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The Paradox of Thrift in an Inegalitarian Neoclassical Economy
Mabrouk, Mohamed
Ecole Supérieure de Statistiques et d’Analyse de l’Information (Tunis), 6 rue des métiers, Charguia 2, Tunis, Tunisia
12 December 2016
Online at https://mpra.ub.uni-muenchen.de/76026/
The Paradox of Thrift in an Inegalitarian Neoclassical Economy
version December 12, 2016 Mohamed Mabrouk1 A bst r act
[Schilcht 1975] and [Bourguignon 1981] st udied t he case of a convex sav- ing funct ion in t he [St iglit z 1969] model. T hey have shown t hat if one of t he two proport ions of t he rich or t he poor is below a cert ain t hreshold, t here is a two-class equilibrium. However, t hey have only proved t he exist ence of t his t hreshold. We give here a syst em of equat ions t o calculat e t his t hreshold which we int erpret as t he maximum proport ion of rich for having a st able two-class con…gurat ion. If t he proport ion of rich exceeds t his t hreshold, t he economy en- t ers a phase of decline alt hough t he golden-rule capit al has not yet been reached.
T his decline is due t o a speci…c art iculat ion between t he rat e of decrease in t he product ivity of capit al and t he rat e of increase in t he depreciat ion of capit al.
T he mechanism of t his decline recalls t he descript ion given in [Keynes 1936], of t he decline which happens when t here is t oo much savings in an inegalit arian cont ext . T his is an example of what is known as t he " paradox of t hrift " . It is remarkable t hat t his paradox t akes place in a neoclassical set t ing t hat does not include key Keynesian element s such as sat urat ion of demand, monet izat ion of savings, short -t erm e¤ect s, expect at ion problems, involunt ary unemployment and rigidit ies. Numerical simulat ions are given t o illust rat e and analyze t he mechanisms involved.
Keywords : Paradox of T hrift , Inequality, Saving, Growt h.
1 I nt r oduct ion
[Schilcht 1975] and [Bourguignon 1981] st udied t he case of a convex relat ion- ship between savings and income in t he [St iglit z 1969] model. T he purpose of t he St iglit z model was t o show t he in‡uence of income and wealt h dist ribu- t ion on economic growt h and on t he convergence of social classes. Alt hough t here is no evidence of t he convex or non-convex nat ure of t he relat ionship be- tween savings and income at t he aggregat e level, at t he individual and st at ic level t he convexity hypot hesis is t he most likely [Dynan-Skinner-Zeldes 2004, Boushey-Hersh 2012]. T herefore, t he present st udy is based on t his hypot he- sis of convexity. [Schilcht 1975] has shown t hat if t his hypot hesis is adopt ed inst ead of t he concavity or linearity of t he relat ionship between individual sav- ings and income, and if t he proport ion of one of t he social classes is less t han a cert ain t hreshold, t he convergence of social classes no longer t akes place and
1Ecole Supérieure de St at ist iques et d’A nalyse de l’I nformat ion (Tunis), 6 rue des mét iers,
t he syst em evolves t owards an inegalit arian equilibrium wit h two social classes.
T herefore, t he spont aneous and generally observable t rend t owards a rich / poor social st ruct ure rat her t han an egalit arian st ruct ure is furt her con…rmat ion of t he convexity hypot hesis. [Bourguignon 1981] shows t hat under t his hypot hesis, inegalit arian equilibria Paret o-dominat e t he egalit arian equilibrium.
In t his paper we give a syst em of equat ions which allows calculat ing t his t hreshold and we int erpret it as t he maximum proport ion of rich t o have a st able two-class con…gurat ion. If t he proport ion of rich people exceeds t his t hreshold, t he economy ent ers a phase of decline.
T he purpose of t his paper is also t o examine in det ail t his decline in t he light of t he descript ion in [Keynes 1936] of t he economic decline caused by an excess of savings in a cont ext of inequality.
We begin in sect ion 2 by present ing t he charact erist ics of t he model and t he main result s obt ained by [Schilcht 1975] and [Bourguignon 1981]. We most ly keep t he not at ions and met hod of [Bourguignon 1981].
In Sect ion 3, we give t he equat ions for calculat ing t he maximal sust ainable proport ion of rich (MSPOR). We calculat e t he MSPOR from numerical values proposed for t he rat e of capit al depreciat ion and for product ion and saving funct ions. T he calculat ion is carried out for di¤erent values of t he propensity t o save. T hese numerical values are also used for t he following sect ions t o illust rat e t he …ndings.
In Sect ion 4, we analyze t he dynamics of t he decline. Given a cert ain re- semblance t o t he descript ion in [K eynes 1936], we refer t o it as t he " Keynesian decline" .
We t hen discuss t he following quest ions:
- How does t he equilibrium of t he economy behave according t o t he dist rib- ut ion of wealt h? (Sect ion 5) T his sect ion shows t hat a t iny proport ion of rich people makes it possible t o push a locked economy int o insu¢ cient savings and egalit arian poverty t owards a level close t o t he golden-rule. On t he ot her hand, it also shows t hat t he increase in t his proport ion is harmful.
- How does equilibrium behave according t o t he propensity t o save, for a given dist ribut ion of wealt h? (Sect ion 6). T his sect ion highlight s t he phe- nomenon of " paradox of t hrift " alt hough t he model does not include st rict ly K eynesian element s, such as sat urat ion of demand, monet izat ion of savings, short -t erm e¤ect s, expect at ion problems, unemployment and rigidit ies.
Sect ion 7 concludes and present s possible direct ions for furt her st udy.
2 N ot at ion and m ain feat ur es of t he m odel
Wemainly use t he assumpt ions, not at ions, met hod and result s of [Bourguignon 1981].
T he economy is represent ed by a per capit a product ion funct ion() where
is t he average capit al per capit a. is increasing, concave and twice di¤eren- t iable. Individual savings are assumed t o depend on income according t o t he
funct ion() where is t he income of t he individual concerned. is convex, increasing, di¤erent iable and checks lim
! 1 0() = 1.
T he capit al undergoes depreciat ion at a rat e per unit of t ime and capit al.
T his depreciat ion plays t he same roleas populat ion growt h in [Bourguignon 1981].
We have t hought t hat it would be more appropriat e, in modern economic con- dit ions, t o speak of depreciat ion of capit al rat her t han demography. But t he int erpret at ion ofas a populat ion growt h rat e remains valid.
We assume t hat t he economy has a unique st able egalit arian equilibrium
0. Mat hemat ically, t his condit ion is equivalent t o saying t hat 0 is t he unique solut ion of t he equat ion(()) = and t hat 0((0))0(0) .
We denot e by ¤ t he per capit a capit al of t he golden-rule de…ned by
0(¤) =
T he society is composed of 2 classes: t he poor, in proport ion 1 and t he rich in proport ion2 = 1 ¡ 1. In a t heoret ical perspect ive, t his assumpt ion is not rest rict ive because t he convexity of saving implies t hat t he equilibrium has at most two classes [Bourguignon 1981]. In t he spirit of t he present st udy, t he concept of " poor class" includes t he middle class. Consequent ly, t he poor class is t he majority. So, we have2 1. We will assume t his for all t he following.
T he capit al st ock per capit a is1 for t he poor and2 for t he rich. T he average per capit a capit al t herefore sat is…es= 11+ 22.
As st at ed by t he neoclassical t heory of dist ribut ion, capit al is paid for ac- cording t o it s marginal product ivity. T he remunerat ion of per capit a capit al is t herefore: 0(). By deduct ion, t he per capit a wage is() ¡ 0(). All in- dividuals receive t he same payment in exchange for t heir cont ribut ions t o work, i.e. () ¡ 0(). For capit al, individuals are remunerat ed according t o t he shares of capit al t hey hold. T hus, an individual of class (wit h = 1 or 2) receives0() in exchange for his cont ribut ion t o capit al. Moreover, he bears t he share of t he depreciat ion of t he capit al he owns: .
T he equat ion of capit al evolut ion for classis t herefore
¢= [() ¡ 0() + 0()] ¡
T he equilibrium is t hus charact erized by t he following 3 equat ions for t he 3 unknowns1 2 and:
[() + (1¡ )0()] ¡ 1 = 0 (1)
[() + (2¡ )0()] ¡ 2 = 0
= 11+ 22
Denot e t he inverse funct ion of . We have
01 000 and
lim0() = 1
Let () be t he curve in t he space ( ) de…ned by t he equat ion :
[() + (¡ )0()] =
or, equivalent ly:
() + (¡ )0() = ()
T he curve () int ersect s t he line ( = ) at t he point s sat isfying() =
(). By assumpt ion, t his equat ion is veri…ed only in0. T herefore t he curve () int ersect s t he line (= ) only in0.
[Bourguignon 1981] shows t hat a necessary condit ion for an equilibrium wit h two social classes is0 ¤. In t his case and for2 [0 ¤[ he est ablishes t hat t he equat ion() + (¡ )0() = () admit s two solut ions1() and2() such t hat 1() 6 and2() . T hese two solut ions are candidat es for per capit a capit al values of t he two social classes at equilibrium.
All det ails and just i…cat ions concerning t he elaborat ion of t he curve (), t he phase plan and t he dynamics of t he syst em can be found in [Bourguignon 1981].
We have reproduced here t he not at ions of [Bourguignon 1981] in order t o facil- it at e t he consult at ion of t his reference at t he same t ime as t he present paper.
We assume for all t he following t hat t he condit ion0 ¤is checked because wit hout it all social classes would necessarily converge. Indeed, in cont inuity wit h t he work of [Schilcht 1975] and [Bourguignon 1981], our concern is t o st udy
t he consequences of a persist ent inequality, a pat t ern t hat seems t o be more realist ic.
If we consider t he product ion paramet ers as given (i.e. t he product ion func- t ion and t he depreciat ion coe¢ cient ) t hen t he posit ion of0 wit h respect t o¤ depends on t he saving behavior, t hat is, on t he funct ion. T he int uit ive eco- nomic int erpret at ion of t he condit ion0 ¤ is t hat t he poor class, if it where alone, would not have t he su¢ cient saving propensity t o reach t he golden-rule
2.
De…ning t he funct ion() by t he equat ion (1¡ ())1()+()2() = , [Bourguignon 1981] shows t hat is posit ive and cont inuous over ]0 ¤[, t hat
(0) = 0 and lim
! ¤() = 0. It follows t hat() admit s a maximum on ]0 ¤[ denot ed , and t hat under t he condit ion 0 ¤, for a st able inegalit arian equilibrium t o exist , we must have inf (1 2) . T his condit ion is also su¢ cient3 and t he inegalit arian equilibrium Paret o-dominat es t he egalit arian equilibrium.
Since we have assumed 2 1, t he necessary and su¢ cient condit ion be- comes2 . Let us observe t hat t he social class which was init ially poor will never be able t o surpass t he rich class. Indeed, assuming t hat t he syst em invert s t he sit uat ions along t he way, t hen, by cont inuity of t he st at e variables1
and2, it would be necessary t hat at a cert ain dat e t hese two variables become equal. Equat ions (1) show t hat t hese two variables would t hen always remain equal from t his dat e on.
We deduce t hat 2 const it ut es t he proport ion of t he rich class at t he begin- ning and at t he end. One can t herefore reformulat e t he necessary and su¢ cient condit ion for t he exist ence of a st able inegalit arian equilibrium by saying t hat t he proport ion of rich must be less t han.
3 T he m ax im al sust ainable pr opor t ion of r ich
If t he proport ion of rich exceeds, [Bourguignon 1981] shows t hat t here can be only an egalit arian equilibrium Paret o-dominat ed by t he inegalit arian equilibria achievable wit h proport ions of rich less t han. As soon as t he proport ion of rich exceeds, we will see t hat t he economy ends up being t rapped in a decline.
For t his reason we refer t o as t he maximal sust ainable proport ion of rich (MSPOR).
In t his sect ion we est ablish a syst em of equat ions for calculat ing. T hen, as example, di¤erent values of corresponding t o di¤erent values of cert ain paramet ers are calculat ed.
2T his int uit ive int erpret at ion of0 ¤ ent ails t hat0 increases wit h t he saving propen- sit y, what is checked in all t he following. T he precise de…nit ion of t he saving propensit y is given in next sect ion.
3I n fact , Bourguignon assert s t hat inf (1 2) 6 is a necessary and su¢ cient condit ion, but if inf (1 2) = , t he st abilit y is lost .
To calculat e, we st art from t he syst em (1) replacing 2 by (). From now on, it is assumed t hat t he syst em (1) is smoot h enough for t he funct ions
1() 2() and() t o be di¤erent iable. T hen we derive t he 3 equat ions wit h respect t oand we writ e t hat = 0.
We have
= (1 ¡ ())1+ ()2
Deriving wit h respect t o, we get : 1 = ()(2¡ 1)
+ (2¡ 1)
+ 1
We writ e t hat = 0 at , which gives:
= 1 ¡ 1
2
¡ 1 (2)
Furt hermore, t he derivat ives of1 and2 wit h respect t o are obt ained by deriving t he …rst two equat ions of t he syst em (1):
1
= " ()(1¡ )
0(1) ¡ 0() (3)
2
= " ()(2¡ )
0(2) ¡ 0() (4) Last ly, t he t hird equat ion of t he syst em (1) provides:
= ¡ 1
2¡ 1
(5) We obt ain equat ions (2) t o (5) for t he unknowns: 12 and . By adding t he …rst two equat ions of t he syst em (1), we obt ain 6 equat ions for t he 6 unknowns1 212 and.
It is not ewort hy t hat t he value ofdepends only on t he product ion funct ion, t he rat e of depreciat ion and t he saving funct ion, and not on t he init ial st at e of t he economy (i.e. init ial capit al and wealt h dist ribut ion).
Since t here is no explicit formula for , we have t hought useful t o t ake numerical values for t hese 3 dat a (product ion funct ion, depreciat ion rat e and saving funct ion) t o illust rat e our point and get an idea of t he order of magnit ude of for t hese numerical values. It is not argued t hat t he following calculat ions express t he act ual sit uat ion of a part icular count ry4.
T he product ion funct ion is chosen so t hat t he marginal product ivity of cap- it al can decrease rapidly. T he choice is a Cobb-Douglas wit h a share of t he capit al income equal t o 0.3. T he paramet ers of t he product ion funct ion have been adjust ed so t hat t he capit al coe¢ cient is 2.5 for an average per capit a
4T he model is st ill at t he rudiment ary st age t o lend it self t o empirical work.
income normalized t o 1. Consequent ly, t he product ion funct ion per capit a is
() = 3403.
An analyt ic form has been adjust ed for t he individual saving funct ion t o ensure t hat it is increasing, convex and t hat t he limit of t he marginal propensity t o save equal t o 1:
() = + 1
2(1 + )(¡ ) + 1 ¡ 1 + s
0+
· 1
2(1 + )(¡ )
¸2
T his form checks t he request ed condit ions. T he coe¢ cient s and0are adjust ed t o have t he following values for individual savings rat es at di¤erent levels of income:
income 0.1 1 1.5 2
savings rat e 7% 15% 20% 30%
By minimizing t he sum of t he absolut e values of t he deviat ions, t he adjust ed values for and0are:
= 17105249
= 00255809
= 00677230
0 = 01889504
T he t erm " social propensity t o save" is used hereaft er t o indicat e t he general st at e of mind of society about t he willingness t o save. If funct ion represent s t he saving behavior, t he change in t he level of t he social propensity t o save can be obt ained by t he form:
() = 1
()
T he variat ion of t he coe¢ cient t hus represent s t he variat ion of t he overall willingness t o save of society (see t he following graph). If increases, t he willingness t o save increases. is referred t o as t he " social propensity t o save" . It is obvious, however, t hat t he variat ion of t he coe¢ cient can not in it self represent all t he possibilit ies of modifying t he pro…le of t he willingness t o save.
For example, one can t hink of an increase in t he willingness t o save among t he poor at t he same t ime as a decrease among t he rich. Such a change is not capt ured by t he paramet erand is not considered in t he present st udy.
If 1 t he propensity t o save increases for all incomes. It decreases if
1:
income 0.1 1 1.5 2
savings rat e wit h= 12 7.8% 16.7% 25.3% 37.1%
savings rat e wit h= 08 5.8% 13.6% 16.7% 21.8%
We obt ain t he following curves t hat give t he individual savings rat e as a funct ion of income for = 08 = 1 and = 12:
S( y ) S( 1 . 2 y ) / 1 . 2
S( 0 . 8 y ) / 0 . 8
Last ly, t he annual capit al depreciat ion rat e is set at 3.7%.
Wit h t he various paramet ers speci…ed above, t he following result s for as a funct ion ofare obt ained by comput er:
1 1.2 1.1 0.9 0.8
5.44% 1.33% 4.45% 5.35% 4.85%
We see t hat t he MSPOR decreases quit e sharply if t he social propensity t o save increases from t he reference sit uat ion= 1.
For each value of and wit h a proport ion of rich equal t o, values of per capit a and per class capit al and out put at inegalit arian equilibrium are given as well as per capit a capit al and out put at egalit arian equilibrium:
1 1.2 1.1 0.9 0.8
average per capit a capit al 8.89 11.75 9.89 8.29 7.86
average per capit a income 1.44 1.57 1.49 1.41 1.39
per capit a capit al of t he poor 6.51 10.73 7.98 5.62 5.01 per capit a capit al of t he rich 50.2 87.5 50.86 55.53 63.72 per capit a income of the poor 1.33 1.53 1.41 1.28 1.24 per capit a income of the rich 3.46 4.61 3.35 3.83 4.36 per capit a capit al at egalit arian equilibrium 6.25 10.66 7.77 5.35 4.75 per capit a income at egalitarian equilibrium 1.30 1.53 1.39 1.24 1.2 per capit a capit al at t he golden-rule 13.18 13.18 13.18 13.18 13.18
We see t hat t he best sit uat ion for bot h t he poor and t he rich is t he sit uat ion
= 12, where t he social propensity t o save is high and t he proport ion of wealt hy low. T he most damaging sit uat ion for t he poor is t he sit uat ion= 08 where t he social propensity t o save is low and t he proport ion of rich is quit e high.
We now give t he savings rat es()at equilibrium by social class and for society as a whole, for each value of.
1 1.2 1.1 0.9 0.8
savings rat e of the poor 18.1% 26% 21% 16.3% 15%
savings rat e of the rich 53.7% 70.2% 56.2% 53.6% 54%
aggregat e savins rat e 22.7% 27.7% 24.5% 21.7% 20.9%
We see t hat , apart from t he case = 12, t he aggregat e savings rat es are relat ively close. However, t he social propensit ies t o save, individual savings rat es and equilibrium incomes di¤er signi…cant ly. In fact t he aggregat e savings rat e is a paramet er which, considered alone, does not re‡ect t he saving behavior.
Ot her charact erist ics are import ant such as t he level of average income, t he dist ribut ion of wealt h and income, or t he posit ion in t he accumulat ion process (more or less close t o equilibrium). For example, t he aggregat e savings rat e may increase due t o a higher concent rat ion of income while t he average income falls. T his may explain t he inconclusive result s of t he st udies on t he relat ionship between aggregat e savings rat es and income [Dynan-Skinner-Zeldes 2004]. But it should not be concluded t hat at t he individual level, t he savings rat e does not increase as income increases.
4 T he K eynesian decline
We are int erest ed here in what happens when t he proport ion of rich exceeds
. Aft er a period of growt h, t he economy declines t owards t he egalit arian con…gurat ion which happens t o be Paret o-dominat ed by inegalit arian equilibria, as showed by [Bourguignon 1981]. We t ry t o see t he mechanisms of t his decline t hrough a numerical example.
T he paramet ers of t he sect ion 3 are used again: product ion funct ion, saving funct ion wit h= 1 and depreciat ion rat e. T he following …gure shows t he phase plan if we t ake a proport ion of rich of 3%, less t han t he MSPOR which is 5.44%
for = 1. T he init ial per capit a capit al of t he rich class is given t he values
02 = 5 and 6 t hen 100, and t he init ial per capit a capit al of t he poor class t he value01= 06. T he following t raject ories (in green) are obt ained:
c
1c
20
2
c
1
0
c
0
100
2
c
0
6
2
c
0
5
2
c
%
2 3 a
We observe t hat if 02= 6, t he economy is freed from from t he pat h t o t he poor egalit arian equilibrium and grows t owards t he rich inegalit arian equilib- rium. Whereas if one begins wit h02 = 5, t he income of t he rich class is not su¢ cient t o allow a saving capable t o release t he economy from t he pat h of egalit arian poverty. T his conclusion is not surprising. It is consist ent wit h t he int uit ion t hat capit al weakness can t rap t he economy int o poverty.
It is less immediat e t o admit t hat an excess of capit al can lead t o t rapping t he economy in poverty. Yet , if we t ake a proport ion of rich above t he MSPOR, t his is what we observe. T his is t he case t hat is int erest ing t o analyze.
We t ake2= 6%. T he curves n ¢
1= 0 o
and n ¢
2= 0 o
int ersect only in t he poor egalit arian equilibrium. T he following t raject ory is obt ained for 02 = 50 and01= 08:
2
0
c c
1 0
c
2c
1%
2
6 a
In t his set t ing, t he rich begin wit h a per capit a capit al of 50. T hey t hen climb t o more t han 90 t o …nally plummet t o 625 which is t he capit al per capit a of t he poor egalit arian equilibrium. T he poor also experience a drop at t he end of t he t raject ory from 690 t o 625. But t his decline is less marked and t he overall balance is posit ive for t hem: from 08 t o 625.
To underst and t he reason for t his decline, we are int erest ed in what governs t he capit al dynamics for t he rich, t hat is, t heir savings on t he one hand and t he depreciat ion of t heir capit al on t he ot her.
time axis (years) capit al depreciat ion
of t he rich savings of t he rich
At t he st art , bot h classes t ake advant age of t he exist ence of inequality. In- deed, t he poor bene…t from a good level of product ion made possible by t he capit al of t he rich, whereas t he rich pro…t from a good product ivity of t heir capit al t hanks t o t he labor of t he poor, or in ot her words, t hanks t o a st ill modest macroeconomic capit al per capit a rat io. T he economy is growing con- siderably.
T his st rong growt h has t he e¤ect of an increase in t he capit al st ock and a rapid decline in capit al marginal product ivity. T his decline doubly a¤ect s t he income of t he rich in comparison wit h t he case of an equal dist ribut ion of wealt h.
Indeed, it curbs t he increase in product ion, as is also t he case in an egalit arian society where capit al st ock is growing. But in addit ion t o t his, it diminishes t he income share of t he wealt hy acquired t hrough t he exist ence of inequalit ies.
In t he above graph, t he income and savings of t he rich begin t o decline aft er about 20 years. However, t heir savings remain abundant . T heir capit al t herefore cont inues t o rise and it begins t o fall only aft er about 50 years of t he dat e of t he decline in income. T his discrepancy is t he cause of an excessive accumulat ion which leads t o a sit uat ion where it is no longer possible t o cover t he depreciat ion of capit al by saving. T he decline t hen begins and it is no longer recoverable. In fact , t his dynamic depends on t he comparison between
t he decline in t he product ivity of capit al and t he increase in t he depreciat ion of capit al. It should be not ed t hat at t he macroeconomic level, average per capit a capit al does not reach t he golden-rule st age beyond which capit al product ivity falls below depreciat ion rat e. T hus, inequality makes t he economic growt h st op before reaching t he golden-rule st age. But it will be seen below (sect ion 5) t hat inequality can also make it possible t o approach t he golden-rule by compensat ing t he weakness of t he savings of t he poor class.
T hus t he init ial abundance of wealt h is t he very cause of subsequent decline.
A smaller proport ion of rich in t he beginning could have delayed capit al growt h and marginal product ivity decline so t hat t he economy st abilizes wit hout t um- bling int o poverty, as t he case2= 3% shows.
T hemechanism of t hisdecline remindsone of t hedescript ion in [Keynes 1936], of t he decline t hat occurs when t here is t oo much unevenly dist ribut ed wealt h.
T hat ’s what he calls " t he paradox of poverty in t he midst of plenty, where exces- sive wealt h and saving of t he rich can lead t o a decline in bot h aggregat e wealt h and savings" [Keynes 1936, chapt er 3, sect ion I I]. In t his regard, he assert s t hat :
“ . . . t he richer t he community, t he wider will t end t o be t he gap between it s act ual and it s pot ent ial product ion; and t herefore t he more obvious and out rageous t he defect s of t he economic syst em.
For a poor community will be prone t o consume by far t he great er part of it s out put , so t hat a very modest measure of invest ment will be su¢ cient t o provide full employment ; whereas a wealt hy commu- nity will have t o discover much ampler opport unit ies for invest ment if t he saving propensit ies of it s wealt hier members are t o be compat - ible wit h t he employment of it s poorer members. If in a pot ent ially wealt hy community t he inducement t o invest is weak, t hen, in spit e of it s pot ent ial wealt h, t he working of t he principle of e¤ect ive de- mand will compel it t o reduce it s act ual out put , unt il, in spit e of it s pot ent ial wealt h, it has become so poor t hat it s surplus over it s con- sumpt ion is su¢ cient ly diminished t o correspond t o t he weakness of t he inducement t o invest .”
T he decline in invest ment opport unit ies in t his paragraph of K eynes corre- sponds in t he present model t o declining product ivity as capit al accumulat ion progresses. However, t here is no quest ion of capit al depreciat ion in t his para- graph of K eynes, but of underemployment .
Ot her element s generally present in Keynesian economics, such as demand- driven economy, monet izat ion of savings, short -t erm e¤ect s, expect at ion prob- lems and rigidit ies, are not included in t he present model of neoclassical essence.
It is remarkable t hat , despit e t his, t he decline does occur anyway.
5 T he pr opor t ion of r ich and t he aggr egat e sav- ings r at e
A number of economist s share t he view t hat great er inequality, by shift ing in- come t oward more saving agent s, increases t he aggregat e savings rat e, t hus accelerat ing capit al accumulat ion and growt h. T his idea can be found, for ex- ample, in [Barro 2000].
On t he cont rary, more recent opinions reconnect wit h t he vision expressed in K eynes‘s quot e (sect ion 4) and at t ribut e a less posit ive role t o inequalit ies wit h respect t o t heir impact on t he economy and consequent ly saving [St iglit z 2011, Ost ry-Berg-T sangarides 2014].
It should be not ed t hat what is generally referred t o as " inequality" is meant t o describe a sit uat ion wit h a large income gap between rich and poor. T his con- cept of inequality is not only dependent on t he proport ion of rich. It can evolve even in t he opposit e direct ion t o t he proport ion of rich if one keeps personal in- comes const ant and if one measures inequality by t he Gini index. However, t his sect ion only examines t he relat ionship between t he proport ion of rich and t he ag- gregat e savings rat e, what is nevert heless a t opical issue as t he number of billion- aires has doubled since 2008 …nancial crisis [Oxfam report “ Even It Up” 2014].
Wit hin t he present framework, we show t hat if we st art from an egalit arian sit uat ion and int roduce a t iny proport ion of rich people, t he aggregat e savings rat e at equilibrium improves signi…cant ly. But if we st art from a sit uat ion where t here are already some rich people, t he addit ion of new rich people det eriorat es t he income and t he aggregat e savings rat e at equilibrium.
[Bourguignon 1981] shows t hat , for a given proport ion of rich 2 sat isfying 0 2 , t he possible equilibria are pairs (1 2) each consist ing of an un- st able equilibrium1and a st able equilibrium2 wit h2 1. We deduce t hat t he equilibrium det ermined by (2) = sup f() = 2g is a st able equilib- rium. As st at ed in [Bourguignon 1981], t he equilibriumParet o-dominat es all t he ot her equilibria where t he proport ion of rich is2.
Let us show t hat t he equilibrium capit al (2) and t he aggregat e savings rat e are decreasing as funct ions of2 as long as20:
At equilibrium, aggregat e savings are necessarily equal t o t he depreciat ion of t he t ot al capit al:
=
T he aggregat e savings rat e as a funct ion of is t herefore() =
(). It is easily checked t hat() is an increasing funct ion ofbecause is concave and posit ive. We now prove t hat (2) is a decreasing funct ion of 2, which will est ablish t he decrease of t he aggregat e savings rat e as a funct ion of2.
Suppose not . T here would be two real numbers in ]01[ such t hat , which would check () () or () = (). Suppose() (). De…ne t he funct ion () = () ¡ on t he int erval £
() ¤¤
. T he funct ion () is
assumed t o be cont inuous on [0 ¤] (by set t ing(¤) = 0 - for t he de…nit ion and propert ies of A, see sect ions 2 and 3).
We have(()) = (()) ¡ = ¡ 0 and(¤) = (¤) ¡ = ¡ 0.
T he funct ion being cont inuous, t here would exist in £
() ¤¤
such t hat
() = 0. We would have() ¡ = 0, wit h> () (). T his cont radict s t he de…nit ion of() = sup f() = g. We t hus have() > (). In fact , we have() (). Indeed, since is cont inuous, we have(()) = . If we suppose() = (), t hen (()) = (()), which implies = and cont radict s . QED
We now show t hat t he limit of (2) when2! 0 is¤:
Since t he funct ion() is decreasing, it has a limit when2! 0. We show t hat t his limit is¤. Consider an increasing sequence which t ends t o¤from t he left . Denot e = (). T he funct ion being posit ive on ]0 ¤[ and cont inuous on [0 ¤], t he sequence is posit ive and t ends t o 0. Moreover
() > . So() ! ¤, what shows t hat t he limit of when2 t end t o 0 from t he right is¤. QED
T herefore, when t he proport ion of rich decreases, t he average per capit a capit al t ends t o t he golden-rule level, where t he average net income is maximum.
T he limit of t he aggregat e savings rat e is t he golden-rule’s savings rat e(¤) =
¤
(¤). In t he Cobb-Douglas case, t his rat e is equal t o t he share of capit al, which is 03 in our numerical simulat ion.
It is remarkable t hat t his result does not depend on t he saving funct ion, provided it is increasing and convex. It should be not ed t hat in our inegali- t arian economy, t his rat e does not correspond t o t he individual savings rat es.
T he poor save less and t he rich save much more. But it happens t hat capit al is dist ribut ed mechanically during t he growt h process so t hat t he equilibrium approaches spont aneously t he golden-rule.
However, while it is t rue t hat t he decline in t he proport ion of rich increases t he aggregat e savings rat e and brings t he economy closer t o t he golden-rule, t he t ot al suppression of t he rich reduces t his rat e and drops t he economy in a sit uat ion Paret o-dominat ed by all t he inegalit arian sit uat ions.
In t he case = 1, t he following graphs represent t he aggregat e savings rat e, t he rat ios of income and capit al between rich and poor as funct ions of t he proport ion of rich2:
a2
aggregat e savings rat e at equilibrium as a funct ion of a2
a2
rat io of income between rich and poor at equilibrium as a funct ion of a2
a2
rat io of capit al between rich and poor at equilibrium as a funct ion of a2
T he relat ionship between t he aggregat e savings rat e at equilibrium and t he proport ion of rich is t hus cont rary t o t he immediat e impression t hat wealt h increases savings. T his …nding support s t he idea, suggest ed by K eynes’s quot e (sect ion 4), t hat excessive wealt h creat es poverty. However, it does not advocat e egalit arianism since it also shows t hat a su¢ cient ly small proport ion of rich makes it possible t o approach t he level of savings of t he golden-rule and t o rescue t he economy from egalit arian poverty.
6 T he par adox of t hr ift
What has been called " Keynesian decline" presupposes t hat above a cert ain point , saving plays a count erproduct ive role. T his phenomenon is known as t he
" paradox of t hrift " . According t o [K eynes 1936, Chapt er 23, Sect ion VI I], t he exist ence of t his paradox has been t he subject of cont roversy between econo- mist s. Indeed, it is not easy t o admit t hat abundance can creat e scarcity when one is accust omed t o reasoning in t erms of supply-demand balance.
[Keynes 1936, Chapt er 23, Sect ion VI I] present s t he fable of t he bees of Mandeville where he sought t o explain t he count erproduct ive e¤ect of an excess of savings, as well as t he host ile react ions of some English aut hors of t he 18t h cent ury. St ill according t o [Keynes 1936], cont roversy cont inued in t he 19t h cent ury between Ricardo and Malt hus in t he form of a debat e on t he possibility of a sit uat ion of overproduct ion, which amount s t o a debat e on t he paradox of t hrift . Indeed, if t here is overproduct ion, t here is under-consumpt ion t herefore over-saving. Aft er t he First World War, Hayek and Schumpet er st ood up against
st ill seem t o be divided. On t he side of t he paradox, we …nd for example [Krugman 2009] and t he sept ic side we …nd [Barro 2000].
In t his sect ion we examine numerically t he relat ionship between t he social propensity t o save and t he average income at equilibrium, t he proport ion of rich being …xed. T he social propensity t o save varies from 08 t o 12. T he proport ion of rich is2= 3%. We obt ain t he following graph for t he equilibrium net income= () ¡ :
social propensity to save net income at equilibrium
income increases with the propensity to save
income decreases with the propensity to save
inequality equality
We observe t hat t here is an opt imal value for t he social propensity t o save
¤ = 1064 (wit h a maximum error of 10¡ 3). T he net equilibrium income for
= ¤ is t hen = 11356. T his income is slight ly less t han t he net income of t he golden-rule¤ = 11380. But it is much higher t han t he egalit arian net income for= ¤, which is0(¤) = 10880.
Beyond ¤and before reaching = 1155 net income declines alt hough t he economy remains in t he inegalit arian and rich part . At = 1155, t he economy crashes sharply in t he egalit arian and poor area. It looks like t he Marxist t ransit ion from capit alism t o socialism, but wit hout t he class st ruggle!
To recover once in t he egalit arian st ruct ure, it would require a social propensity t o save of about = 126, what means an increase in aggregat e savings rat e
from 23% t o 30%. Also observe t hat t he decline in t he inegalit arian sit uat ion begins at (¤) = 11652, whereas in t he egalit arian sit uat ion it begins only when t he economy is overaccumulat ed, i.e. when capit al exceeds t he golden- rule level ¤= 13182
To sum up, if t he social propensity t o save is not very high (less t han¤), t he int roduct ion of a proport ion of rich by 3% makes it possible t o signi…cant ly exceed t he egalit arian net income and t o approach t he net income of t he golden- rule. But t his gain may quickly vanish if t he social propensity t o save increases.
7 Conclusion
T his st udy highlight ed one aspect of t he consequences of inequality on t he macroeconomic relat ionship between savings and income in a basic neoclassical model. Inequality is at t he same t ime useful and harmful. It is useful because it makes it possible t o achieve an aggregat e income out of reach if t he savings of t he majority class is insu¢ cient . It is harmful in t he sense t hat it renders t he economic equilibrium t hat it has achieved fragile. Indeed, t he economy risks a great decline if t he size of t he rich class or t he social propensity t o save exceeds cert ain t hresholds. T his decline is due t o a speci…c art iculat ion between t he rat e of decline in t he product ivity of capit al and t he rat e of increase in t he de- preciat ion of capit al. T he dynamics of such a decline reminds one of Keynes’s descript ion of t he consequences of excess savings in a cont ext of inequality. It is not ewort hy t hat t his decline t akes place in a neoclassical model t hat does not include key Keynesian element s such as sat urat ion of demand, monet izat ion of savings, short -t erm e¤ect s, expect at ion problems, involunt ary unemployment and rigidit ies. It is remarkable t hat t he decline does occur anyway.
T he following direct ions should be furt her explored: t aking int o account t axat ion, t echnical progress, imperfect compet it ion and rent seeking behavior.
R efer ences
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