Aiyagari Models, Study notes of Law

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Aiyagari Models

Jes´us Fern´andez-Villaverde^1 April 12, 2022 (^1) University of Pennsylvania

A trio of models

• Different GE heterogenous household models with incomplete markets make different assumptions about how to interpret the assets in the household consumption-savings problem, and how they are supplied: 1. Huggett model: private IOUs in zero net supply (Huggett, 1993). 2. Bewley model: money or bonds in positive net supply (Imrohoro˘glu, 1989). 3. Aiyagari model: capital in positive net supply (Aiyagari, 1994).
• That is why the model is sometimes called the Bewley-Huggett-Aiyagari model.
• We will mainly focus on the (canonical) Aiyagari model.
• Later, we will say a few things about the other two models.

Building blocks

• Continuum of households (vs. models with finite number/types of agents).
• One firm renting aggregate capital.
• No aggregate uncertainty.
• Individuals are subject to idiosyncratic shocks to their labor income.
• Incomplete markets.

Households

• Continuum of measure 1 of households.
• Preferences for household i: E 0

X^ ∞

t=

βt^ u(ct )

• Budget constraint: ct + at+1 = wt yt + (1 + rt )at
• We could consider a hand-to-mouth (i.e., autarky) variation: ct = wt yt.
• Initial conditions y 0 , a 0 ≥ 0.
• Borrowing constraint at+1 ≥ 0.

Firm

• Perfectly competitive firm with neoclassical technology: Yt = F (Kt , Lt )
• Depreciation rate: 0 < δ < 1.
• Aggregate resource constraint: Ct + Kt+1 − (1 − δ)Kt = F (Kt , Lt )
• The only net asset in economy is physical capital.
• No state-contingent claims (i.e. incomplete markets).
• Remark: ownership of the firm.

Recursive formulation, I

• (a, y ): household state.
• Φ(a, y ): aggregate state variable.
• A = [0, ∞): set of possible asset holdings.
• B(A): Borel σ-algebra of A.
• Y : set of possible labor endowment realizations.
• P(Y ): power set of Y.
• Z = A × Y and B(Z ) = P(Y ) × B(A).
• M the set of all probability measures on the measurable space (Z , B(Z )).

Recursive competitive equilibrium

A RCE is value function v : Z × M → R, household policy functions a′, c : Z × M → R, firm policy functions K , L : M → R, pricing functions r , w : M → R and law of motion H : M → M s.t.

1. v , a′, c are measurable with respect to B(Z ), v satisfies Bellman equation and a′, c are the policy functions, given r () and w ().
2. K , L satisfy, given r () and w () r (Φ) = FK (K (Φ), L(Φ)) − δ w (Φ) = FL(K (Φ), L(Φ))
3. For all Φ ∈ M, L(Φ) =

R

ydΦ and

K ′(Φ′) = K (H(Φ)) =

Z

a′(a, y ; Φ)dΦ Z c(a, y ; Φ)dΦ +

Z

a′(a, y ; Φ)dΦ = F (K (Φ), L(Φ)) + (1 − δ)K (Φ)

1. Aggregate law of motion H is generated by π and a′. 8

Transition functions

• Define transition function QΦ : Z × B(Z ) → [0, 1] by

QΦ((a, y ), (A, Y)) =

X

y ′∈Y

π(y ′|y ) if a′(a, y ; Φ) ∈ A 0 else

for all (a, y ) ∈ Z and all (A, Y) ∈ B(Z ).

• QΦ((a, y ), (A, Y)) is the probability that an agent with current assets a and current income y ends up with assets a′^ in A tomorrow and income y ′^ in Y tomorrow.
• Hence

Φ′(A, Y) = (H(Φ)) (A, Y)

Z

QΦ((a, y ), (A, Y))Φ(da × dy )

Characterizing the stationary RCE

• Recall that L is exogenously given.
• Thus, from r = Fk (K , L) − δ w = FL(K , L) we can get w as a function of r (with w ′^ (r ) < 0).
• Example: Y = K αL^1 −α with: r = αK α−^1 L^1 −α^ − δ ⇒ K =

r + δ α

 (^) α (^1) − 1 L and w = (1 − α) K αL^1 −α^ = (1 − α) α 1 −αα (r + δ) αα− 1 L

Existence and uniqueness

• By Walras’ law, we can forget about goods market and we only need to check input market clearing.
• Define asset market clearing condition: K = K (r ) =

Z

a′(a, y )dΦ ≡ Ea(r )

• Then: r = Fk (K (r ), L) − δ
• Existence and uniqueness of stationary RCE boils down to one equation in one unknown.
• From assumptions on production function, K (r ) is continuous, strictly decreasing function on r ∈ (−δ, ∞) with

r →−limδ K^ (r^ )^ =^ ∞ r lim→∞ K^ (r^ )^ =^0

A fixed point problem, I

• From now on assume ∃¯a s.t. a′(¯a, yN ) = ¯a and a′(a, y ) ≤ a¯ for all y ∈ Y and all a ∈ [0, a¯]. State space Z = [0, a¯] × Y and optimal policy a′ r (a, y ) defined on Z , indexed by r.
• Asset demand Ea(r ) =

Z

a′ r (a, y )dΦr

• Need Φr that satisfies Φr (A, Y) =

Z

Qr ((a, y ), (A, Y))dΦr

where Qr is the Markov transition function defined by ar as

Qr ((a, y ), (A, Y)) =

X

y ′∈Y

π(y ′|y ) if a′ r (a, y ) ∈ A 0 else

A fixed point problem, II

• Need to establish that operator T (^) r∗ : M → M defined by

(T (^) r∗ (Φ)) (A, Y) =

Z

Qr ((a, y ), (A, Y))dΦ

has a unique fixed point.

Existence of stationary distributions, I

• Assumption 1 requires that Qr is transition function, i.e., Qr (z, .) is probability measure on (Z , B(Z )) for all z ∈ Z and Qr (., Z ) is B(Z )-measurable ∀Z ∈ B(Z ). Use that a′(a, y ) is continuous.
• The assumption that Qr is increasing requires that for any nondecreasing function f : Z → R we have that (Tf ) (z) =

Z

f (z′)Qr (z, dz′)

is also nondecreasing. Note that a′(a, y ) is increasing in (a, y ).

• Monotone mixing condition 3. satisfied? Pick z∗^ = ( 12 (a′(0, yN ) + ¯a) , y 1 ). Start at d with a sequence of bad shocks y 1 and from c with a sequence of good shocks yN.

Existence of stationary distributions, II

• Conclusion of the theorem assures existence of a unique invariant measure Φr which can be found by iterating on the operator T ∗.
• Convergence is in the weak sense: for every continuous and bounded real-valued function f on Z , we have nlim→∞

Z

f (z)dΦn =

Z

f (z)dΦr