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Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Inequality in a monetary dynamicmacroeconomic model
M. R. Grasselli
Mathematics and Statistics, McMaster UniversityJoint with Gael Giraud (AFD, CNRS, ENPC)
CFMAR 2017University of California, Santa Barbara, May 19, 2017
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
The book
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Opening salvo
To put it bluntly, the discipline of economics has yet to getover its childish passion for mathematics and for purely
theoretical and often highly ideological speculation, at theexpense of historical research and collaboration with the other
social sciences.
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Piketty’s methodology
Pikettys model is not a deterministic system from which heattempts to predict all future economic history, but rather asystem of interacting mathematical regularities and patterns,themselves directly measurable from the statisticalanalysis of historical data, intended to give a good match toempirically observed results, and from which we can then makesome predictions about the future by extrapolating the mostrobust trends and incorporating what we know of presenteconomic conditions. (Dan Kervik, Rugged Egalitarianism)
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Key definitions
Yn = (Yn −W ) + W (total income equals capital incomeplus labor income)
rk = (Yn−W )pK (rate of return on capital)
αk = Yn−WYn
(capital share of total income)
βk = pKYn
(capital-to-income ratio)
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Output growth
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Rate of return on capital - Britain
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Capital share
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Capital-to-Income ratio - Britain
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
The argument in a nutshell
First Law of Capitalism:
αk =(Yn −W )
Yn=
(Yn −W )
pK
pK
Yn= rkβk
Second Law of Capitalism:
βk →s
g
Therefore, if rk > g , wealth and income inequality tend toincrease in time.
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Underpants Gnome’s Business Plans
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Closing Fanfare
The inequality r > g implies that wealth accumulated in thepast grows more rapidly than output and wages. This
inequality expresses a fundamental logical contradiction. Theentrepreneur inevitably tends to become a rentier, more andmore dominant over those who own nothing but their labor.Once constituted , capital reproduces itself faster than output
increases. The past devours the future.
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Criticisms of Piketty
Validity of the Second Law of Capitalism
Stability of the relationship rk > g
Cambridge Capital Controversies
Representative Agent
Nevertheless . . .
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Capital-to-Income - World
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Return on capital versus growth
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Income inequality - top 1%
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Income inequality - top 0.1%
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Wealth inequality
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Inheritance
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
SFC table for the dual Keen model
Households Firms Banks Sum
Balance sheetCapital stock +pK pKDeposits +Mh +Mf −(Mh + Mf ) 0Loans −Lh −Lf +(Lh + Lf ) 0
Sum (Net worth) Xh Xf Xb X
Transactions Current CapitalConsumption −pCh +pC −pCb 0Investment +pI −pI 0Accounting memo [GDP] [pY ]Depreciation −pδK +pδK 0Wages +w` −w` 0Interest on loans −rLh −rLf +r(Lh + Lf ) 0Interest on deposits +rMh +rMf −r(Mh + Mf ) 0Dividends +∆b −∆b 0
Financial balances Sh Sf −pI + pδK Sb 0
Flows of fundsChange in capital stock +p(I − δK ) +p(I − δK )
Change in deposits +Mh +Mf −(Mh + Mf ) 0
Change in loans −Lh −Lf +(Lh + Lf ) 0
Column sum Sh Sf Sb +p(I − δK )
Change in net worth Xh = Sh Xf = Sf + pK Xb = Sb X = pK + pK
Table: SFC table for the dual Keen model.
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Dual Keen model - definitions
Let Dh = Lh −Mh and Df = Lf −Mf and assume that∆b = r(Dh + Df ) and Cb = 0.This leads to Sb = 0, so we take Xb = x0 = 0, so thatDh = −Df .Therefore
Dh = pCh − w`+ rDh − r(Dh + Df )
= pY − pI − w`− rDf = −Df .
Denoting ω = W /(pY ), dh = Dh/(pY ), assume thatconsumption is given be C := c(ω − rd)Y for a function cof disposable income (ω − rd).Letting I = Y − C , we have that
K = Y − C − δK =
(1− c(ω − rd)
ν− δ)K
where ν = K/Y is a constant capital-to-output ratio.
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Dual Keen model - Differential Equations
Assume further a wage-price dynamics of the form
w
w= Φ(λ) + γ
(p
p
)i(ω) =
p
p= ηp(mω − 1),
for a constant mark-up factor m ≥ 1.
The model can now be described by the following system
ω = ω[Φ(λ)− α− (1− γ)i(ω)]
λ = λ[1−c(ω−rdh)
ν − (α + β + δ)]
dh = dh
[r − 1−c(ω−rdh)
ν + δ − i(ω)]
+ c(ω − rdh
)− ω.
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Dual Keen model - Equilibria
Analogously to the original Keen model, this modelexhibits a good equilibrium characterized by
ω1 = η + r[1− η − ν(α + β + δ)
α + β + i(ω1)
].
λ1 = Φ−1(α + (1− γ)i(ω1)
).
d1 =1− η − ν(α + β + δ)
α + β + i(ω1),
where η1 := c−1(1− ν(α + β + δ)
).
It also exhibits a bad equilibrium of the form (0, 0,+∞).
Both equilibria can be locally stable for some parametervalues, but not at the same time.
There’s also an equilibrium of the form (ω3, 0, dh3).
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Example 1: convergence to the interior equilibrium(phase space)
Figure: ν = 3, ηp = 0.35, γ = 0.8
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Example 1: convergence to the interior equilibrium(time)
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Example 2: business cycles (phase space)
Figure: ν = 3, ηp = 0.45, γ = 0.96
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Example 2: business cycles (time)
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Example 3: convergence to explosive equilibrium(phase)
Figure: ν = 15, ηp = 0.35, γ = 0.8
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Workers versus investors - motivation
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Workers versus investors - modelling
Consider now two different classes of households, namelyworkers and investors, with wealth given by
Xw = −Dw
Xi = Ef + Eb − Di .
It follows from the budget constraint that
Dw = pCw − w`+ rDw
Di = pCi − rkpK −∆b + rDi .
Finally, assume that consumption is of the formCw = cw (yw )Y and Ci = ci (yi )Y with
∂cw∂yw
(ω − rdw ) >∂ci∂yi
(rkν − rdi ).
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
SFC table for the two-class Keen model
Workers Investors Firms Banks Sum
Balance sheetCapital stock +pK pKDeposits +Mw +Mi +Mf −(Mw + Mi + Mf ) 0Loans −Lw −Li −Lf +(Lw + Li + Lf ) 0Equities +peE −peE 0
Sum (Net worth) Xw Xi Xf Xb X
Transactions Current CapitalConsumption −pCw −pCi +pC −pCb 0Investment +pI −pI 0Accounting memo [GDP] [pY ]Wages +w` −w` 0Depreciation −pδK +pδK 0Interest on loans −rLw −rLi −rLf +r(Lw + Li + Lf ) 0Interest on deposits +rMw +rMi +rMf −r(Mw + Mi + Mf ) 0Dividends +rkpK + ∆b −rkpK −∆b 0
Financial balances Sw Si Sf −pI + pδK Sb 0
Flows of fundsChange in capital stock +p(I − δK ) p(I − δK )
Change in deposits +Mw +Mi +Mf −(Mw + Mi + Mf ) 0
Change in loans −Lw −Li −Li +(Lw + Li + Lf ) 0
Change in equities +pe E −pe E 0
Column sum Sw Si Sf Sb p(I − δK )
Change in net worth Xw = Sw Xi = Si + peE Xf = Sf − peE + pK Xb = Sb X = pK + pK
Table: SFC table for the workers and investors model.
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Return on capital and external financing
We assume the firms retain profits according to a constantretention rate Θ, leading to an endogenous return oncapital given by
rk := rk(ω, dw , di ) =Θ(pY − w`− rDf − pδK )
pK
=Θ
ν(1− ω + r(dw + di )− δν) ,
Savings by the firms are then given by
Sf = (1−Θ)(pY−w`−rDf−pδK ) = pY−w`−rDf−pδK−rkpK
Therefore, the amount to be raised externally by firms is
p(I − δK )− Sf = pI − pY + w`+ rDf + rkpK
=(ω − r(di + dw )− c + rkν
)pY ,
As in the Keen model, we assume that this is raised solelythrough new loans from the banking sector.
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
The main dynamical system
Define
c(·) ≡ c(ω, dw , di ) = cw (ω − rdw ) + ci (rkν − rdi ),
We then get
ω = ω[Φ(λ)− α− (1− γ)i(ω)]
λ = λ[1−c(·)ν − (α + β + δ)
]dw = dw
[r + δ − 1−c(·)
ν − i(ω)]
+ cw (ω − rdw )− ω
di = di
[r + δ − 1−c(·)
ν − i(ω)]
+ ci (rkν − rdi )− rkν
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Equilibria
With considerable more work, it is possible to show thatthe system exhibits a class of good equilibria of the form(ω1, λ1, dw1, d i1) typically (but not always) satisfyingdw1 > 0 and d i1 < 0.
In addition, the system admits a class of bad equilibria tothe form (ω2, λ2, dw2, d i2) = (0, 0,±∞,±∞)
Finally, it also exhibits equilibria of the form(ω3, 0, dw3, d i3), where dw3 and d i3 can be either finite ofinfinite.
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Example 4: convergence to the interior equilibrium(phase space)
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Example 4: convergence to the interior equilibrium(time)
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Example 5: business cycles (phase space)
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Example 5: business cycles (time)
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Example 6: convergence to explosive equilibrium(phase)
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Example 6: convergence to explosive equilibrium(time)
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Long-run inequality
The income shares of nominal output for workers,investors, and firms are defined as
yw =Y nw
pY= ω − rdw
yi =Y ni
pY= rkν − rdi
πr =Πr
pY= (1−Θ)(1− ω − rdf − δν),
from which we obtain that the income share of capital is
yc = yi + πr = 1− ω + rdω − δν = 1− yω − δν.
It is easy to see that the growth rate of real income for allthree sectors coincide at the interior equilibrium and equalα + β.
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Inequality as a hallmark of inefficiency
However, at each of the equilibria(ω2, λ2, dw2, d i2) = (0, 0,±∞,±∞) we observedivergence in income between workers and capitalists.
For example, if dw → +∞ and di → −∞, thenyw → −∞, yi → +∞, πr → −∞, whereas yc → +∞.
Similarly, whenever dw → +∞, we have xw → −∞ andxi → +∞.
At the deflationary equilibrium (ω3, 0, dw3, d i3), theincome shares are rkν − rd i3 and ω3 − rdw3.
But this is an artifact of the fact that prices are fallingfaster than real output Y → λ3N/a = 0.
In other words, nominal income ratios remain artificiallyconstant, but the real income of both populationscollapse, so both types of households end up ruined!
Inequality in amonetarydynamicmacroeco-nomicmodel
M. R. Grasselli
Introduction
Review ofPiketty
Dual Keenmodel
Inequality andspeculation
Conclusions
Concluding remarks
We provided a stock-flow consistent model for debtdynamics of workers and investors.
When the economy converges to an equilibrium with finitedebt ratios, the income ratio between the two classes isconstant.
Increasing income (and wealth) inequality is a signature ofconvergence to the bad equilibrium with infinite debtratios.
In future work we explore the effects of default, variablecapacity utilization, and of migration between classes a laAcemoglu (2014).
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