There are overlapping generations and an agent born in periodtlives for two periods.
There is no population growth. The endowment is exogenous, stochastic and non storable. There is an exogenous stochastic processst∈S={s1, . . . , sn}that is a stationary, first-order Markov chain. A young agent has a non storable endowment w1 : S → W = [w,w¯], wherew > 0 andw <¯ ∞. Old agents at timethave an endowmentw2 : S →∈ [0,w¯]. The total endowment in state st isw(st) = w1(st) +w2(st).
The one-step transition probability isπi,j = prob(st+1 = sj | st = si)for i, j= 1, . . . , n. The history of realizations up to timetisst= (s0, s1, . . . , st)and defineπt(st)as the probability ofst. Then×nmatrix of transition probabilities is Π, with(i, j)-elementπ(sj |si)such that summation over a row equals one. Let ˆ
π(s)denote the unconditional probability of being in states, equal to the sum of a
column of the matrixΠ. By assumption, the Markov chain is stationary, so thatˆπis the eigenvector of the transition matrix and the eigenvalue is1.
The preferences of a young agent born in periodtare U(ctt) +
st+1
π(st+1|st)V(ctt+1), (1)
wherectt is the timetconsumption of a young agent born in periodtandctt+1 is the timet+ 1 consumption of an agent born in periodt. The functionsU, V are increasing, strictly concave, twice continuously differentiable and satisfy the Inada conditionslimc→0U(c) =∞andlimc→∞U(c) = 0forU =U, V.
To ensure that a transfer of a unit of consumption from youth to old-age is always welfare improving, the following assumption is made.
Assumption 1. Leta >0such thatw > a≥0. Asa→0,
−U1(w1(s)−a) +
j
π(sj |s)V1(w2(sj) +a)>0.
Denote w¯j = n
i=1ˆπ(si)wj(si) forj = 1,2. The unconditional means of the endowment processes satisfy
U1( ¯w1) =V1( ¯w2). (2) LetV1be convex, so thatV1( ¯w2))<
jπˆjV1(w2(sj)).
The restriction on the marginal utilities ensures that young agents wish to save in the stochastic environment. The restriction on the unconditional means of the endowment process ensures that the deterministic competitive equilibrium is Pareto optimal. The convexity ofV1is assumed so that the assumptions on endowment processes in the deterministic and stochastic environments are consistent.
A feasible solution for the consumption of the young agent is a function of the formc :S →W. Using the resource allocation constraint (and assuming non satiation), consumption of the old isw(s)−c(s). The marginal utility of consumption for a young agent in statesisU1(c(s))and the marginal utility for an old agent is V1(w(s)−c(s)). The intertemporal marginal rate of substitution between statessi andsjfor a young agent born in statesiis
m(sj, si)≡V1(w(sj)−c(sj)) U1(c(si)) .
There are basically two matrices of interest in studying asset pricing and dynamic efficiency: first, the contingent claims pricing matrix which depends on the IMRS for a single agent over time and the transition probabilities, and second, the ma- trix associated with the central planner’s problem, which depends on the transition probabilities and the MRS across agents within a period.
1.1 Contingent claims matrix
Only stationary equilibria will be examined.1The price of a contingent claim to one unit of consumption in statesjwhen the current state issiis
q(si, sj)≡π(sj |si)m(sj, si). (3) Define then×ndiagonal matrixU(c)with(i, i)-th elementU1(c(si))along the diagonal and zeroes elsewhere. Next define then×nmatrixV(c)with(i, j)element V1(w(sj)−c(sj))along the diagonal and zeroes elsewhere, and denoteVT as its transpose. Define0as ann×1vector of zeroes. Define then×nmatrix of contingent claims prices by
Q≡(U)−1ΠV. (4)
The elements of the matrixQare the contingent claims prices that support the con- sumption allocationc(s), w(s)−c(s).
Aiyagari and Peled [1991] find a necessary and sufficient condition for a Pareto optimal allocation to exist. They 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>0. The Perron-Frobenius Theorem (see Strang p. 271) can be applied to determine if [I−Q]−1exists. Letηdenote the dominant root ofQ. Ifη >1then the inverse fails to be nonnegative. Ifη= 1, then the inverse fails to exist. Ifη <1then
(I−Q)−1=I+Q+Q2+· · · (5) is a convergent sequence. Hence any arbitrary but feasible consumption function may not result in a dominant root less than unity.
To see why the invertibility of the matrix is important, letWbe then-dimensional endowment vector withi-th elementw(si). The expected discounted present value of wealth of the economy
ΠˆT[I+Q+Q2+· · ·]W is finite, if the sum converges.
A feasible consumption allocationc(s)can be used to define the MRS across agents within a period
K=V1(w(s)−c(s))
U1(c(s)) . (6)
Given a functionKsuch thatK:S→(0,∞), the inverse function theorem can be used to determine the value ofcsatisfying
K(s) = V1(w(s)−c) U1(c) .
1I assume that the initial old generation has preferences over consumption that take the form V, so that the initial old have the same utility as old agents at later dates. This allows me to focus on stationary solutions that are identical to the steady state. See Aiyagari and Peled [1991], Peled [1984], and Labadie [2004] for further discussion.
Since the right side is strictly increasing inc, define the functionHby
c(s) =H(s, K(s)). (7) Hence solving forc(s)is equivalent to solving forK(s)and conversely.
It will be useful to further decompose the matrixQ. LetKdenote an×nmatrix with elementK(s)on the diagonal and zeroes elsewhere. The matrix Qcan be written as
Q=U−1ΠKU. (8)
The SDF can be expressed as
m(sj, si) =V(w(sj)−c(sj, K(sj))
U(c(si, K(si)) =K(sj)G(si, sj) where
G(si, sj)≡ U(c(sj, K(sj))) U((c(si, K(si))).
This is the IMRS across different generations of young agents, specifically the ratio of the marginal utility of young agents at timet+ 1in statesjto the marginal utility of young agents in periodt, statesi. In a representative agent model, the SDF would beG(si, sj)and if the representative agent were infinitely lived, then utility would generally be discounted over time. In the OG model studied here, the SDF is the product of the MRS times the ratio of marginal utilities of young tomorrow relative to young today. By construction,KandGare negatively correlated. The implications for the SDF are studied below.
1.2 Pareto optimal allocation
Letφbe an×1vector of Pareto weights. Peled [1984] and Labadie [2004] show that the first-order condition for the central planning problem is
Kφ(s)≡ φ(s)
st−1φ(st−1)π(s|st−1). (9) The Pareto-optimal consumption allocation is
cφ=H(s, Kφ(s)).
The solution has the property that the consumption of the current old is invariant with respect to last period’s aggregate shock.
The first-order condition for the Pareto-optimal solution is expressed in matrix notation as
0 = [I−(U)−1VΠT]φ
= [I−(U)−1KUΠT]φ
= [I− KΠT]φ
Observe that the matrixP ≡ KΠThas all positive elements. Once again, the Perron- Frobenius Theorem can be applied to determine if[I− P]−1exists. Observe that this is a homogeneous system of equations so that, given the allocationcφand the matrixK, the solutionφis not unique. This becomes apparent by dividing each of thenequations byφ(si)and solving for then−1values
φ(sj) φ(si)
.
Letηp denote the dominant root of P. Ifηp > 1 then the inverse fails to be nonnegative, which cannot be a solution since all elements ofφmust be positive. If ηp= 1, then the inverse fails to exist. Ifηp<1then
(I− P)−1=I+P+P2+· · ·
is a convergent sequence. Hence the existence of a Pareto optimal solution will depend on the dominant root of the matrixP, which depends on the transition matrix Π, and the properties of the diagonal matrixK. As mentioned earlier, Aiyagari and Peled show that a necessary and sufficient condition for a Pareto optimal allocation to exist is that the matrixQhas a dominant root that is less than or equal to unity.
A corollary established here is that the associated matrix of within period MRS, the matrixK, must have a dominant root less than or equal to unity for a Pareto optimal allocation to exist.
I now explore the asset-pricing implications by examining the autarkic solution, and then addressing the ET-PO and CPO economies.