Financial Structure, Growth and the Distribution of Wealth

Transcription

1 Financial Srucure, Growh and he Disribuion of Wealh Davide Fiaschi Universiy of Pisa. Diparimeno di Scienze Economiche Riccarda Longarei Universiy of Milano - Bicocca. Diparimeno di Economia Poliica riccarda.longarei@unimib.i May 005 Preliary Draf Absrac A large heoreical and empirical lieraure has documened he relaion exising beween financial secor, economic growh and capial accumulaion. The causal relaionships among hese variables are no clear hough and reasonably i does no work in a unique direcion: financial developmen can be boh he cause and he effec of economic growh or, alernaively, a mere feaure of he growh process; he same consideraion holds for inflaion. In his paper we argue ha inequaliy plays a role in explaining hese relaionships and propose an OLG model wih heerogeneous alruisic agens, credi and money. Agens are heerogeneous in wealh and alens. They mus choose wheher being workers-lenders or enrepreneurs (eiher be self-financed or liquidiy consrained). Individual occupaional choice depends boh on individual s wealh and abiliies, bu also on he real ineres raes on deb and on credi. Financial developmen is measured by he gap beween he ineres rae on deb and credi. Credi is also supplied by a cenral bank ha provides money o he financial secor and his, in urn finances he invesmen projecs. The cenral bank hen ransfers equally o each and every agen he reurn on he supply of credi. This can be a source of inflaion. 1

2 1 INTRODUCTION In equilibrium he real ineres raes on deb and on credi, he individual occupaional choice and he share of enrepreneurs in he populaion, i.e. he disribuion of income and wealh, heir average and aggregae levels, are affeced by he inflaion rae and he degree of financial developmen. More precisely we show ha an increase in he coss of financial ransacion has basically wo effecs. On one hand, reducing he gain o be a lender, i increases he relaive share of enrepreneurs in he populaion (subsiuion effec). On he oher hand, i increases he ineres rae and herefore decreases he enrepreneurial share of he populaion since i reduces he available pool of loanable funds (scale effec). The ne effec depends on which effec prevails. Moreover, in his framework money is no neural. The higher he money sock, he lower, for any given inflaion rae, he equilibrium noal ineres rae. In urn, his implies a lower share of he populaion wih liquidiy consrains and herefore an increase in he enrepreneurial share of he populaion and in he average wealh. Finally we demonsrae ha financial developmen and moneary policy have asymmeric effecs hroughou he populaion. Liquidiy consrained agens are definiely made worse off by boh an increase in he coss of financial inermediaion and by an increase in inflaion rae. This provides a possible jusificaion for he direc relaionship beween inequaliy and high inflaion ha is shown by empirical esimaions. 1 Inroducion The causal relaionships among financial secor, economic growh and capial accumulaion are no clear, hough analysed by heoreical lieraure and empirically esimaed. Reasonably i does no work in a unique direcion: financial developmen can be boh he cause and he effec of economic growh or, alernaively, a mere feaure of he growh process; he same consideraion holds for inflaion ( Gerler (1988), Levine (1997) and Pagano (1993)) In his paper we argue ha wealh and income inequaliy, resuling from heerogeneiy in alens, plays a role in explaining hese relaionships. Following Longarei and Delli Gai (005), we propose an OLG model wih heerogeneous alruisic agens and money, and we model, in his framework, he credi marke. Agens are heerogeneous in wealh, alens and consequenly in income. They mus choose wheher being workers-lenders or enrepreneurs, underaking an invesmen projec wih a fix inpu requiremen. Enrepreneurs may be eiher self-financed

3 1 INTRODUCTION or liquidiy consrained. In he former case hey need o borrow he inpu requiremen in excess o heir own wealh. Self-financed enrepreneurs, on he oher hand, lend heir wealh in excess o he inpu requiremen. The occupaional choice depends boh on he individual wealh and abiliies, bu also on he real ineres raes on deb and on credi. Financial developmen is measured he resources consumed by he financial secor or equivalenly he coss of financial ransacion. This generaes a gap beween he ineres rae on deb and credi. Apar from individuals, credi is also supplied by a cenral bank ha provides money o he financial secor and his, in urn finances he invesmen projecs. The cenral banker hen ransfers equally among individuals he profi earned. This can be a source of inflaion. In equilibrium he real ineres raes on deb and on credi, he individual occupaional choice and he share of enrepreneurs in he populaion, i.e. he disribuion of income and wealh, heir average and aggregae levels, are affeced by boh he inflaion rae and he degree of financial developmen. An increase in he coss of financial ransacion has basically wo effecs. On one hand i widens he gap beween he rae of reurn on deb and he rae of reurn on credi for any given ineres rae. This can produce as a sor of subsiuion effec ha increases he relaive share of enrepreneurs in he populaion (he gain o be a lender is lower). On he oher hand, he lower efficiency of he financial secor reduces he available pool of loanable funds causing an increase in he ineres rae and herefore a decrease in he enrepreneurial share of he populaion (scale effec). The ne effec on he equilibrium ineres rae, on he composiion of he populaion per occupaion and on he aggregae wealh depend on which effec prevails. This in urn depends crucially on he average (and aggregae) wealh and on is disribuion. In paricular, he higher he share of populaion wih liquidiy consrains and he lower he average (and he aggregae) wealh, he more he scale effec prevails. Symmerically, he lower he share of populaion wih liquidiy consrains and he higher is he average (and he aggregae) wealh, he more he subsiuion effec prevails. Therefore σ acs as a sor of financial acceleraor widening booms and recessions. Moreover, in his framework money is neiher neural nor superneural. The higher he growh rae of money, he higher (lower) he inflaion (real ineres) rae. Equivalenly he higher he money sock he lower, for any given inflaion rae, he equilibrium noal ineres rae. In urn, his implies a lower share of he populaion wih liquidiy consrains and herefore an increase in he enrepreneurial share of he populaion and in he average wealh. 3

4 THE MODEL Financial markes and moneary policy have synergic ineracion. The more efficien he financial marke, he higher he effecs of moneary policy. Moreover, financial developmen and moneary policy have asymmeric effecs hroughou he populaion, since hey ac via he occupaional choice. Liquidiy consrained agens are made definiely worse off by boh an increase in he coss of financial inermediaion and by an increase in inflaion rae. This provides a possible jusificaion for he direc relaionship beween inequaliy and high inflaion ha is shown by empirical esimaions. The model In he economy here exiss a unique good, he oupu good, ha can be eiher consumed or invesed. There are wo differen producion echnologies in order o produce he oupu good, a labour-inensive echnology and a capial-inensive one. For he sake of simpliciy, we assume ha populaion is consan over ime. Populaion consiss of a mass of young individuals equal o 1 and a mass of old individuals (he young of he previous generaion) equal o 1. Oupu is perishable and herefore wealh canno be sored o be consumed in he fuure. For simpliciy, preferences are uniform across individuals and he young do no receive uiliy from consumpion. Assug inergeneraional alruism, well-behaved preferences may be represened by: U = (c i+1 ) γ (b i+1 ) 1 γ, (1) where c i+1 is consumpion of he agen when old and b i+1 is beques of he old o he young. The i-h young a ime is endowed by Naure wih one uni of labour and a some level of alen ρ i. Moreover, he receives a moneary beques equal o P b i from his faher. He can allocae his beques in wo ways: eiher offer i o he financial secor, becog a lender, and/or inves a fix amoun P x in he enrepreneurial projec ogeher wih his endowmen of labour. Since he amoun of invesmen is fixed, an enrepreneurial agen may be self financed (i.e. an enrepreneur-lender) and, in his case, he offers he remaining amoun of his beques o he financial secor. Oherwise he may borrow from he financial secor becog an enrepreneur-borrower. Finally here are all non-enrepreneurs agens offer inelasically heir labour endowmen, becog workers, a he fix real wage w < 1. 1 Sumg up here exis hree ypes of agens: workers-lenders (WL 1 This consrain on real wage is no source of limiaion in he analysis because is essenially due o 4

5 THE MODEL hereafer), enrepreneurs-lenders (EL hereafer) and enrepreneurs-borrower (EB hereafer). The financial secor consumes σ unis of oupu per uni of loans. Therefore σ can be hough of as a ransacion cos or, equivalenly, a measure of he efficiency of he financial secor. The lower σ he more efficien he financial secor. Therefore he supply of credi by he financial secor is a share (1 σ) of he lenders bequess. Therefore he supply and demand of agens are given by: 3 L P,S i = (1 σ)p b i if i is WL; = (1 σ) (P b i P x) if i is EL; L P,S i L D i = P x P b i if i is EB. () where L P,S i (L P,D i ) is he noal supply (demand) of credi of agen i and index P sands for privae supply of credi, as i will see below, since here exiss also a public supply of credi by a cenral banker In he economy here exiss also a cenral banker ha gives a sock of money a ime H o he financial secor in order o be len (his is equivalen o a public supply of credi). A ime + 1 he financial secors pays back o he cenral banker an amoun equal o H R +1 (1 σ). This is ransferred o he old agens wih equal ransfers hroughou he economy. Since he populaion is of mass 1, H R +1 (1 σ) represens boh aggregae and individual ransfers. 4 Producion akes place when agens are old. 5 If an agen knows he will become an enrepreneur, he i-h old-enrepreneur ges a gross real reurn ρ i on he invesmen projec. We assume ha ρ i is drawn from a ime invarian uniform disribuion beween 0 and 1, i.e.: ρ i U (0, 1). (3) Every borrowers pays back his deb o lenders a he gross noal ineres rae R. he assumpion ha agens abiliies are on he range [0,1]. Noice ha σ can also be hough of as he reserve requiremen raio when financial secor is composed by banks. 3 Pr sands for privae, as i will be clear below, since here exiss also a public supply of credi by a cenral banker. 4 This is equivalen o assume ha every agen ges an equal share of he profis of he financial secor once we ake σ as he uni cos of he financial inermediaion (calculaed on HR and no on H). 5 Noe ha, in order o simplify algebra, we assume ha invesmen, and consequenly occupaional choice, akes place when agens are young, whereas producion akes place when agens are old. 5

6 .1 Occupaional choice THE MODEL Therefore he ineremporal budge consrain is given by: P +1 w + (1 σ) (P b i + H )R +1 = P +1 (c i+1 + b i+1 ) if i is WL; P +1 ρ i (P x P b i )R +1 + (1 σ)h R +1 = P +1 (c i+1 + b i+1 ) if i is EB; P +1 ρ i + (1 σ) (P b i P x + H )R +1 = P +1 (c i+1 + b i+1 ) if i is EL. (4) The coss of financial ransacion creae a wedge beween he ineres rae on debi (R +1 ) and he ineres rae on credi ((1 σ) R +1 ), being he former higher han he laer. The ineremporal budge consrains can be expressed in real erms dividing by P +1, i.e.: w + (1 σ) (b i + h )θ +1 R +1 = c i+1 + b i+1 if i is WL; (5) ρ i (x b i )θ +1 R +1 + (1 σ)h θ +1 R +1 = (c i+1 + b i+1 ) if i is EB; (6) ρ i + (1 σ) (b i x + h )θ +1 R +1 = c i+1 + b i+1 if i is EL, (7) where θ +1 = 1/ (1 + π +1 ) is he real rae of reurn on money, θ +1 R +1 = (1 + r +1 )/(1 + π +1 ) is he real gross ineres rae and h = H /P is he real money supply..1 Occupaional choice Maximizaion of uiliy (1) corresponds o maximize he income of he old agen. 6 Therefore, from Eqq. (5) and (7), when agen i is sufficienly endowed, i.e. b i x, he chooses o become an enrepreneur (i.e. an EL) if: y WL +1 y EL i+1 w + (1 σ) (b i + h )θ +1 R +1 ρ i + (1 σ) (b i x + h )θ +1 R +1, ha is if: ρ i ρ EL +1 = w + (1 σ) xθ +1 R +1. (8) Eq. (8) represens he hreshold of he reurn on invesmen (individual alen) above which he bes endowed agens choose o become enrepreneurs. Given he 6 Le y i+1 = c i+1 + b i+1 be he income of old agen. Then maximizaion of U leads o: U max = (γy i+1 ) γ [ y i+1 ] 1 γ, since: c i+1 = γy i+1 ; b i+1 = y i+1. 6

7 THE MODEL.1 Occupaional choice disribuion of ρ i, ρ EL represens he share of workers ou of self-financed agens. I is quie sraighforward ha his is increasing in he real wage w, in he ineres rae on credi R +1 and in he inpu requiremen x. I is decreasing in he parameer reflecing he ransacion coss of financial inermediaion σ since i decreases he ne real ineres rae on credi. In oher words he higher he real wage, he ineres rae on credi and he inpu requiremen, he more profiable is for self financed agens o become workers and lend heir bequess via he financial secor. From Eqq. (5) and (6) he less endowed agens, i.e. agens such ha b i < x, chooses o become an enrepreneur (i.e. an EB) if: y WL +1 y EB i+1 w+(1 σ) (b i + h )θ +1 R +1 ρ i (x b i )θ +1 R +1 +(1 σ) h θ +1 R +1, ha is if: ρ i ρ EB +1 = w + R +1 θ +1 x σr +1 θ +1 b i = ρ EL + σθ +1 R +1 (x b i ). (9) Finally, if b i = x hen ρ EB +1 = ρ EL +1. Eq. (9) highlighs ha ρ EB +1 ρ EL +1 > 0 and he laer difference is decreasing in b i and increasing boh in σ and in R +1 θ +1. I is worh noing ha hreshold ρ EB +1 is no he share of workers ou of liquidiy consrained agens, since i depends on b i. In oher words ρ EB is only he hreshold of alen in order o become EB. Wihou lack of generaliy assug ha bequess are disribued on he suppor ( ) b,b max according o he densiy funcion f (b ), he share of WL ou of liquidiy consrained agens is equal o x ρ EB f (b )db. Figure 1 provides a graphical represenaion of b he pariion of he all populaion, afer normalizing he suppor of he disribuion of wealh/beques. bi b max b b 1 WL EL x b max b b WL WL EB ˆρ EL 1 ρ i Figure 1: Pariion of he populaion according o individual wealh and alen In order o simplify he noaion le α LC = x f (b )db be he share of he popula- = ( ) 1 α LC b max = f (b x )db be he ion wih (possible) liquidiy consrains and α SF 7 b

8 3 CREDIT AND MONEY MARKETS (possible) self-financed share of he populaion a ime a ime. Therefore he shares of he hree differen ypes of agens in he populaion, β EL +1, β EB +1 and β WL +1, are given by: 7 β+1 EL = ( ) 1 ˆρ EL +1 α SF = ( )( ) 1 ˆρ EL +1 1 α LC ; (10) [ ( ) ( )] β+1 EB = αlc 1 ˆρ EL +1 σθ+1 R +1 x b ; (11) β WL +1 = ˆρ EL +1 + αlc ( σθ +1 R +1 x b ). (1) 3 Credi and money markes To calculae he supply and demand of credi we mus calculae he endowmen of he SF self-financed and liquidiy-consrained agens. Le b = b max b x f (b )db be he average wealh/beques of he (possible) self-financed agens and = x b f (b ) db he average wealh/beques of he (possible) liquidiy-consrained agens. I is sraighforward ha he average wealh/beques of economy is: b = α SF bsf b LC b + α LC blc (13) From Eqq. () he aggregae (and average) supply of credi (he sum of privae and public supply of credi) is given by: L S = L P,S + P h = P (1 σ) [ β+1 EL ( bsf x ) + ˆρ EL b + ( ] β+1 WL ˆρ EL) blc + h, while he demand of credi is equal o: ( ) L D = P β+1 EB x blc. Equilibrium on he credi marke implies ha 8 (1 σ) [ β+1 EL ( bsf x ) + ˆρ EL +1 b + ( β+1 WL ˆρ EL) blc ] ( ) +1 + h = β EB +1 x blc. (14) 7 Since producion akes place when agens are old, we index he each occupaional share of populaion wih + 1, even hough he occupaional choice akes place when young. 8 Noe ha, if σ = 0 and h = 0, equilibrium on he credi marke implies β EL ( ( ) β WL +1 ˆρ EL) blc = β+1 EB x blc I comes ou b x [ 1 β WL] = 0, ha is β WL = 1 b x ( bsf x ) + ˆρ EL b +

9 4 DYNAMICS OF BEQUESTS 3.1 Money marke In order o calculae he equilibrium real ineres rae (θ +1 R +1 ) we mus subsiue Eqq. (8), (10), (11) and (1) ino (14): 9 (θ +1 R +1 ) = (1 σ)x α LC σ (x b ( ) [ α LC x b LC (1 σ) [ x σ α LC ) b α LC b LC ( ) σb LC x b LC x ( ) ] 1 α LC + h ] w + x (1 α LC ). Eq. (15) saes he equilibrium real rae of ineres for (θ +1 R +1 ) 0. If L D < L S (θ +1 R +1 ) 0 hen (θ +1 R +1 ) = 0 will be he equilibrium ineres rae. 10 We will see ha his can happen for a some configuraion of parameers (see Proposiion ). (15) 3.1 Money marke The real quaniy of money is defined by Eq. (14): ( ) h = βeb +1 x blc β EL ( bsf +1 x ) ˆρ EL b ( β+1 WL ˆρ EL) blc 1 σ. (16) We observe ha in equilibrium all variables on he righ are consan and herefore h as well. Assume ha money grow a a consan rae µ, i.e.: H +1 = (1 + µ)h. Dividing by P +1 boh sides we ge h +1 = (1 + µ)θ +1 h, ha is: 4 Dynamics of bequess h +1 h = (1 + µ)θ +1 = 1 + µ 1 + π +1. (17) The dynamics of economy is fully described by he dynamics of he agens bequess. From agen s preferences (1) we have ha: b i+1 = y i+1. 9 See Appendix A for he calculaions. 10 Even if i is possible ha he real ineres rae is negaive (e.g. when he inflaion rae is srongly negaive) we exclude his possibiliy because i appears very unlikely in our long-run analysis. 9

10 5 LONG-RUN EQUILIBRIUM The agen s income differ according o he agen s ype; in paricular: b WL i+1 = [w + (1 σ) (b i + h) (θ +1 R +1 ) ] ; (18) b EL i+1 = [ρ i + (1 σ) (b i x + h) (θ +1 R +1 ) ] ; (19) b EB i+1 = [ρ i (x b i ) (θ +1 R +1 ) + (1 σ) h (θ +1 R +1 ) ]. (0) An analyical characerizaion of he dynamics is difficul because he shape of he accumulaion equaions (18)-(0) depens on he equilibrium real ineres rae (θ +1 R +1 ), and he laer, in urn, depends on he disribuion of bequess, which evolves nonlinearly. 5 Long-run equilibrium In his secion we characerize he properies of he possible long-run equilibria. Firs of all we idenify he possible average bequess for he hree ypes of agens. Proposiion 1 Le (θr) LRE be he long-run real ineres rae. Assume ha economy does no show a posiive long-run growh, ha is: 11 (1 σ) (θr) LRE < 1. (1) Then he average long-run equilibrium levels of bequess for he hree differen ypes of agens are: bwl = bel = beb = [w + (1 σ)h(θr) LRE] ; () 1 (1 σ) (θr) LRE [ ρ EL + (1 σ) (h x) (θr) LRE] ; (3) 1 (1 σ) (θr) LRE [ ρ EB + (1 σ)h(θr) LRE x (θr) LRE] ; (4) 1 (θr) LRE where ρ EL and ρ EB are respecively he average alen of EL and EB agens, while he average long-run equilibrium beques b LRE is given by: blre = β WL bwl + β EL bel + β EB beb (5) 11 This condiion ensures ha WL and EL agens canno increase forever heir wealh. The dynamics of agens EB is no relevan because, even if heir beques could increase forever, hen once over x hey become EL agens. 10

11 5 LONG-RUN EQUILIBRIUM Proof. I direcly follows from Eqq. (18)-(0) and from Definiions (10)-(1). However, i is possible ha for a some configuraion of parameers no all hree ypes of agens are presen. In paricular hree differen ypes of long-run equilibria are possible: i) all agens are lenders, i.e. here is no EB agens (i.e. β EB = 0); ii) all agens are liquidiy consrained, i.e. here is no EL agens (i.e. β EL = 0) and iii) all hree ypes of agens are presen in he economy (i.e. β WL,β EL,β EB > 0). To have an inuiion of hese hree cases in Figure 5 we repor he accumulaion funcions of he hree ypes of agens given by Eqq. (18)-(0) for a given disribuion of wealh, i.e. aking as consan he ne ineres rae θr. b i+1 b EL b ( ρ = 1) EL ( ρ = ρ ) WL EL = b ˆ b i+1 b b EL b ( ρ = 1) ( ρ = ρ ) EL ( ρ = ρ ) WL EL = b ˆ b i+1 EB b ( ρ = 1) b EB ( ρ = 1) EL ( ρ = 1) ( ρ = ρ ~ ) ( ρ = ρ ) EL ( ρ = ρˆ ) x b x i b x i Case 1 Case Case 3 0 ( γ)w LRE [ ] 1 ( 1 γ) 1+ ( 1 σ) h( θr) Case 1 Case 3 Case x bi Figure : The hree differen cases In he firs case agens have a beques such ha all can inves (noe ha θr = 0); in he second case no agen have a sufficienly high level of beques o be able o inves wihou ask for credi; finally in he hird case some agens are liquidiy consrained, bu oher have a sufficienly high level of beques o be able boh o inves and o supply credi. More deails on Figure 5 will be given below, when we analysis each single case. Proposiion saes more formally he condiions for Case 1. Proposiion Le Case 1 define as he long-run equilibrium where here is no EB agens. The necessary and sufficien condiion because Case 1 happens is given by: 0 < x < w. (6) 11

12 5 LONG-RUN EQUILIBRIUM In he long-run equilibrium i holds ha: (θr) LRE = 0; ( β WL,β EB,β EL) = (w, 0, 1 w) ; ( bwl, b EL,b LRE) = ( w, Proof. See Appendix B. ( 1 + w ), (1 + ) w ). The picure on he lef in Figure 5 represens Case 1. There is a mass of w workers whose beques is w and a mass of 1 w EB agens uniformly disribued on he range [w, 1]. The opposie of Case 1 is an economy where all agens are liquidiy consrained. Proposiion 3 saes he condiion for Case. Proposiion 3 Le Case be he long-run equilibrium where here is no EL agens. The necessary and sufficien condiion because Case happens is given by: [ x 1 + (1 σ)h(θr) LRE]. Appendix D repors he values of (θr) LRE, b WL, b EB, b LRE, β WL and β EB for his case. Proof. See Appendix C. The inuiion of Proposiion 3 can be found in he cenral picure of Figure 5; in fac, he necessary and sufficien condiion for Case consiss in excluding he possibiliy ha he mos alened agen has a long-run beques greaer han x (he highes accumulaion funcion in he picure). An analyical soluion for (θr) LRE is no easily available, excep for he rivial case σ = 0; however i is sraighforward o observe ha: Remark 4 A necessary condiion for Case is ha x. I is worh noing ha also for x > is possible ha Case does no happen (as we will see below we are in Case 3). This is because for x = θr is equal o 0 and, even if θr is increasing in x, ex-ane i is no clear which beween x and θr increases quicker when x is around. However, for x very large i is plausible ha condiion for Case holds. For w < x < we are in Case 3. Proposiion 5 saes he condiions for Case 3 more precisely. 1

13 5 LONG-RUN EQUILIBRIUM 5.1 Financial secor Proposiion 5 Le Case 3 define as he long-run equilibrium where here are all hree ypes of agens. The necessary and sufficien condiion because Case 3 happens is given by: [ w < x < 1 + (1 σ) h (θr) LRE]. Appendix D repors he values of (θr) LRE, b WL, b EB, b EL, b LRE, β WL, β EB and β EL for his case. Proof. See Appendix D. A direc implicaion of Proposiion 5 is ha: Remark 6 A sufficien condiion for Case 3 is ha w x. The inspecion of he picure on[ he righ of Figure[ 5 provides an inuiion of Proposiion 5. We observe ha for x w, 1 + (1 σ)h(θr) LRE]] all hree ypes of agens are presen in he economy because for some paricularly alened agen (i.e. wih ρ > ρ) he long-run level of beques is over x. A he same ime here are some EB agens, whose alen belongs o he range [ ρ, ρ ]. Finally, WL agens have he lowes alen, i.e. in he range [ 0,ρ ]. Proposiions, 3 and 5 characerize he saic properies of he long-run equilibrium, while i remains difficul o analyse he dynamics of wealh/beques disribuion. In paricular, we conjecure ha disribuion dynamic is ergodic (accumulaion equaions (18)-(19) are coninuous and piecewise linear). Moreover, we expeced ha an increase in he efficiency of financial secor and/or an increase in he real sock of money can increase boh he average long-run level of wealh/beques and decrease he inequaliy in he disribuion. In he nex Secion we use numerical simulaions o es hese conjecures. 5.1 Efficiency of he financial secor In his Secion we analyse he effec of a change in he efficiency of financial secor, i.e. a decrease in σ. On one hand i reduces he gap beween he rae of reurn on deb and he rae of reurn on credi for any given ineres rae. This can produce as a sor of subsiuion effec ha decreases he relaive share of enrepreneurs in he populaion (he gain o be a lender is lower) (subsiuion effec). On he oher hand, he higher efficiency of he financial secor increases he amoun of loanable funds, causing a decrease in he ineres rae and, herefore, an increase in he enrepreneurial share of he populaion (scale effec). The ne effec on he equilibrium ineres rae, on he composiion of he 13

14 6 NUMERICAL SIMULATIONS populaion and on he aggregae wealh depend on which effec prevails. This in urn depends crucially on he average (and aggregae) wealh and on is disribuion. In paricular, he higher he share of populaion wih liquidiy consrains and he lower he average (and he aggregae) wealh, he more he scale effec prevails. Symmerically, he lower he share of populaion wih liquidiy consrains and he higher is he average (and he aggregae) wealh, he more he subsiuion effec prevails. In Case 3 scale effec prevails. To have an inuiion consider Figure 5.1, where a change in σ is accompanied by a change in α LC from α LC 0 o α LC 1 (his reflec he change in he composiion of populaion). b i+1 ( ρ = 1) ( ρ =1) x h A A B C C D G D G x ( ρ = 0) ( ρ = 0) ~ ( ρ ) 0 ~ ( ρ ) 1 ( ρ 0 ) EL ( ˆρ 0 ) ( ρ 1 ) EL ( ˆρ 1 ) ( ρ ( σ )) 0 b i Figure 3: The effec of an increase of he efficiency of financial secor The bold lines are paramerized o he new σ. I is possible o show ha AA = GG = ( ) α1 LC α0 LC and herefore AG = A G. Quie inuiively i comes ou ha α LC 0 = BG AG and αlc 1 = BG A G = BG > AG αlc 0. Following he same raionale ρ EL 0 = DG AG 0. This shows ha he and ρ EL 1 = D G = D G < A G AG ρel 0 ; ρ 1 = C G = C G < CG < A G AG AG ρ scale effec should prevail. We es his predicion by numerical simulaions below. 6 Numerical simulaions In all numerical simulaions, if no diversely specified, we assume he following configuraion of parameers: σ = 0.1, γ = 0.8, w = 0., h = 0.05, θ = 1, i.e. µ = 0. Parameers h and θ should be endogenous deered ou of equilibrium, bu o speed up he analysis from he begin we ake hem a heir long-run equilibrium levels. Proposiions -5 sugges o focus on he value of x. In paricular we change he value of he laer o reproduce all hree Cases. In every simulaion we consider

15 6 NUMERICAL SIMULATIONS agens, whose iniial bequess are drawn by a log-normal disribuion wih mean equal o and sandard deviaion equal o 1 on he log scale. 1 For every simulaion we consider 50 periods, which appears a sufficien number o reach he long-run equilibrium. Finally, for each case we perform 0 simulaions wih a differen iniial disribuion in order o check ha disribuion dynamics is ergodic. Table 1 repors some descripive saisics of hree configuraion of parameers corresponding o he hree cases (b and sd b are respecively he average of he means and of he sandard deviaions of he 0 disribuions resuling from he 0 simulaions). x Case b (0) sd b (0) b LRE sd LRE b (θr) LRE ( β WL,β EB,β EL) (0.0, 0, 0.80) (0.70, 0.9, 0) (0.44, 0.54, 0.013) Table 1: Numerical simulaions of he hree cases We find a confirmaion of he resul of our Proposiions -5. I is o remark ha he maximum long-run wealh is relaed o Case 3, and no o Case 1 where x is he lowes. This resul is explained in he zero level of ineres rae, which does no allow he riches agens o ge an adequae reurn of heir wealh. However a oo high ineres rae, as in Case, makes for many alened bu poor agens no convenien o become an enrepreneur and herefore hur he average wealh. This sugges ha he relaionship beween average wealh and real ineres rae has an invered-u shape. Taking as saring poin we ake Case 3 and we analyse how changes in s affecs he resuls. In Table we repor he resul of simulaions (for compuaional convenience we limi o simulaions for each configuraion of parameers). 1 Simulaions are made by R. Codes are available on wwwdse.ec.unipi.i/fiaschi/workingpapers.hm. 15

16 7 CONCLUSIONS x σ Case b (0) sd b (0) b LRE sd LRE b (θr) LRE ( β WL,β EB,β EL) (0.4, 0.57, 0.01) (0.4, 0.56, 0.01) (0.44, 0.54, 0.01) (0.48, 0.51, 0.01) (0.5, 0.47, 0.01) (0.57, 0.4, 0.01) (0.64, 0.36, 0.001) Table : Numerical simulaions for differen levels of efficiency of he financial secor These resuls broadly suppor our inuiions: an increase in σ (a decrease in he efficiency of he financial secor) increases he equilibrium real ineres rae θr and decrease he lon-run average wealh (a change in σ from 0 o 0.5 decreases b of almos 10%). This is caused by a change in he occupaional shares, where workers share is increasing while enrepreneurs share is decreasing. 7 Conclusions We presened an OLG model wih heerogeneous alruisic agens, money and credi in order o invesigae he relaion beween financial secor and he firs and higher momens of he disribuion of wealh. The occupaional choice made by agens, ha differ from one anoher in alens and wealh, make he real ineres rae and he disribuion of wealh (is mean, variance, suppor) depending on he efficiency of he financial secor and on he money sock and on he rae of money growh. For any given real ineres rae, an increase in he efficiency of he financial secor makes lending aciviy more aracive. In oher words, for any given ineres rae he enrepreneurial share of he populaion decreases. On he oher hand, for any given composiion of populaion by occupaion, he available pool of loanable funds increase. This lower he equilibrium real ineres rae ha changes he composiion of he populaion increasing he enrepreneurial share and increasing he liquidiy consrained share of populaion (since EL lower heir accumulaion). The ne effec on he composiion of he populaion depends on which effec prevails. The higher he share of populaion wih liquidiy consrains and he lower he average (and he aggregae) wealh, he more he scale effec prevails. Money is no neural bu affecs he real secor via he occupaional choice and he 16

17 REFERENCES REFERENCES consequenly real ineres rae ha clears he credi marke. The higher he growh rae of money, he higher (lower) he inflaion (real ineres) rae. Equivalenly he higher he money sock he lower, for any given inflaion rae, he equilibrium noal ineres rae. In urn, his implies a lower share of he populaion wih liquidiy consrains and herefore an increase in he enrepreneurial share of he populaion and in he average wealh. Moneary policy is amplified by financial markes efficiency since money is supplied o he agens via he financial secor. References Gerler, M. (1988) Financial Srucure and Aggregae Economic Aciviy: Overview. Journal of Money, Credi and Banking, 0, an Levine, R. (1997). Financial Developmen and Economic Growh: Agenda. Journal of Economic Lieraure, 35, Views and Longarei, R., D. Delli Gai, (005). Moneary policy and he Disribuion of Wealh in OLG Economy wih Heerogeneous Agens, Money and Bequess, in The Complex Dynamics of Economic Ineracion (Gallegai, M, Kirman, A., Marsili, M. eds). Springer. Pagano, M. (1993). Financial Markes and Growh: an Overview. European Economic Review, 37,

18 B PROOF OF PROPOSITION Appendix A Calculaions of he equilibrium real ineres rae in Eq. (15) From he equilibrium of credi marke we ge: α LC ) ( σb LC ) x ] + σr ( ρ EL +1θ +1 x b = [ x σ α LC b LC + x (1 α LC ) [ (1 σ) b α LC b LC x ( 1 α LC) ] + h [ ] + x σ α LC b LC + x (1 α LC ) ( ) α LC x b LC + [ ], x σ α LC b LC + x (1 α LC ) which ogeher wih Eq. (8) leads o Eq. (15). B Proof of Proposiion If w > x hen for every iniial disribuion of resource he long-run disribuion will be such ha b > x. In his case he whole populaion in he long-run equilibrium is self financed, i.e. α LC = 0. In he credi marke he demand for credi is zero, bu supply is posiive, so ha he equilibrium real ineres rae (θr) is zero. Given ha (θr) = 0 we have ha β WL = ρ EL = w and β EL = 1 ρ EL = 1 w. From Proposiion 1 i is sraighforward o calculae: bwl = w; ( ) 1 + w bel = ; b LRE = (1 + w ), since ρ EL = (1 + w)/ (EL agens are uniformely disribued in he range [w, 1]). QED 18

19 C PROOF OF PROPOSITION 3 C Proof of Proposiion 3 A necessary and sufficien condiion o have zero EL agens is ha even he mos alened agen has a long-run beques lower han x, i.e: bel (ρ i = 1) x. This leads o he condiion in he proposiion. The syem of equaions is derived by Eqq. (15), (10)-(1), ()-(5) once we se β EL = 0, α LC = 1, b LC = b LRE, b = and β WL is idenified by he level of alen of he marginal individual being indifferen beween becog WL or EB (in Figure 5 indicaed by ρ ), i.e.: [w + (1 σ)h(θr) LRE] = 1 (1 σ) (θr) LRE [β WL + (1 σ) h (θr) LRE x (θr) LRE]. 1 (θr) LRE Finally, given he mass of EB agens equal o 1 β WL, he average alen is given by (EB agens are uniformerly disribued in in range [ β WL, 1 ] ): b WL ρ EB = 1 + βwl. Therefore (θr) LRE, b WL, b EB, b LRE, β WL and β EB are he soluion of he following sysem: (θ +1 R +1 ) LRE = bwl = [( x b LRE) (1 σ)h w ( x σ b LRE)] ( x σ blre )[ (1 σ)x σ ( x b WL)( σ b LRE x )] [w + (1 σ)h(θr) LRE] ; 1 (1 σ) (θr) LRE blre = β WL bwl + (1 b WL ) b EB ; [ 1+β WL + (1 σ) h (θr) LRE x (θr) LRE] beb = ; 1 (θr) LRE β WL = [(θr) LRE] w + [x w] (θr) LRE (1 σ) (x + σh) ; 1 (1 σ) (θr) LRE β EB = 1 β WL ; blre = β WL bwl + β EB beb. QED 19

20 D PROOF OF PROPOSITION 5 D Proof of Proposiion 5 The condiion of Case 3 direcly follows as inermediae case beween Case 1 and Case. Wih he help of Figure 5 we see ha he mass of EL agens β EL is equal o 1 ρ, where ρ is he imum level of alen which allows an agen o be a EL, i.e. from Eq. (10) ρ : [ ρ + (1 σ) (h x) (θr) LRE] = x. 1 (1 σ) (θr) LRE Therefore: β EL = 1 ρ = [ 1 + (1 σ)h(θr) LRE] x. 1 γ From Eq. (10) i is possible o ge α LC : α LC = x [w + (1 σ) (h + x) (θr) LRE] [1 w (1 σ)x(θr) LRE]. Again wih he help of Figure 5 we see ha he mass of EB agens β EB is equal o ρ ρ, where ρ is he imum alen which is convenien o be an (borrower) enrepreneur insead of a worker. In paricular from Eqq. (4) and (): ρ : [ρ + (1 σ)h(θr) LRE x (θr) LRE] = 1 (θr) LRE [w + (1 σ)h(θr) LRE], 1 (1 σ) (θr) LRE ha is: ρ = [w + (1 σ)h(θr) LRE][ 1 (θr) LRE] [1 (1 σ) (θr) LRE] +[x (1 σ)h] (θr)lre. Finally β WL = ρ. Therefore i is easy o calculae he average alen of EB and EL agens, i.e.: ρ EL = 1 + ρ and ρ EB ρ + ρ =. 0

21 D PROOF OF PROPOSITION 5 From Eqq. () and (4) he average wealh/beques of (possible) liquidiy consrained agens is given by: [ ( ) b LC ρ w + (1 σ)h(θr) LRE] = + ρ 1 (1 σ) (θr) LRE [ ( ) ρ ρ ( ρ ) + ρ / + (1 σ)h(θr) LRE x (θr) LRE] +, ρ 1 (θr) LRE while from Eqq. ()-(5) we ge he long-run equilibrium level of wealh/beques: ρ [w + (1 σ)h(θr) LRE] blre = + 1 (1 σ) (θr) LRE (1 ρ) [(1 + ρ)/ + (1 σ) (h x) (θr) LRE] (1 σ) (θr) LRE ( ρ ) ρ [ ( ρ ) + ρ / + (1 σ)h(θr) LRE x (θr) LRE] 1 (θr) LRE. Finally he equilibrium ineres rae is equal o: (θr) LRE = ( (1 σ)x α LC σ x b WL) ( ) σb LC x α (x LC b LC) (1 σ)[ blre α LC b LC x ( 1 α LC) ] + h [ ] w x σ α LC b LC + x (1 α LC ), where: QED b WL = [w + (1 σ)h(θr) LRE]. 1 (1 σ) (θr) LRE 1