i, )k i, w k i, + s y i, A k ε i, k i, + s w i, 47) k A which is equivalen o equaion 42). Taking he expeced value of equaion 42), using ha k i, is independen o he random vecor ε i,, δ i,, λ i,, s y i,, sw i,), and using ha he expeced value of λ i, is given by L +1 /L = 1+l, i follows ha l)1 + a)k +1 = 1 δ)k w + s y y + s w w 48) A A Using equaion 14) and ha 1 + l)1 + a) = 1 + g, i follows ha 1 + g)k +1 = 1 δ)k 1 α)y 1 which is equivalen o equaion 43), observing ha y w + s y y + s w 1 α)y 1 49) sk )fk ) = s y y 1 s w )1 α)y 1 50) 4 Convergence o he seady sae In his secion, we analyze he convergence of he derended capial sock o he saionary sae boh in he aggregae and disaggregae levels. The proof of he following general resul of convergence is sraighforward and i is omied. Proposiion 4.1 Assume ha 1 + g)k +1 = 1 δ)k + sk )fk ) 51) and ha here exis consans ˆk > k > 0 such ha s k)f k) δ + g) k > 0 > sˆk)fˆk) δ + g)ˆk 52) If he funcion sk )fk ) has posiive firs derivaive and negaive second derivaive for all k k, hen he derended per capia aggregae capial sock k ends in he long run o he unique soluion, for k k, of he equaion δ + g)k = sk)fk) 53) whenever he iniial derended per capia aggregae capial sock k 0 k. 12

13 We now prove he convergence resul o he saionary sae for he derended capial sock boh in he aggregae and disaggregae levels. Proposiion 4.2 Assume ha 1, ha δ i, < α 1/2, ha and ha s y 1 s w )1 α) > δ + g > 0 54) s y 1 s > 1 w 55) If he iniial derended per capia aggregae capial sock k 0 1, hen he derended per capia aggregae capial sock k ends in he long run o he unique soluion k of he equaion 53), for k k, and he disribuion of he derended capial sock k i, of family i in he period beween and + 1 ends in he long run o a unique saionary disribuion, which is independen of he disribuion of he iniial derended capial sock k i,0. The upper ail of he saionary disribuion is approximaed by a Pareo disribuion whose exponen P is he unique posiive consan such ha 1 δi, + r 1 s y i, ε ) ) P i, y/k 56) 1 + a)λ i, has expeced value equal o one, where r and y are, respecively, he average rae of reurn on capial and he derended per capia aggregae produc a he seady sae. Proof: The convergence of k o k follows from Proposiion 4.1, since sk ) and fk ) ha appear in Proposiion 3.3 saisfy is assumpions, as proved in Subsecion A.1 of he appendix. For he convergence of k i,, we firs noe ha he equaion 42) can be wrien as k i,+1 = µ i, k i, + ν i, 57) where and µ i, = 1 δ i, w /A + s y i, k ε y i, k 58) 1 + a)λ i, ν i, = sw i,w /A ) 1 + a)λ i, 59) 13

14 Since k ends o k, by equaions 14) and 15), we have ha y ends o y and ha w /A ends o a limi w/a. Hence µ i, converges in mean o µ i, = 1 δ i, w/a + s y i, k ε y i, k 60) 1 + a)λ i, while ν i, converges in mean o ν i, = sw i,w/a) 1 + a)λ i, 61) Taking he expeced value of equaion 57), we have ha k +1 = µ k + ν 62) where µ and ν are he expeced values of, respecively, µ i, and ν i,. Hence i follows ha µ = k +1 k ν k 63) On one hand, we have ha µ converges o µ, he expeced value of µ i,. On he oher hand, we have ha µ converges o 1 ν/k, where ν is he expeced value of ν i,. This shows ha k i, is an asympoic Kesen process, as defined in appendix. By Proposiions A.1 and A.2, i follows ha k i, ends in he long run o a unique saionary disribuion, which is independen of k i,0, and ha he upper ail of he saionary disribuion is approximaed by a Pareo disribuion whose coefficien P is he unique posiive consan such ha µ P i, has expeced value equal o one. I remains o show ha µ P i, is equivalen o he expression appearing in 56). By equaion 37), we have ha w y A 1 α 64) so ha w /A k = w y A y k 1 α ) y k = y k r 65) which ends o w/a y k k r 66) The resul hus follows replacing his expression in he equaion 60). 14

15 Noe ha he Pareo exponen P given by he previous proposiion is a decreasing funcion of he difference r g, since 1+r appears in he numeraor and 1 + a)λ i,, whose expeced value is 1 + g, appears in he denominaor of he basis of he random variable which deermines he Pareo exponen P. We now deermine he relaions beween Lorenz curves and Gini indexes of he disribuions of oal income and capial sock. Proposiion 4.3 We have ha and ha L Y x) = α L K x) + 1 α )x 67) G Y = α G K 68) where L Y, L K are he Lorenz curves and G Y, G K are he Gini indexes of he disribuions, respecively, of oal gross income and of capial sock, while α is he capial share of he gross produc. The same resul holds replacing gross quaniies by ne quaniies. Proof: By equaion 13), he rae of reurn on capial sock of family i in he period beween and +1 is a se of random variables which are independen and idenically disribued across families and imes. Now consider he se of families whose respecive capial sock K i, is approximaely equal o a given K. Hence he sum of he respecive capial incomes is given by r i, K, while he sum of he respecive capial socks is given by NK, where N is he number of hese families. The average rae of reurn on capial sock of hese families is given by ri, K NK = ri, 69) N which, by he Law of Large Numbers, is approximaely equal o he expeced value of r i,, given by r = f k ) by equaion 35). Thus i follows ha he rae of reurn on capial sock is consan relaive o he capial sock. Besides his, as he wages are he same for all families, i follows ha he orders of capial socks and wages coincide, ha L w x) = x and ha G w = 0. The resul hus follows from Proposion

16 5 Income and inheriance axes In his secion, we inroduce redisribuion policies based on income and inheriance axes. The income ax is applied jus on profis, so ha he family i pays τ y Y i, w L i, ) 70) of income ax a ime, where τ y is he unique income ax rae. Noe ha he income ax does no influence he decisions of hiring labor ime, since he new maximizaion problem has he same soluion ha he original one. Beside his, when a family suffers a loss, he income ax acs as a ransfer, aenuaing very unfavorable produciviy shocks. The inheriance ax is applied on he inheried capial sock, so ha he heirs of he family i pay τ b b i, K i, 71) of inheriance ax a ime, where τ b is he unique inheriance ax rae and b i, is equal o one when here is an inheriance process and is equal o zero when here is jus an accumulaion process, forming a sequence of random variables which are independen and idenically disribued across families and imes. The redisribuion policies affec dynamics in boh he aggregae and disaggregae levels. Proposiion 5.1 Assume ha 1, ha δ i, +τ b b i, < α, and ha τ y < s y i,, whenever ε i, w A y. If he colleced axes are equally disribued among he families and saved wih a saving rae s τ i,, hen he dynamics of he derended capial sock k i, of family i in he period beween and + 1 is given by 1 + a)λ i, k i,+1 = 1 δ i, τ b b i, τ y ε i, y w ) w ) + s y i, k A A k ε y i, k i, + k +s τ i, τ b bk + τ y y w )) + s w w i, 72) A A where b is he expeced value of b i,, which is equal o he probabiliy of a inheriance process happening, while he dynamics of he derended per capia aggregae capial sock k is similar o he dynamics described in Proposiion 3.3, being only necessary o replace he depreciaion rae and he produc and wage saving raes, respecively, by δ τ = δ + 1 s τ )τ b b s y τ = s y 1 s τ )τ y s w τ = s w + 1 s τ )τ y 73) 16

17 where s τ is he expeced value of s τ i,. Furhermore, if we also assume ha α 1/2, hen he derended per capia aggregae capial sock k ends in he long run o he unique soluion k τ of he equaion 53), for k τ k, and he disribuion of he derended capial sock k i, of family i in he period beween and + 1 ends in he long run o a unique saionary disribuion, which is independen of he disribuion of he iniial derended capial sock k i,0, whenever he inequaliies 54) and 55) hold and whenever he iniial derended per capia aggregae capial sock k 0 1. The upper ail of he saionary disribuion is approximaed by a Pareo disribuion whose exponen P is he unique posiive consan such ha 1 δi, τ b b i, + 1 τ y )r τ 1 s y i, ε ) P i,)y τ /k τ ) 74) 1 + a)λ i, has expeced value equal o one, where r τ and y τ are, respecively, he average rae of reurn on capial and he derended per capia aggregae produc a he seady sae. In paricular, when s τ = 1, he Pareo exponen P is an increasing funcion of he inheriance ax rae τ b. Proof: Since he ax collecion is equally disribued among he families, each family receives L 1 τ y Y i, w L i, ) 75) L i=1 due o income ax redisribuion, as well 1 L L i=1 τ b b i, K i, 76) due o inheriance ax redisribuion. By he Law of Large Numbers, hese incomes are approximaely equal o heir respecive expeced value, given by τ y A y w ) and τ b ba k 77) Taking ino accoun wha he family i pays and receives due o he reribuive policies, he accumulaion or inheriance process is given by λ i, K i,+1 = 1 δ i, )K i, τ b b i, K i, w L i, + s y i, Y i, τ y Y i, w L i, ) + +s w i,w + s τ i, τ b ba k + τ y A y w ) ) 78) 17

18 so ha we obain he dynamics of he derended capial sock k i, of family i in he period beween and + 1 dividing he above equaion by A, using equaions 11) and 12), and remembering ha A +1 = 1 + a)a. Taking he expeced value of equaion 72), i follows ha 1 + a)1 + l)k +1 = which can be rewrien as 1 δ τ b b τ y k y w ) A +s τ τ b bk + τ y y w )) A w + s y y ) k + A k k + s w w A 79) 1 + g)k +1 = 1 δ 1 s τ )τ b b ) k + s y 1 s τ )τ y ) y 1 s w 1 s τ )τ y ) w A 80) so ha, using equaions 14) and 15), he dynamics of he derended per capia aggregae capial sock k is given by where 1 + g)k +1 = 1 δ τ ) k + sk )fk ) 81) sk ) = s y τ 1 s w τ ) 1 α)fk ) ) The convergence resuls and he condiion on he Pareo exponen P are obained exacly as in he proof of Proposiion 4.2. Finally, when s τ = 1, i follows ha k τ = k, ha r τ = r, and ha y τ = y, so ha he Pareo exponen P is a increasing funcion of he inheriance ax rae τ b, since an increase in τ b decreases he numeraor of he basis of he random variable which deermines he Pareo exponen P. 6 Comparaive saics In his secion, we presen he comparaive saics in order o obain he relaion beween he growh rae and he income and wealh inequaliies in he seady sae. We focus on he case where he gross elasiciy of subsiuion is equal o one or wo. We firs obain explici expressions for he derended per capia aggregae capial sock in he seady sae. 18

19 Proposiion 6.1 Assume ha 1, ha δ i, < α 1/2, ha he inequaliies 54) and 55) hold, and ha iniial derended per capia aggregae capial sock k 0 1. Then he derended per capia aggregae capial sock k ends in he long run o s y 1 s w )1 α) k = δ + g ) 1 1 α 83) when = 1, while i ends in he long run o 1 s w s y k = 2s y 1 s w ) ) 2 1 α 84) α when = 2 and δ + g = s y α 2, and i ends in he long run o 2s y 1 s w ))α + ) 2 1 s k = w )α 2 4δ + g)1 s w s y ) 1 α) 2δ + g s y α 2 ) 85) when = 2 and δ + g s y α 2. Proof: When = 1, equaion 53) is given by so ha δ + g)k = s y 1 s w )1 α))k α 86) k 1 α = sy 1 s w )1 α) δ + g 87) which is equilvalen o equaion 83). When = 2, equaion 53) is given by δ + g)k = s y fk) 1 s w )1 α)fk) ) Since in his case fk) = αx + 1 α) 2, where x = k 1 2, i follows ha he above equaion is equivalen o δ + g)x 2 = s y αx + 1 α) 2 1 s w )1 α)αx + 1 α) 89) which can be rewrien as δ + g s y α 2 )x s w 2s y )α1 α)x + 1 s w s y )1 α) 2 = 0 90) 19

20 If δ + g = s y α 2, his is a linear equaion, which has a unique soluion given by x = 1 sw s y 1 α 91) 2s y 1 s w ) α If δ + g s y α 2, his is a quadraic equaion, which has a unique soluion greaer han one given by x = 2sy 1 s w ))α + 1 s w )α 2 4δ + g)1 s w s y ) 1 α) 92) 2δ + g s y α 2 ) If we square equaions 91) and 92), we obain equaions, respecively, 84) and 85), since k = x 2. Figure 1: Ne elasiciy of subsiuion versus growh rae g Now we focus in he following se up in order o make he analysis more similar o he analysis presened in Pikey e Zucman, 2015). There are no 20

21 redisribuion policies and he depreciaion rae is assumed o be a consan δ i, = δ < α. All families have exacly wo children, so ha λ i, = 1, which implies ha he labor force growh rae is zero, l = 0, and ha he growh rae is equal o he labor produciviy growh rae, g = a. As in Rognlie, 2014, page 6), we se ha δ = We also se ha s y = 0.8, ha s w = 0.1, ha α = 1/3, for = 1, and ha α = 1/5, for = 2. The growh rae g varies in he inerval [0, 0.04]. These numerical values produce he graphics of ne elasiciy of subsiuion, of ne saving rae s, and of ne capial share of produc α, which are presened in Figures, respecively, 1, 2, 3. The ne elasiciy of subsiuion varies in he inerval [0.5, 0.8], when = 1, and varies in he inerval [1.1, 1.6], when = 2, which coincides wih he inerval menioned in Pikey e Zucman, 2015, page 1351). Figure 2: Ne saving rae s versus growh rae g The ne saving rae s varies in he inerval [0.0, 0.09] and is increasing, when = 1, and varies in he inerval [0.0, 0.13] and is non-monoonic, when = 2. The ne capial share of produc α varies in he inerval [0.17, 0.27] 21

22 Figure 3: Ne capial share α versus growh rae g and is increasing, when = 1, and varies in he inerval [0.20, 0.43] and is decreasing, when = 2. The graphics of he ne capial over produc raio β and of he ne rae of reurn on capial r are omied. The ne capial over produc raio β varies in he inerval [2.3, 5.0] and is decreasing, when = 1, and varies in he inerval [1.7, 36.4] and is also decreasing, when = 2. The ne rae of reurn on capial r varies in he inerval [0.04, 0.12] and is increasing, when = 1, and varies in he inerval [0.01, 0.12] and is also increasing, when = 2. Now we look a he relaion beween he growh rae and he wealh inequaliy in he seady sae. We assume ha he random variable s y i, ε i, is equal o s y /p wih probabiliy p and is equal o zero oherwise. Here we can hink, as in Nirei, 2009; S. Aoki and M. Nirei, 2016), ha s y i, is a consan and ha ε i, is a random variable, or we can hink, as in Pikey e Zucman, 2015), ha ε i, is a consan and s y i, is a random variable. The condiion ha deermines he Pareo exponen P of he upper ail of he derended capial 22

23 sock disribuion in Proposiion 4.2 is hen given by 1 δi, + r 1 s y ) P ) P /p) y/k 1 δi, + r y/k p +1 p) = 1 93) 1 + g 1 + g where k is given by Proposiion 6.1, while y = fk) and r = αfk)/k) 1. In Figure 4, we presen he graphics of Pareo exponen P versus growh rae g for = 1, where he probabiliy can be equal o p = 0.5, o p = 0.6, o p = 0.7, and o p = 0.8. In all hese cases, he Pareo exponen P is a decreasing funcion of he growh rae g. This implies ha he inequaliy is an increasing funcion of he growh rae g. I is also eviden ha, when he probabiliy p increases, which is he same as he variance decreases, hen he Pareo exponen increases, which is he same as he op inequaliy decreases. Figure 4: Pareo exponen P versus growh rae g for = 1 In Figure 5, we presen he graphics of Pareo exponen P versus growh rae g now for = 2, where he probabiliy, as in he previous case, can be 23

24 equal o p = 0.5, o p = 0.6, o p = 0.7, and o p = 0.8. Now, in all hese cases, he Pareo exponen P is a non-monoonic funcion of he growh rae g. I is an increasing funcion for growh raes lower han approximaely and i is an decreasing funcion for growh raes greaer han approximaely Hence, only for exremely low values, he op wealh inequaliy increases wih a reducion of he growh rae. On he oher hand, since he ne capial share srongly increases wih a reducion of he growh rae, as shown by Figure 3, he inequaliy of income disribuion should also increase wih a reducion of he growh rae, due o Proposiion 4.3. Figure 5: Pareo exponen P versus growh rae g for = 2 All he formulas used o elaborae he graphics which are presened in he Figures 1, 2, 3 are presened in he Subsecion A.3 of he appendix. 24

25 7 Nex seps We conclude his aricle presening a concise and non-exhausive selecion of possible seps for fuure research: 1. Deermine he degree of social mobiliy inside he presened model. 2. Invesigae, hrough compuaional simulaions, wha happens ouside he upper ail of he derended capial sock disribuion. 3. Perform, hrough compuaional simulaions, he comparaive saics in order o deermine he impac of he disribuive policies over he income and wealh disribuions and inequaliies. 4. Analyze if he empirical daa approximaely saisfy he assumpions of Proposiion 1.1. If affirmaive, find possible heoreical explanaions for his fac. 5. Develop a muli-secor version of he presened model, as suggesed, for example, in Sigliz, 2015) and in Pikey, 2015, page 81). Sar including a non-produced and non-depreciable good, such as land, in order o address he criicism presened, for example, in Rognlie, 2014) and in Bonne e ali, 2014), abou he idenificaion beween capial and wealh, and he role of capial gains in he recen growh of he wealh over income raio. 6. Creae an open economy version of he presened model in order o analyze he possible effecs of globalizaion in he evoluion of income and wealh inequaliies. 7. Inroduce some degree of monopoly ino he model in order o explain he sagnaion of wages in he las hree decades in USA even wih posiive labor produciviy growh rae, as suggesed in Sigliz, 2015). A Appendix A.1 Convergence o he seady sae In his secion, we prove ha he funcions sk ) and fk ), which appear in Proposiion 3.3, saisfy he assumpions of Proposiion

26 Proof: We have ha boh sk ) and fk ) are smooh funcions. For k 1, i follows ha fk ) 1 94) and we have ha We also have ha so ha I follows ha ) 1 f fk ) k ) = α > 0 95) k sk )fk ) = s y fk ) 1 s w )1 α)fk ) 1 96) sk )fk )) = s y f k ) 1 s w )1 α) 1 fk ) 1 1 f k ) 97) sk )fk )) = s y 1 s w )1 α) 1 fk ) 1 1 ) f k ) > 0 98) where he above inequaliy follows from 1 and from inequaliies 54), 94), and 95), since s y 1 s w )1 α) 1 fk ) 1 1 > s y 1 s w )1 α) > 0 99) By equaions 95) and 97), we have ha sk )fk )) = s y αfk ) 1 k 1 1 s w )1 α) α fk ) 2 1 k 1 100) so ha sk )fk )) = s y α fk ) 1 1 αfk ) 1 k 1 +s y αfk ) 1 1 ) k 1 1 k ) ) 1 s w )1 α) α s w )1 α) α fk ) fk ) 2 2 αfk ) 1 k 1 ) k 1 1 k 1

27 Hence i follows ha Since i follows ha and also ha so ha sk )fk )) = α fk ) 3 2 k 1 1 [ s y αfk ) 1 +1 k ) s y fk ) 2 +2 ) 2 1 s w )1 α) 1 αk s w )1 α) 1 ] fk ) = 1 103) fk ) 1 +1 = αk α 104) fk ) 2 +2 = sk )fk )) = α fk ) 3 2 k 1 1 s y α 2 k s w )1 α) αk α) 2 105) [ s y α + 2αk s w )1 α) 1 Simplifying his expression, we have ha sk )fk )) = α fk ) 3 2 k 1 1 αk 1 ) + 1 α k 1 106) ) 1 α) + 1 α) 2 ) 1 αk 1 + ) αk ] α [ s y αk s w )1 α) 1 +1 s w )1 α) 1 ] 1 α) ) 1 α) + 1 α) 2 107) ) αk

28 so ha sk )fk )) = α fk ) 3 2 k α) [ 108) ) ) s w ) s y αk s w ) 1 ) ] sy 1 α) By inequaliy 55), i follows ha ) 1 1 s w ) s y < 0 109) so ha since k 1 sk )fk )) α fk ) 3 2 k α) [ 110) ) ) s w ) s y α s w ) 1 ) ] sy 1 α) 1. Simplifying his expression, we have ha [ 1 s w ) sk )fk )) α fk ) 3 2 k α) Since α 1/2 and 1, i follows ha α + 1 2α ) s y ] 111) sk )fk )) α fk ) 3 2 k α) [1 s w )1 α) s y ] < 0 112) where he las inequaliy follows from inequaliy 54). Seing k = 1, by inequaliy 54), i follows ha s k)f k) δ + g) k = s y 1 s w )1 α) δ + g) > 0 113) On he oher hand, by equaions 95) and 97), i follows ha sk )fk ) δ + g)k ) = sy α β 1 α1 sw )1 α) δ + g) 114) fk ) 1 1 β 1 28

29 which ends o δ + g) < 0, when k goes o. Hence i follows ha so ha here exiss ˆk > k > 0 such ha lim sk )fk ) δ + g)k = 115) k sˆk)fˆk) δ + g)ˆk < 0 116) A.2 Kesen process A sequence of random variables k i, is called a Kesen process if k i,+1 = µ i, ki, + ν i, 117) where µ i,, ν i, ) is a sequence of independen and idenically disribued nonnegaive random vecors, where he expeced value µ of µ i, is lower han one and where he expeced value ν of ν i, is finie. The following resul is known as Kesen Theorem. Proposiion A.1 If k i, is a Kesen process, hen is disribuion ends in he long run o a unique saionary disribuion, which is independen of k i,0, and ha upper ail of he saionary disribuion is approximaed by a Pareo disribuion whose coefficien P is he unique posiive consan such ha µ P i, has expeced value equal o one. The proof of Kesen Theorem and he omied addiional echnical assumpions can be found in Kesen, 1973 and 1974) and in Goldie, 1991). A sequence of random variables k i, is called a asympoic Kesen process if k i,+1 = µ i, k i, + ν i, 118) where he sequence µ i, converges in mean o he sequence µ i,, he sequence ν i, converges in mean o he sequence ν i,, and he sequence k i,, given by k i,+1 = µ i, ki, + ν i, 119) is a Kesen process, called he associaed Kesen process. 29

30 Proposiion A.2 If k i, is an asympoic Kesen process, hen is disribuion ends in he long run o he saionary disribuion of is associaed Kesen process. Proof: Firs we show ha he expeced value k of k i, is bounded. The expeced value µ of µ i, ends o µ in he long run, since µ i, converges in mean o µ i,. Analogously, he expeced value ν of ν i, ends o ν in he long run, since ν i, converges in mean o ν i,. Hence here exis T > 0, µ < µ < 1, and ν < ν such ha µ µ and ν ν for all T. Taking he expeced value of equaion 118), i follows ha k +1 = µ k + ν µk + ν 120) for all T. Thus we can show, by a direc inducion on, ha for all > T, so ha k µ T k T + ν T 1 n=0 µ n 121) k µ T k T + ν 1 µ T 1 µ < k T + ν 1 µ 122) for all > T, which shows ha k is a bounded sequence. Now, subracing equaion 118) and 119), i follows ha so ha k i,+1 k i,+1 = µ i, k i, µ i, ki, + ν i, ν i, = µ i, k i, k i, ) + µ i, µ i, ) k i, + ν i, ν i, 123) k i,+1 k i,+1 µ i, k i, k i, + µ i, µ i, k i, + ν i, ν i, 124) Taking firs he expeced value of his inequaliy and hen he upper limi when ends o infiniy, using he independence beween µ i, and k i, k i,, he independence beween µ i, µ i, and k i,, ha µ i, converges in mean o µ i,, and ha ν i, converges in mean o ν i,, i follows ha 0 D µd 125) 30

31 where D denoes he upper limi of he expeced value of k i, k i, when ends o infiniy. Taking firs he expeced value and hen he upper limi when ends o infiniy of he inequaliy k i, k i, k i, + k i, 126) i follows ha D is finie. By inequaliy 125) and since µ is lower han one, i follows ha D is equal o zero, which shows ha k i, and k i, approach o each oher in mean. Since convergence in mean implies convergen in disribuion and since he disribuion of k i, ends o is saionary disribuion, he resul follows. A.3 Comparaive saics In his secion, we presen all he formulas used o elaborae he graphics presened in Figures 1, 2, 3. The formula of he ne elasiciy of subsiuion is given by 1 δ/r) = 127) 1 δk/y while he formula of he ne saving rae is given by s = sk) δk/y 1 δk/y 128) and he formula of he ne capial share of produc is given by α = 1 1 δ/r)αk/y) 1 δk/y 129) where k is given by Proposiion 6.1, while sk) is given by Proposiion 3.3, y = fk) is given by Proposiion 3.2, and r = αy/k) 1. A.4 Alernaive saving heory In his secion, we presen and alernaive saving heory which can be easily accommodaed in he model presened in his paper. Insead of s y i, be he saving rae of he produc, i could be he saving rae of he profi, provided 31

32 ha s y i, is equal o one, and consequenly sw i, is lower han one, whenever a loss happens. The accumulaion or inheriance process is hen replaced by λ i, K i,+1 = 1 δ i, )K i, + s y i, Y i, w L i, ) + s w i,w 130) I easy o show ha he dynamics of he derended capial sock k i, of family i in he period beween and + 1 is hen given by 1 + a)λ i, k i,+1 = 1 δ i, + s y y i, ε i, w )) k i, + s w w i, 131) k A k A ha he Pareo exponen P is he unique posiive consan such ha 1 δi, + s y i, r 1 ε ) i,) y/k) P 132) 1 + a)λ i, has expeced value equal o one, and ha he dynamics of he derended per capia aggregae capial sock k and he condiions ensuring convergence are he same of he ones presened in Proposiion 3.3 and in Proposiion 4.2, jus replacing he expression 1 s w by he expression s y s w. The following resul provides a sufficien condiion for he consisence of he alernaive saving heory inside he model. Proposiion A.3 If s y i, is equal o one whenever ε i, < 1 α, hen s y i, is equal o one whenever a loss happens. Proof: The resul follows direcly from he following Y i, w L i, = A y i, w ) k i, A = A y k i, k = A y k i, k k ε i, w ) A y ) ε i, 1 α y 1 1 > A y k i, k ε i, 1 α)) 133) where he firs equaliy follows from equaion 11), he second equaliy follows from equaion 12), he hird equaliy follows from equaion 14), and he las inequaliy follows from y 1 and from

33 A.5 Nirei model In his secion, we show ha he assumpion of he previous knowledge of he produciviy shocks before he beginning of he producion process may rise an inconsisence in he Nirei model. The producion funcion is given by Y i, = K α i,ɛ i, A L i, ) 1 α 134) where ɛ i, is he labor produciviy shock, and his expression is equivalen o Y i, = ε i, K α i,a L i, ) 1 α 135) where ε i, = ɛ 1 α i, 136) is he oal produciviy shock. Each family i hires a amoun of labor ime L i, so ha i maximizes is profis, given by The accumulaion process is given by Y i, w L i, 137) K i,+1 = 1 δ)k i, + sy i, w L i, ) + sw 138) where δ and s are he consan, respecively, depreciaion and saving raes. Le d be he depreciaion rae of he capial when i is no used in he producion process. I is very reasonable assume ha d < δ. Proposiion A.4 If he labor produciviy shock is hen ) α δ d 1 α ɛ i, < y 139) α 1 d)k i, > 1 δ)k i, + Y i, w L i, 140) so ha i is raional no o engage in he producion process. Proof: The firs order condiion of he maximizaion problem faced by family i is given by Y i, 1 α) ɛ i, A w = 0 141) ɛ i, A L i, 33

34 so ha y i, L i, = w 1 α)a = y 142) where he las equaliy follows from muliplying he firs equaion by L i, and hen aking he expeced value. On he oher hand, dividing equaion 134) by A L i,, i follows ha ) α y i, ki, = ɛ 1 α i, 143) L i, L i, so ha, by equaions 142) and 143), i follows ha L i, = ɛ 1 α α i, y 1 α K i, A 144) By equaions 142) and 144), we have ha and also ha Hence i follows ha w L i, = ɛ 1 α α i, y 1 α w A K i, = 1 α) Y i, = y A L i, = 1 δ)k i, + Y i, w L i, = ɛi, y ɛi, y ) 1 α α Ki, 145) ) 1 α α Ki, 146) 1 δ + α ɛi, y ) 1 α ) α K i, < )) δ d 1 δ + α K i, α = 1 d)k i, 147) If i is always raional o engage in he producion process, i is easy o show ha he dynamics of he derended per capia aggregae capial sock k was given by 1 + a)k +1 = 1 δ)k + sck α 148) 34

35 where c is he expeced value of ɛ 1 α α i, raised o α, as saed in Nirei, 2009; S. Aoki and M. Nirei, 2016). Hence he producion in he seady sae would be given by s y = c ) α 1 α 149) δ + a and he condiion in he previous proposiion a he seady sae would be given by ) α δ d s 1 α ɛ i, < c 150) α δ + a If we assume ha c = 1, ha δ = 0.04, ha d = 0.01, ha α = 1/3, ha s = 0.2, and ha a = 0.02, his condiion would be given by ɛ i, < 0.55, and if assume ha d = a = 0, hen i would be given by ɛ i, < If he variance of he labor produciviy shocks are reasonably large, he Nirei model hus presen an inconsisence. A.6 Income, capial and wages inequaliies In his secion, we prove Proposiion 1.1, which seems a knew resul, since a leas i no appears in Lamber, 2001), which is a reference book in his subjec. Proof: We pu he individuals in increasing order of capial sock, so ha, for each x in he inerval [0, 1], he inerval [0, x] corresponds o he individuals who are among he x percen associaed o he lower values of capial sock. Since capial sock and wages orders coincide, he inerval [0, x] also corresponds o he individuals who are among he x percen associaed o he lower values of wages. Denoe by Kx) and by W x) he sum, respecively, of capial sock and of wages of he individuals corresponding o he individuals associaed o he inerval [0, x]. Since he rae of reurn on capial r is consan relaive o capial sock, we have ha r Kx) is he sum of capial income corresponding o he individuals associaed o he inerval [0, x], so ha Y x) = r Kx) + W x) 151) is he sum of oal income corresponding o he individuals associaed o he inerval [0, x]. On he oher hand, by definiion, we have ha L Y x) = Y x) Y, L K x) = Kx) K, L w x) = W x) W 152) 35

36 where Y = Y 100%), K = K100%), W = W 100%) 153) By equaions 151) and 152), i follows ha Y L Y x) = r K L K x) + W L w x) 154) so ha L Y x) = r K L K x) + W L w x) 155) Y Y The resul follows, since α = r K /Y and since 1 α = W /Y. The derivaion of he resul for he Gini indexes from he resul for he Lorenz curves is sraighforward and is omied. References [1] D. Acemoglu e J. Robinson 2015). The Rise and Fall of General Laws of Capialism. Journal of Economic Perspecives, vol. 29, 328. [2] S. Aoki e M. Nirei 2016). Pareo Disribuion of Income in Neoclassical Growh Models. Review of Economic Dynamics, vol. 20, [3] J. Benhabib, A. Bisin e S. Zhu 2011). The disribuion of wealh and fiscal policy in economies wih finiely lived agens. Economerica, vol. 79, [4] J. Benhabib, A. Bisin e S. Zhu 2015). The wealh disribuion in Bewley economies wih capial income risk. Journal of Economic Theory, vol. 159, [5] O. Bonne e ali 2014). Does housing capial conribue o inequaliy? A commen on Thomas Pikeys Capial in he 21s Cenury. Sciences Po., Discussion Paper [6] F. Cowell 1998). Inheriance and he Disribuion of Wealh. LSE STICERD, Research Paper 34, hp://ssrn.com/absrac=

37 [7] C. Goldie 1991). Implici renewal heory and ails of soluions of random equaions. Annals of Applied Probabiliy, vol. 1, [8] P. Lamber 2001). The disribuion and redisribuion of income. Mancheser Universiy Press, Mancheser an New York. [9] H. Kesen 1973). Random difference equaions and renewal heory for producs of random marices. Aca Mahemaica, vol. 131, [10] H. Kesen 1974). Renewal Theory for Funcionals of a Markov Chain wih General Sae Space. The Annals of Probabiliy, vol. 2, [11] J. Hubmer, P. Krusell and A. Smih Jr 2016). The Hisorical Evoluion of he Wealh Disribuion: A Quaniaive-Theoreic Invesigaion. NBER Working Paper No Issued in December [12] T. Pikey 2014). Capial in he 21s Cenury. Harvard Universiy Press, Cambridge. [13] T. Pikey 2015). Puing Disribuion Back a he Cener of Economics: Reflecions on Capial in he Tweny-Firs Cenury. Journal of Economic Perspecives, vol. 29, [14] T. Pikey e G. Zucman 2015). Wealh and Inheriance in he Long Run. Handbook of Income Disribuion, vol. 2, [15] M. Rognlie 2014). A noe on Pikey and diminishing reurns o capial. Deparmen of Economics, MIT, hp:// mrognlie/pikey diminishing reurns.pdf [16] R. Solow 1956). A Conribuion o he Theory of Economic Growh. Quarerly Journal of Economics, [17] J. Sigliz 2015). New Theoreical Perspecives on he Disribuion of Income and Wealh among Individuals, Working Paper 21189, hp:// [18] T. Swan 1956). Economic growh and capial accumulaion. Economic Record,