2 Description of the economy

Each time period there are two types of agents in the economy: young and old.

There is no population growth and all variables are expressed as per capita. The exogenous endowments follow a stationary, first-order Markov chain. Lets_t∈S = {s₁, . . . , s_n}. A young agent has a nonstorable endowmentw¹:S→W = [w,w¯], wherew > 0. Old agents have an endowmentw² : S → W₂ = [0,w¯]. Denote w(s) =w¹(s) +w²(s)as total endowment in states.

Defineπ_i,j=prob(s_t+1 =s_j |s_t=s_i)fori, j= 1, . . . , n. DefineΠas then× nmatrix of transition probabilities with(i, j)-elementπ(s_j|s_i), where summation across a row equals one. Finally, letˆπ(s)denote the unconditional probability of being in states, equal to the sum of a column of the matrixΠ. LetΠˆ denote the vector of unconditional probabilities. Lets^t = (s₁, . . . , s_t)be the history of realizations up to timetand letπ_t(s^t)denote the probability ofs^t, wheres^t∈S^t=S × · · · × S

t

. Denotec^t_ias the periodiconsumption of an agent born in periodt, wherei=t, t+1. The preferences of a young agent born in periodtare

U(c^t_t) +

st+1

π(s_t+1|s_t)V(c^t_t+1), (1) whereU, V are increasing, strictly concave, and twice continuously differentiable.

Let U₁, V₁ denote the first derivatives and assume the Inada conditions hold:

limc→0U1(c) =∞andlimc→∞U1(c) = 0forU=U, V. Assumption 1. Leta >0such thatw > a≥0. Asa→0,

−U₁(w¹(s)−a) +

j

π(s_j |s)V₁(w²(s_j) +a)>0. Denote w¯^j = _n

i=1ˆπ(s_i)w^j(s_i) forj = 1,2. The unconditional means of the endowment processes satisfy

U₁( ¯w¹) =V₁( ¯w²). (2) LetV₁be convex, so thatV₁( ¯w²))<

jπˆ_jV₁(w²(s_j)).

This assumption ensures that young agents wish to save in the stochastic environ- ment. The restriction on the unconditional means of the endowment process ensures that the deterministic competitive equilibrium is Pareto optimal in autarky (this is dis- cussed later). The convexity ofV₁is assumed so that the assumptions on endowment processes in the deterministic and stochastic environments are consistent.

It is useful at this point to define a function used repeatedly in solving the model.

LetK∈ ⁺be finite and given. Forw≤w≤w¯, letcsolve V₁(w−c)

U₁(c) =K.

Under the assumptions onU, V, the left side is strictly increasing inc. Asc → 0, U₁ → ∞whileV₁ → V₁(w) >0so the ratio converges to0. Asc → w,U₁ → U₁(w)whileV₁→ ∞; hencecis the unique solution. The inverse function theorem can be applied to define a functionG:W × ⁺→W, such that

c=G(w, K). (3)

The Pareto optimal solution is discussed next and then the competitive equilibrium is constructed.

2.1 Central planning problem

Letφ_t(s^t)>0for eachs^t∈S^tdenote the Pareto weight associated with a young agent born at timetin states^t, and letφ₀(s)be the Pareto weight associated with the initial old at time1 in states. Since I focus on stationary solutions, assume φ_t(s^t) =β^tφ(s_t), where0< β ≤1. The resource constraint is

w(s) =c^t_t+c^t−1_t . (4) The central planner solves

{c⁰₁max,c^t_t,c^t_t+1}

s1

φ₀(s₁)V(c⁰₁)

+ ∞ t=1

s^t∈S^t

β^tφ(s_t) U(c^t_t) +

st+1

π(s_t+1|s_t)V(c^t_t+1)

!

+λ_t(s^t)[w(s_t)−c^t_t−c^t−1_t ]

, (5)

whereλ_t(s^t)is the Lagrange multiplier for the resource constraint (4) at timet.¹The first-order conditions with respect to{c¹₀, c^t_t, c^t_t+1}are

φ₀(s₁)V₁(c⁰₁) =λ₁(s¹), (6) β^tφ(s_t)U₁(c^t_t) =λ_t(s^t), (7) β^tφ(s_t)π(s_t+1|s_t)V₁(c^t_t+1) =λ_t+1(s^t+1). (8) To find a stationary solution, restrict the decision variables soc : S → ⁺ and c₂ : S ×S → ⁺, such that c^t_t = c(s_t) andc^t_t+1 = c₂(s_t+1, s_t). With these restrictions on the functionsc, c₂, observe that the resource constraint is a function of(s_t, s_t−1) only and not the entire history s^t. Hence define the current-period Lagrange multiplier as

λ(s_t, s_t−1)≡β^−t

s^t−2

λ_t(s^t).

The first-order conditions are modified as

φ₀(s₁)V₁(c⁰₁) =βλ₁(s₁), (9) φ(s_t)U₁(c(s_t)) =

st−1

λ(s_t, s_t−1), (10) φ(s_t)π(s_t+1|s_t)V₁(c₂(s_t+1, s_t)) =βλ(s_t+1, s_t). (11)

1This formulation follows Abel et al. (1989). See the appendix of their paper.

Decrease the time subscript in (11) by one, divide both sides byβ, sum overs_t−1, equate the right side of (11) to the right side of (10), incorporate (4), and rewrite to obtain

βφ(s_t)U₁(c(s_t)) =V₁(w(s_t)−c(s_t))

st−1

φ(s_t−1)π(s_t|s_t−1). (12) This equation corresponds to Equation (3) in Peled (1984) or Equations (4) and (5) in Aiyagari and Peled (1991). Letφ₀satisfy

φ₀(s) =

kφ(s_k)π(s_i|s_k)

β .

Under this assumption, there is a stationary solution.² Rewrite (12) as

V₁(w(s)−c(s))

U₁(c(s)) = βφ(s)

st−1φ(s_t−1)π(s|s_t−1). (13) The solution to the central planning problem is

c(s;φ) =G w(s), βφ(s)

st−1φ(s_t−1)π(s|s_t−1)

!

. (14)

The solution has the property that the consumption of the old is invariant with respect to last period’s aggregate shock.

Given a Pareto vectorφ and the associated solution c(s;φ), define the n× ndiagonal matrixU(c) with(i, i)-th element U₁(w¹(s_i)−c(s_i, φ))along the diagonal and zeroes elsewhere. Next define then×nmatrixV(c)with(i, j)element π(s_j |s_i)V₁(w(s_j)−c(s_j, φ))and denoteV^Tas its transpose. Define0as ann×1 vector of zeroes and letφ be then×1 vector of Pareto weights. The first-order condition is expressed in matrix notation as

0 = [U(c(φ))−β⁻¹V^T(c(φ))]φ, (15) which is a homogeneous system of equations. Given the allocationcand the matrices V,U, the solutionφis not unique. This becomes apparent by dividing each of then equations byφ(s_i)and solving for then−1values

φ(sj) φ(si)

.

2Observe that my formulation is equivalent to that of Aiyagari and Peled under certain conditions. They have an explicit constraint incorporating the utility of the initial old agents, and they allow the initial old to have a different utility function from agents who are old at later dates. The consumption of the initial old generation constrains the consumption of all subsequent generations and, as they point out, leads to a stationary solution that differs from the steady state. Peled ignores the utility of the initial old generation, thereby focusing on the steady state. By settingβ≤1, and hence discounting the utility of future agents relative to the current generation, and restrictingφ0(s), the economy reaches the stationary solution of Aiyargari and Peled and the steady state solution of Peled, depending on the parameterβ.

The functional dependence of the matrices will be suppressed for convenience in the discussion below. Multiply both sides of (15) by(U)⁻¹(the inverse matrix of U) to obtain

0 = [I−β⁻¹(U)⁻¹V^T]φ.

Observe that the matrixM ≡β⁻¹(U)⁻¹V^T has all positive elements. The Perron- Frobenius Theorem (see Strang p. 271) can be applied to determine if[I− M]⁻¹ exists.

Letη_mdenote the dominant root ofM. Ifη_m >1then the inverse fails to be nonnegative, which cannot be a solution since all elements ofφmust be positive. If η_m= 1, then the inverse fails to exist. Ifη_m<1then

(I− M)⁻¹=I+M+M²+· · ·

is a convergent sequence. Hence any arbitrary but feasible set of weights may not result in a dominant root less than unity, implying that there is an additional condition that must be satisfied for the allocationcto be Pareto-optimal.

Aiyagari and Peled find a necessary and sufficient condition for a Pareto optimal allocation to exist. They define an×nmatrix

Q≡(U)⁻¹V (16)

and show that an allocationcis Pareto optimal if and only if the matrixQ, which has all positive elements, has a dominant root that is less than or equal to unity. If the dominant root is less than unity, then(I−Q)⁻¹>0and, by the Perron-Frobenius theorem

(I−Q)⁻¹=I+Q+Q²+Q³+· · ·,

which converges to a fixed matrix. LetW be then-dimensional endowment vector withi-th elementw(s_i). In this case, the expected discounted present value of wealth of the economy

Πˆ^T[I+Q+Q²+· · ·]W

is finite. The elements of the matrixQare the contingent claims prices that support the consumption allocation.

Conditional and equal-treatment Pareto-optimal solutions. The Pareto-optimal allocation derived above results in a marginal rate of substitution across agents at a point in time (Eq. 13) that typically varies across statess_t. A solution with this property is said to beconditionally Pareto optimal(CPO).

In a static setting, a property of full risk-sharing is a constant marginal rate of substitution across states for all agents. Theequal-treatment Pareto optimal solution (ET-PO) exhibits constant marginal rate of substitution across states, or

V₁(w(s)−c(s))

U₁(c(s)) =β, . (17)

In this case, the Pareto weights must satisfy φˆ(s_j) =

si

π(s_j|s_i) ˆφ(s_i),

so each Pareto weightφ(s_t)is proportional to its unconditional probabilityπˆ(s_t). Observe thatβ= 1is the golden rule allocation.