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Banque Nationale De Belgique Sa — Technical Presentation (December 2008)
42 slides from Banque Nationale De Belgique Sa's technical presentation, Dec 11, 2008
Slide 1: The distribution of wealth and fiscal policy in economies with finitely lived agents
The distribution of wealth and …scal policy in economies with …nitely lived agents Jess Benhabib Alberto Bisin New York University December 11, 2008 Jess Benhabib Alberto Bisin (New York University) The distribution of wealth and …scal policy in economies December with …nitely 11,lived 2008 agents 1
Slide 2: U.S. Wealth Distribution
U.S. Wealth Distribution Survey of Consumer Finances, 2004 Jess Benhabib Alberto Bisin (New York University) The distribution of wealth and …scal policy in economies December with …nitely 11,lived 2008 agents 2 / 42
Slide 3: Motivations
Motivations We set up a …nite OLG model with stochactic and capital labor income across …nitely-lived generations. We solve it analytically to characterize the tail of wealth distribution which exhibits: Right skewness Heavy upper tail We study the persistence of talent across generations. We explor
Slide 4: Mechanism to produce a Pareto distribution
Mechanism to produce a Pareto distribution Pareto distribution: f (x ) x β x x A standard old mechanism to generate Pareto distribution is to construct a stochastic process with negative drift and a lower re‡ecting barrier, as in Champernowne (1953), or using a birth and death process as in Wold and
Slide 5: History
History After Pareto (1897), Cantelli (1921), Kalecki (1945), Champernowne (1953), Rutherford (1955), Wold and Whittle (1957), among others, explored the economics of wealth dynamics through stochastic processes that converged to skewed distributions. Mincer (1958) pleaded for microfoundations and m
Slide 6: History (cont'd)
History (cont’d) With the advent of simulation techniques, stochastic features like luck in labor endowments, together with optimizing agents, were reintroduced to analyze income and wealth distributions. Calibrated models aimed at reproducing the wealth distribution in the data through consumption-
Slide 7: Jess Benhabib Alberto Bisin (New York University) The distribution of wealth and …scal policy
Jess Benhabib Alberto Bisin (New York University) The distribution of wealth and …scal policy in economies December with …nitely 11,lived 2008 agents 7 / 42
Slide 8: Jess Benhabib Alberto Bisin (New York University) The distribution of wealth and …scal policy
Jess Benhabib Alberto Bisin (New York University) The distribution of wealth and …scal policy in economies December with …nitely 11,lived 2008 agents 8 / 42
Slide 9: Idiosyncratic investment risk
Idiosyncratic investment risk Bertaut and Starr-McCluer (2002) (BSM), using the Survey of Consumer Finances (SCF) in 1998, document that 19.2% of households have direct investment in stocks as a …nancial asset, and 89.9% of households have non…nancial assets most of which with highly nondiversi…ed r
Slide 10: Idiosyncratic investment risk-cont'd
Idiosyncratic investment risk-cont’d The catalogue of non…nancial assets consists of primary residence, investment real estate, business equity, and other non…nancial assets. 66.3% of households choose to own rather than rent their primary residence. 18.6% of households have investments in real esta
Slide 11: Benhabib-Zhu Model
Benhabib-Zhu Model There is a continuum of agents in the economy. Constant death rate p. When the agent dies, one child is born. Investment opportunity riskless asset dQ (t ) = Q (t )rdt risky asset dS (t ) = S (t )αdt + S (t )σdB (t ) where r < α. Joy-of-giving’bequest motive Z (s, t ) denotes the
Slide 12: A portfolio selection problem with uncertain lifespan-Richard (1975)
A portfolio selection problem with uncertain lifespan-Richard (1975) The agent’s utility maximization problem is Z +∞ 1 γ (s, v ) t) C ((1 ζ )Z (s, v ))1 γ max Et e (θ +p )(v [ + pχ ]dv C ,ω,P t 1 γ 1 γ (1) subject to dW (s, t ) = [(r τ )W (s, t ) + (α r )ω (s, t )W (s, t ) C (s, t ) P (s, t )]dt +
Slide 13: Problems
Problems No stochastic labor income. Perpetual Youth (buys stationarity). Agents born with too low inheritance maybe because parents had bad luck, are topped up. Alternatively everyone receives a transfer at birth–which may be reinterpreted as discounted value of non-stochastic lifetime earning stre
Slide 14: This Model
This Model Agents: Z s +T c (s, t )1 σ [(1 b ) w (s, s + T )]1 σ U= e ρt + χe ρT dt t =s 1 σ 1 σ subject to ẇ (s, t ) = r (t ) w (s, t ) + y (t ) c (s, t ) Simpli…cation Idiosyncratic rates of returns and labor income are drawn from a distribution at birth, possibly correlated with those of the par
Slide 15: Human capital and optimal consumption proposition
Let an individual’s age be denoted τ = t s. Let human capital of an agent born at s at time t, h(s, t ), be de…ned as: Z s +T (r ) τ h (s, t ) = y ( τ )e dτ t Proposition The optimal consumption path satis…es c (s, t ) = m (τ ) (w (s, t ) + h (s, t )) , where the propensity to consume out of …nancia
Slide 16: Solution to Agent's Problem
Solution to Agent’s Problem Substituting the optimal consumption path into the budget constraint, we can write the dynamics of individual wealth as a function of age τ, to obtain the following linear di¤erential equation with variable coe¢ cients: ẇ (τ ) = r̃ (τ ) w (τ ) + yq (τ ) The dynamics of i
Slide 17: Wealth of an individual of age τ
Proposition The wealth of an individual of age τ is: e A (r )T + ( A ( r ) B ( b ) 1 ) e A (r ) τ w (τ ) = e A (r )(T τ )+r τ (A(r )B (b ) 1 ) e A (r )T ! (A(r )B (b ) 1) A ( r ) T w (0) + y (Q T (r ) 0 Q (r )) e + A(r )B (b ) 1 where Q T (r ) = fQ (r , τ )gτ =T , Q 0 (r ) = fQ (r , τ )gτ =0 , and Z
Slide 18: Exploiting the solution for w(T) in terms of w(0)
Exploiting the solution for w (T ) in terms of w (0) , we can construct a discrete time map for each dynasty equating post-tax bequests from parents with initial wealth of children. Let wn = w (0, nT ) be the initial wealth of the n’th dynasty. A generic individual of the n’th dynasty faces constant
Slide 19: Stochastic difference equation for initial wealth of dynasties
We can then construct a stochastic di¤erence equation for the initial wealth of dynasties, mapping wn 1 into wn . It is in fact convenient to work with discounted variables: 0 n 0 n 1 zn = e g wn , zn 1 = e g wn 1 We thus obtain the following stochastic di¤erence equation: zn +1 = αn zn + βn where A
Slide 20: Assumptions
Assumptions Let r̄ = fr 1 , ..., r m g denote the state space of (rn )n , and P its transition matrix. Similarly, let ȳ = fy 1 , ..., y l g denote the state space of (yn )n . and α(r̄ ) denote the state space of (αn )n as induced by the map α (rn ) . Assumption r̄ = fr 1 , ..., r m g, ȳ = fy 1 , .
Slide 21: Some technical assumptions to avoid degeneracies
Some technical assumptions to avoid degeneracies Assumption The process (αs , βs )n∞=s is independent of the realization of βs 1 , all s. The assumption would be satis…ed, for instance, if a single Markov process (xn )n corresponding e.g., to say productivity shocks, drove parameters of the distribu
Slide 22: Recall the stochastic difference equation
Recall the stochastic di¤erence equation zn = αn zn 1 + βn where zn is the discounted initial wealth of generation n. The multiplicative component αn can be interpreted as the e¤ective lifetime rate of return on discounted initial wealth (after subtracting consumption spending and incorporating the
Slide 23: Theorem (Saporta 2005)
Let the shorthand ᾱ= fα (r 1 ) , ..., α (r m )g = fα1 , ...αm g denote induced state space of (αn )n and β̄ = β1 , ...βlm the state space of ( βn )n . Let A be the diagonal matrix with diagonal elements αi . Theorem (Saporta (2005),Thm 1).Consider zn = αn 1 zn 1 + βn 1 , z0 > 0 Let (αn , βn )n sati
Slide 24: Remark
Remark The analysis of this section holds more generally, when (αn , βn )n is not restricted to be a …nite Markov chain. In fact for Markov Chains, the power law exponent µ > 1 satis…es ! N1 N 1 λ A P µ 0 = lim E ∏ (α n ) µ = 1. N !∞ n =0 Note also that, when (αn )n is i.i.d., condition ! N1 N 1 lim
Slide 25: Aggregation
Aggregation The wealth at age τ of an agent born with wealth z (0), return r , and income y is z ( τ ) = σ w (r , τ )z (0) + σ y (r )y Let f0 (z ) denote the density of the stationary distribution of initial wealth z. Then a simple change of variable gives the distribution fτ (z ) at age τ, with the
Slide 26: Tail of population wealth distribution proposition
Given the linear map above of initial wealth to wealth at age τ, since the asymptotic power law property with the same power µ for each age is preserved under integration, it is now straightforward to show the following: Proposition Suppose the tail of the limiting distribution of initial wealth z 0
Slide 27: Tail of wealth distribution with reflective process
The Theorem above characterizes the tail of the wealth distribution when the process (αn , βn )n is re‡ective. It follows from the Theorem that, as long as (αn , βn )n is re‡ective, the stochastic properties of labor income risk, ( βn )n , have no e¤ect on the tail stationary distribution of wealth.
Slide 28: Gini coefficient and mean preserving spread
Let G be the Gini coe¢ cient. Then 1 G = 2µ 1 De…nition De…ne a mean preserving spread on the distribution of the stochastic process of (αn )n as any change of the state space ᾱ and/or the transition ! N1 N 1 matrix P which increases the variance of limN !∞ ∏α n while n =0 keeping the mean constant
Slide 29: Proposition: inequality and mean preserving spread
Proposition Wealth inequality, as measured by the Gini coe¢ cient of the tail G , increases with a mean preserving spread on the distribution of the stochastic process of (αn )n . Jess Benhabib Alberto Bisin (New York University) The distribution of wealth and …scal policy in economiesDecember with
Slide 30: Effect of Bequest Motive on Inequality
E¤ect of Bequest Motive on Inequality Wealth inequality depends on the bequest motive, as measured by the preference parameter χ. An agent with a higher preference for bequests will save more and accumulate wealth faster with a higher αn , which in turn will lead to higher wealth inequality. Proposi
Slide 31: Calibration with iid returns
Suppose (αn , βn )n is iid; there is no transmission of talent across generations. Calibrate the stationary return rn with a discrete uniform distribution, taking values 0.01, 0.11 and 0.21, with a mean return r̄ = 0.11. Set σ = 1, g 0 = 0.01, ρ = 0.04, and the estate tax rate b = 0.20. Finally, set
Slide 32: Effects of Fiscal Policies on Inequality
E¤ects of Fiscal Policies on Inequality Fiscal policies in our economy are captured by the parameters b and ζ, representing, respectively, the estate tax and the capital income tax. Proposition Wealth inequality, as measured by the Gini coe¢ cient of the tail G , is decreasing in the estate tax b an
Slide 33: Effects of taxes on the Gini coefficient table
The Table below illustrates instead the e¤ects of taxes on the Gini coe¢ cient G of the tail of the distribution of wealth. We calibrate the economy as before, under perfect social mobility. We also …x χ = 0.2. The table shows G as we vary b and ζ. b nζ 0.15 0.175 0.20 0.10 G = 0.91 G = 0.57 G = 0.3
Slide 34: Taxes dampen the tail of wealth distribution
The results above indicate that taxes have a dampening e¤ect on the tail of wealth distribution. This is the case despite the presence of bequest motives. Becker and Tomes (1979), on the contrary, …nd that tax increases not only have a negligible in‡uence on the stationary wealth distribution, but m
Slide 35: Estate taxation policy experiments
"Finally, as far as the policy experiment of abolishing estate taxation is concerned, we …nd that the steady-state implications of this policy change are to increase output by 0.35 percent and the stock of capital by 0.87 percent, and that its distributional implications are very small." Castaneda,
Slide 36: Fiscal policy and the effect of luck on wealth
In our model …scal policy is used for government consumption and does not have any direct redistributive and wealth equalizing e¤ect. Nonetheless, both capital income and estate taxes dampen the e¤ect of luck acting through the stochastic returns on capital. The e¤ect of a streak of luck acting mult
Slide 37: Social Mobility
Social Mobility De…nition If (αn ,βn ) are independent over generations we have perfect social mobility. De…nition If (αn ,βn ) are positively correlated over generations, we have moderate social mobility. De…nition If (αn ,βn ) are negatively correlated over generations, we have cyclic social mobil
Slide 38: Social Mobility Continued
Social Mobility Continued We consider here an example for the case in which social mobility is moderate to study how social mobility a¤ects the wealth inequality index G . Consider the simple case in which αn is an two-state irreducible Markov Chain on[αl , αh ], with αl < 1 < αh , and ρl 1 ρl P= 1
Slide 39: Social Mobility Continued
Social Mobility Continued Let ρl = ρh = ρ, αh = 1.15 and αl = 0.8, and note that Condition (1 ρh ) ln αl + (1 ρl ) ln αh < 0 is always satis…ed for ρ 2 (0, 1) since ln 1.15 + ln 0.8 < 0. The Figure below illustrates dG d ρ > 0. Jess Benhabib Alberto Bisin (New York University) The distribution of we
Slide 40: Intuition
Intuition For simplicity take βn = β constant (but constant support will do as well). Then, letting P (z ) denote the stationary distribution of (zn )n , Z z β P (z ) = P π (α) d α α where π (αn ) is the density of αn . For large z,in the tail, we can approximate the solution by the solution to Z z
Slide 41: Conclusions
Conclusions The main conclusion of this paper is that capital income risk, that is, idiosyncratic returns on wealth, have a fundamental role in understanding the distribution of wealth. Capital income risk appears crucial in generating the thick tails observed in wealth distributions across a large
Slide 42: Pareto on stochastic processes and skewed wealth
Pareto explicitly noted that an identical stochastic process for wealth across agents will not induce the skewed wealth distribution (Cours d.Economie Politique, 1897, Notes to #962, p. 416). : C’est à dire, le hasard seul ne su¢ t pas pour expliquer la répartition des revenues, il faut, en outre, a
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