3 Competitive equilibrium

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.

Letµ_tdenote the Lagrange multiplier for the budget constraint. The first-order con- ditions are

U₁(c^t_t) =µ_tq_t, (20) V₁(c^t_t+1) =µ_tq_t+1. (21) Eliminateµ_t, substitute in the resource constraint, and solve forq_t+1,

q_t+1=V₁(w−c)

U₁(c) q_t. (22)

For the moment, suppose Assumption (2) doesn’t hold and the allocation of the endowment is such thatV₁(w²)> U₁(w¹). Then the autarky solutionc=w¹results inlimq_t+1 =∞ast→ ∞. The deterministic economy in autarky is dynamically inefficient, implying that the time-0 Arrow-Debreu price for delivery of a unit of consumption in the infinite future is infinity.³ As observed by Shell, the dynamic inefficiency in the infinite horizon model is a result of the double infinity of agents and time periods. The inefficiency is eliminated by transferring resources from the current young to the current old. The initial old experience a clear welfare gain from such a transfer and the current young are compensated by receiving a transfer in their old age. Since the economy has an infinite time horizon and an infinite number of agents, there is always a future young generation from which such a transfer can be implemented, unlike the finite horizon version of the model. In a finite horizon version of the economy, the autarky solution is Pareto efficient. Young agents will not transfer resources to the current old because the terminal young are always worse off under such a transfer scheme.

Now impose Assumption (2) and recall thatw¯¹=G( ¯w,1). Under this assump- tion,

q_t+1 =V( ¯w²) U( ¯w¹)q_t=q_t,

so that prices are constant and the competitive equilibrium is dynamically efficient in autarky. The discounted present value of the economy’s endowment is infinite.

Return now to the stochastic version of the model. LetQ_a denote the matrix (U)⁻¹Vunder autarky. Suppose thatw²(s_j) = 0for somes_j. Then

[I−(U(w¹)⁻¹V(w²)]⁻¹

will fail to exist and the competitive equilibrium in autarky is dynamically inefficient, even thought the deterministic version of this economy is dynamically efficient. It is not necessary thatw²(s_j) = 0for somes_j, only that the dominant root of the matrix Q_abe greater than unity. Hence the issue of inefficient risk-sharing in a model of incomplete participation is an issue of the distribution of income over agents over a point of time, and is not just a matter of the double infinity in the economy.

3If an economy is dynamically inefficient, then the discounted present value of the endow- ment stream is infinite.

Conditional futures market. Under this formulation, an agent is characterized by the datetin which he is born in addition to the states^t. The timing is as follows:

the aggregate shock is realized and then young agents are born. Hence any trading between young and old agents will be conditional on the historys^t. A young agent can insure against old-age endowment risk but has no opportunity to insure against first-period endowment risk.

The lifetime budget constraint of agent(t, s^t)is 0 = [w¹(s_t)−c^t_t]q_t(s^t) +

st+1

q_t+1(s^t+1)[w²(s_t+1)−c^t_t+1], (23) which holds for each historys^t. Agent(t, s^t)solves

{cmax^tt,c^t_t+1}

⎡

⎣U(c^t_t) +

st+1

π(s_t+1|s_t)V(c^t_t+1)

⎤

⎦ (24)

+µ_t(s^t)

⎡

⎣[w¹(s_t)−c^t_t]q_t(s^t) +

st+1

q_t+1(s^t+1)[w²(s_t+1)−c^t_t+1]

⎤

⎦, (25)

whereµ_t(s^t)is the Lagrange multiplier. The first-order conditions are

U₁(c^t_t) =µ_t(s^t)q_t(s^t), (26) π(s_t+1|s_t)V₁(c^t_t+1) =µ_t(s^t)q_t+1(s^t+1). (27) Solve (26) and (27) forµ_t(s^t)and rewrite

π(s_t+1|s_t)V₁(c^t_t+1)

U₁(c^t_t) =q_t+1(s^t+1)

q_t(s^t) . (28)

A stationary competitive equilibrium is a pair of functionsc : S → ⁺ andc₂ : S×S→ ⁺such thatc^t_t=c(s_t)andc^t_t+1=c₂(s_t+1, s_t)and goods markets clear, and a price function, described below. The goods market-clearing condition is

w(s_t) =c(s_t) +c₂(s_t, s_t−1). (29) Under the assumption of stationarity, the budget constraint (23) can be summed over all historiess^t−1

0 = [w¹(s_t)−c(s_t)]

s^t−1

q_t(s^t) +

st+1

[w²(s_t+1)−c₂(s_t+1, s_t)]

s^t−1

q_t+1(s^t+1), (30) where the first term on the right side is a function ofs_tonly and the second term on the right side is a function of(s_t+1, s_t). Hence the Lagrange multiplierµ_t(s^t)can be expressed as a functionµ(s_t)of the current state only. Define

q(s_t+1, s_t)≡

s^t−1q_t+1(s^t+1)

s^t−1q_t(s^t) .

Now (28) can be expressed as

π(s_j, s_i)V₁(w(s_j)−c(s_j))

U₁(c(s_i)) =q(s_j, s_i), (31) fori, j= 1, . . . , n.

Substitute the market-clearing condition and the expression forqinto the budget constraint and rewrite

U₁(c(s_i))[w¹(s_i)−c(s_i)] =

j

π(s_j |s_i)V₁(w(s_j)−c(s_j))[w¹(s_j)−c(s_j)]. (32) This forms a system ofnequations for each states_j innunknownsc^e(s_j). Letc^e denote a solution; a proof of the existence and uniqueness is in Labadie (1986).

A young agent picks current consumption and state-contingent old-age con- sumption such that the weighted marginal utilities are equal. The system above can be defined in matrix notation as

0 = [U(c^e)− V(c^e)]x (33) wherexis anndimensional vector withi-th elementw¹(s_i)−c^e(s_i). Multiply both sides by(U(c^e))⁻¹and rewrite to obtain

0 = [I−Q(c^e)]x, whereQwas defined earlier.

Observe that onceQ(c^e)andc^eare determined, the matrixxis not unique. Since this is a homogeneous system of equations, this is not surprising. The marginal rate of substitution between young and old agents at a point in time will fluctuate with respect to the state. To show that this solution is Pareto optimal, observe the Pareto weight vectorφsolves

0 = [I−β⁻¹(U)⁻¹(c^e)V^T(c^e)]φ.

3.2 Equal treatment futures market

In this section, the timing of the model is modified: At the beginning of periodt, young agents are born and both young and old agents submit excess supplies and demands to the clearing house. Only then is the realization of the aggregate shock is observed by all agents. Letρ_t(s^t)denote the time0price of a unit of consumption delivered at timetin states^t.

The budget constraint of an agent born at time t is now no longer balanced state-by-state but instead balanced when averaged across states, or

0 =

st

⎡

⎣ρ_t(s_t, s^t−1)[w¹(s_t)−c^t_t] +

st+1

ρ_t+1(s_t+1, s^t)[w²(s_t+1)−c^t_t+1]

⎤

⎦, (34)

for any historys^t−1∈S^t−1. A young agent born at timetsolves

{cmax^tt,c^t_t+1}

⎡

⎣

st∈S

⎛

⎝U(c^t_t) +

st+1

π(s_t+1|s_t)V(c^t_t+1)

⎞

⎠

⎤

⎦π(s_t|s^t−1) (35)

+µ_t(s^t−1)

st

⎡

⎣[w¹(s_t)−c^t_t]ρ_t(s^t) +

st+1

ρ_t+1(s^t+1)[w²(s_t+1)−c^t_t+1]

⎤

⎦.(36)

The first-order conditions are

U₁(c^t_t)π_t(s_t, s^t−1) =µ_t(s^t−1)ρ_t(s^t), (37) π(s_t+1|s_t)π_t(s_t, s^t−1)V₁(c^t_t+1) =µ_t(s^t−1)ρ_t+1(s^t+1). (38) Eliminateµ_t(s^t−1)from the first-order conditions and rearrange

π(s_t+1|s_t)V₁(c^t_t+1)

U₁(c^t_t) = ρ_t+1(s^t+1)

ρ_t(s^t) . (39)

To find the stationary solution in which a young agent buys full insurance against endowment risk when young and old, observe that the budget constraint can be expressed as

0 =

st

[w¹(s_t)−c(s_t)]

s^t−1

ρ_t(s^t)

+

st+1

s^t−1

ρ_t+1(s^t+1)[w²(s_t+1)−c₂(s_t+1, s_t)]

!

(40) so thatµ_t=µ_t(s^t)is constant across statess_t. Solve the first-order conditions for µ_t

µ_t= U₁(c(s_t))π(s_t|s_t−1)

s^t−1ρ_t(s^t)

= V₁(c₂(s_t+1, s_t))π(s_t+1|s_t)π(s_t|s_t−1)

s^t−1ρ_t+1(s^t+1) (41)

so that the weighted marginal utility of consumption is equalized for all states. Define ρ(s_t+1, s_t) =

s^t−1ρ_t+1(s^t+1)

s^t−1ρ_t(s^t) . Then (41) can be expressed as

π(s_j|s_i)V₁(w(s_j)−c(s_j))

U₁(c(s_i)) =ρ(s_j, s_i), (42)

where the market-clearing condition has been incorporated. Hence the intertemporal marginal rate of substitution varies over time for an agent.

For old and young agents in the market at timet, decrease the time subscript in (38) by one unit, substitute in the conditions for a stationary solution, solve for the price and use (37) to obtain

s^t−1

ρ_t(s^t) =U₁(c(s_t))

s^t−1π_t(s^t)

µ_t =

s^t−1π_t(s^t)V₁(w(s_t)−c(s_t))

µ_t−1 (43)

which can be rewritten as

V₁(w(s_t)−c(s_t))

U₁(c(s_t)) = µ_t−1

µ_t =β. (44)

Letc^f(s) =G(w(s), β)denote the solution. The lifetime budget constraint is

0 =

i

ˆ π(s_i)

U₁(c^f(s_i))[w¹(s_i)−c^f(s_i)] (45)

−

j

π(s_j|, s_i)V₁(w(s_j)−c^f(s_j))[w¹(s_j)−c^f(s_j)]

.