Optimal policies with heterogeneous agents, Exercises of Literature

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Optimal policies with heterogeneous agents

Jes´us Fern´andez-Villaverde^1 and Galo Nu˜no^2 October 15, 2021 (^1) University of Pennsylvania

(^2) Banco de Espa˜na

Prelude: Infinite dimensional

control

Gˆateaux differential

• Let J (g ) be a functional and let h be arbitrary in L^2 (Φ). If the limit:

δJ (g ; h) = lim α→ 0

J (g + αh) − J (g ) α exists, it is called the Gˆateaux derivative of J at g in the direction h.

• If the limit exists for each h ∈ L^2 (Φ) , the functional J is said to be Gˆateaux differentiable at g.
• If the limit exists, it can be expressed as δJ (g ; h) = (^) ddα J (g + αh) |α=0.

Fr´echet differential

• Let h be arbitrary in L^2 (Φ). If for fixed g ∈ L^2 (Φ) there exists δJ (g ; h) which is linear and continuous with respect to h such that:

lim ‖h‖L (^2) (Φ)→ 0

|J (g + h) − J (g ) − δJ (g ; h)| ‖h‖L (^2) (Φ)

then, J is said to be Fr´echet differentiable at g and δJ (g ; h) is theFr´echet differential of J at g with increment h.

• If the Fr´echet differential of J exists at g , then the Gˆateaux differential exists at g and they are equal.
• See Luenberger (1969, p. 173).

Constrained optimization

• Let H be a mapping from L^2 (Φ) into R.
• If J has a continuous Fr´echet differential, a necessary condition for J to have a maximum at g under the constraint H(g ) = 0 is that there exists a function λ ∈ L^2 (Φ) such that:

L(g ) = J(g ) + 〈λ, H(g )〉Φ

is stationary in g , i.e., δL (g ; h) = 0.

• See Luenberger (1969, p. 243).
• Example

Application 1: Social optima

with heterogeneous agents

How to do it

• Nu˜no and Moll (2018) analyze optimal control problems with a continuum of heterogeneous agents.
• Example: constrained-efficient equilibrium in the Aiyagari model with stochastic lifetimes.

Related literature

• Constrained-efficient problems in discrete-time models with incomplete markets and idiosyncratic risk.
  • D´avila, Hong, Krusell, and R´ıos-Rull (2012).
• Optimal control problems in continuous time:
  • Lucas and Moll (2014) or Afonso and Lagos (2015).
• Mean field control:
  • Bensoussan, Frehse, and Yam (2013).

Households

• Household’s utility: E 0

[∫ ∞

0

e−(ρ+η)t^ c t^1 −χ 1 − χ dt

]

where η is the death arrival (Poisson).

• Asset dynamics (per capita), assuming insurance sector: dat = (wt zt + (rt +η) at − ct ) dt
• Borrowing limit: at ≥ 0
• Idiosyncratic labor productivity: dzt = θ(ˆz − zt )dt + σdBt , zt ∈ [z ¯ , z¯]

The KF equation

∂g ∂t

∂a (s (a, z, wt , rt , c) g )

−

∂z (θ(ˆz^ −^ z)g^ ) +

∂^2

∂z^2

σ^2 g

−ηg + ηδa 0 ,z 0 ,

where −ηgt (a, z) is the outflow of agents due to death and ηδa 0 ,z 0 = ηδ (a) δ (z − z ¯ ) is the inflow of newborn agents with zero assets and productivity z ¯

Constrained-efficient allocation

• The planner chooses individual consumption c (·) in order to maximize:

J (g (0, ·)) = max c(·)∈C(t,a,z)

0

e−ρt

∫ ∫ (^) ¯z

¯^ z

u (c) gt (a, z) dadzdt

subject to the law of motion of the aggregate density, to the factor prices and to the market clearing condition.

• The planner cannot redistribute among agents (she has to respect individual budget constraints).

The Lagrangian

Lce (g , τ, c, j, λ) =

0

e−ρt

u (ct (a, z)) gt (a, z) dadzdt

0

e−ρt

∫ ∫ (^) z¯

¯^ z

jt (a, z)

[

− ∂g ∂t

• A∗gt (a, z) + ηδa 0 ,z 0

]

dadzdt

0

e−ρt^ λt

[

−kt +

0

∫ (^) ¯z

¯^ z

agt (a, z)dadz

]

dt

First-best allocation

• The planner is able to fully redistribute among agents.
• Individual wealth is now given by:

dat = (wt zt + (rt + η) at − ct + τt ) dt,

where τt are transfers across agents.

• The aggregate amount of transfers is zero: ∫ ∫ (^) z¯

¯^ z

τt (a, z)gt (a, z)dzda = 0

The Lagrangian: First best

Lfb (g , τ, c, j, λ) =

0

e−ρt

u (ct (a, z)) gt (a, z) dadzdt

0

e−ρt

∫ ∫ (^) ¯z

¯^ z

jt (a, z)

[

− ∂g ∂t

• A∗gt (a, z) + ηδa 0 ,z 0

]

dadzdt

0

e−ρt^ λt

[

−kt +

0

∫ (^) z¯

¯^ z

agt (a, z)dadz

]

dt

0

e−ρt^ ϕt

[∫ ∞

0

∫ (^) ¯z

¯^ z

τt (a, z)gt (a, z)dadz

]

dt