Discrete-time Recursive Utility
John H. Boyd III
This paper focuses on the fundamentals of discrete-time models using recursive utility. We
examine the relation between preferences, utility, and aggregator, the existence of optimal
paths, and several notions of impatience. In the one-sector model, we characterize optimal
paths and derive a turnpike theorem.1
Topics beyond the scope of this paper include continuous time recursive utility, models
involving uncertainty, the turnpike property in multisector models, and properties of Pareto
optima and equilibrium in multisector models.2
Section 1 discusses the limitations of time additive preferences and some of the benefits of
using a more general recursive utility specification. Section 2 examines the relation between
recursive preferences and the associated aggregator function. A general result on existence
of optimal paths is shown in Section 3. Sections 4 and 5 focus on the one-sector model.
Existence of optimal paths and dynamic programming is considered in Section 4. Section 5
characterizes optimal paths via the Euler equations and then goes on to prove a one-sector
turnpike theorem. Finally, Section 6 takes a brief look at the case where preferences are
both homothetic and recursive.
1. Why Recursive Utility?
Since Ramsey (1928), optimal growth models have primarily focused on the case of time
additive separable (TAS) utility. Reasons for its popularity are easy to find. It is intuitively
simple: We discount each period’s utility at a constant rate before summing over time. It
is often possible to obtain clear-cut analytic results. If a problem is not quite standard, a
large amount of theory developed by mathematicians is readily applicable. It allows the use
of dynamic programming.
In spite of these advantages, TAS utility also has some shortcomings. In particular, it
builds in some assumptions about the marginal rate of substitution between consumption
1
For a more comprehensive treatment of the discrete-time case, see the book by Becker and Boyd (1997).
Epstein (1983) examines when a recursive utility function is also a von Neumann-Morgenstern utility
function. Existence and characterization of optimal paths is studied in Becker, Boyd, and Sung (1989) and
Becker and Boyd (1992). Pareto optima and turnpikes in multisectoral models have been investigated by
Epstein (1987a, 1987b) and Dana and Le Van (1990, 1991a, 1991b).
2
2
JOHN H. BOYD III
in different periods that may not be desirable. This is most obvious when considering consumption paths that are stationary. In that case, the marginal rate of substitution between
consumption today and consumption in the following period is the inverse of the discount
factor. It is unaffected by the level of consumption. If there are multiple consumption goods,
this stationary marginal rate of substitution is also unaffected by the level of consumption
of those other goods.3
This constant marginal rate of substitution severely constrains the long-run behavior of
economic models. For example, a consumer facing a fixed interest rate will try either to save
without limit, or to borrow without limit, except in the knife-edge case where the discount
rate equals the interest rate.
This problem is especially severe when there are heterogeneous households. Unless all of
the households have the same discount rate, the most patient household ends up with all the
capital in the long run, while all other households consume nothing, using their labor income
to service their debt (Becker, 1980). Recursive utility allows for upward (or downward!)
sloping long-run capital supply curves and non-degenerate long-run wealth distributions.
The constant discount rate hypothesis also creates problems for the calculation of welfare
losses arising from capital income taxation. In TAS models, the long-run supply of capital
by households will be perfectly elastic at the discount rate. We are entitled to be a bit
skeptical of the resulting welfare analysis.
When analyzing growing economies, the special behavior of TAS utility on paths that
grow at a constant rate facilitates the construction of tractable models. Interestingly, there
are non-TAS utility functions that exhibit the same behavior (Dolmas, 1996; Farmer and
Lahiri, 2004).
2. Recursive Utility and Aggregators
Alternative methods of aggregating a sequence of period utilities have long been proposed.
Irving Fisher (1907) suggested combining today’s utility and tomorrow’s utility as if they
were two different consumption goods. The result could then be analyzed using indifference
curves over present and future utility. Fisher’s approach was formalized and axiomatized by
Koopmans and his collaborators in the 1960’s and early 1970’s (Koopmans, 1960; Koopmans,
Diamond, and Williamson, 1964; Koopmans, 1972a; Koopmans, 1972b).
3
The use of TAS utility also requires that the intertemporal elasticity of substitution be equal to the
coefficient of relative risk aversion. Epstein and Zin (1989) used techniques borrowed from the recursive
utility literature to construct Kreps-Porteus preferences that relax that restriction.
DISCRETE-TIME RECURSIVE UTILITY
3
Recursive utility is defined in an infinite horizon context. Time is indexed by t = 1, 2, . . . .
Let c = (c1 , c2 , . . . ) be a sequence of consumption bundles ct ∈ Rn in each time period t. We
let S denote the shift operator defined by S(c1 , c2 , . . . ) = (c2 , c3 , . . . ) and π the projection
onto the first co-ordinate, πc = c1 ∈ Rn . Let S denote the space of sequences in Rn and X
be a subset of S such that SX ⊂ X. We give S the product topology. Thus cn → c if and
only if cnt → ct for every t.
Suppose we have continuous preferences % defined on X. We say these preferences have
a recursive utility representation if there is a function W (called the aggregator) and a
subutility function u : S → R obeying
U (c) = W (u(c1 ), U (Sc))
for every c ∈ X. A simple example of recursive preferences is the TAS form U (c) =
P∞
t=1 u(ct ) which has aggregator W (x, y) = x + δy and subutility function u. A non-TAS
Pt
P
4
exp
−[
example of a recursive utility function is given by − ∞
s=1 v(cs )]. This function
t=1
has subutility v and aggregator W (x, y) = (−1 + y) exp(−v(x)). We refer to it as the EH
aggregator.
Koopmans’s Axioms. Koopmans’s axioms are:
(K1) % is a stationary relation: (z, x) % (z, x0 ) for all z ∈ πX if and only if x % x0 .
(K2) % exhibits limited independence: for all z, z 0 ∈ πX and x, x0 ∈ X, (z, x) % (z 0 , x) if
and only if (z, x0 ) % (z 0 , x0 ).
(K3) % is a sensitive relation: there is an x ∈ X and a z, z 0 ∈ πX with (z 0 , x) (z, x).
It is easy to show that any preference order with a recursive representation obeys Koopmans’s Axioms. Koopmans (1960) showed that if a preference order obeys (K1)–(K3) and
has a utility representation U , then the utility representation is recursive. Koopmans also
showed that there was a unique aggregator and subutility associated with U .
Potentially, the aggregator gives the possibility of representing preferences in a compact
form. However, not any function can be an aggregator. This leads to a new question.
Given a would-be aggregator W and subutility u, is there a corresponding recursive utility
function? If so, what domain is it defined over?
2.1. Construction of Recursive Utility from an Aggregator
Lucas and Stokey (1984) provided the first major result concerning construction of the utility
4
This is a discrete-time version of the modified Uzawa (1968) utility used by Epstein and Hynes (1983)
4
JOHN H. BOYD III
function. They used the contraction mapping theorem to construct a utility function defined
on all non-negative sequences under some restrictive conditions on W . Define the Koopmans
operator TW on the continuous functions on S+ by
TW (f )(c) = W (c1 , f (Sc)).
Under Lucas and Stokey’s assumptions, the contraction mapping theorem shows that the
Koopmans operator has a unique fixed point which is the desired recursive utility function.
The most important restriction was that W be bounded. This ruled out many commonly
used TAS utility functions. For example, the aggregator W (x, y) = (1 − σ)−1 x1−σ + δy,
P
t−1 1−σ
which leads to the commonly used utility function U (c) = (1 − σ)−1 ∞
ct does not
t=1 δ
fit into Lucas and Stokey’s framework, even when 0 < σ < 1.
Many authors have proposed solutions to this problem. The most notable are the weighted
contraction method (Boyd, 1990), the “partial sum” method used by Boyd to handle unbounded aggregators, Streufert’s (1990, 1998) biconvergence condition, and the k-local contraction recently used by Rincón-Zapatero and Rodríguez-Palmero (2003a, 2003b).
We will examine the construction of the utility function from an aggregator in detail. For
convenience, we will absorb the subutility into the aggregator for the remainder of the paper.
Thus we write W (x, y) rather than W (u(x), y). Nonetheless, to maintain compatibility with
Koopmans’s Axioms, we will presume our aggregators can be written using a subutility. We
will first follow Boyd’s approach and then comment on biconvergence. Rincón-Zapatero and
Rodríguez-Palmero’s method is akin to Boyd’s, but is able to cope with weaker bounds than
given by (W2) below.
Aggregator. A function W : X × Y → Y is an aggregator if:
(W1) W is continuous on X × Y and increasing in both c and y.
(W2) W obeys a Lipschitz condition of order one: there exists δ > 0 such that |W (x, y) −
W (x, y 0 )| ≤ δ|y − y 0 | for all x in X and y, y 0 in Y.
N
(W3) (TW
y)(c) is concave in c for all N and all constants y ∈ Y.
When W is differentiable the Lipschitz bound in (W2) is given by δ = sup W2 (c, y).
In the TAS case, it coincides with the discount factor. This bound is a strong form of
Koopmans, Diamond, and Williamson’s (1964) concept of time perspective. As viewed from
the present, future utilities appear closer and closer together as they are further out in time,
just as railroad tracks appear to converge in the distance. The Lipschitz bound δ gives
DISCRETE-TIME RECURSIVE UTILITY
5
us our first measure of impatience, which we refer to as the time perspective factor with
corresponding rate δ −1 − 1.
The sole purpose of condition (W3) is to ensure concavity of the utility function. It is not
required for the existence results. Joint concavity of W is not required for the associated
utility function to be concave. Although the EH aggregator is not concave, the correspondP
Pt
00
ing utility function U (c) = − ∞
t=1 exp[−
τ =1 v(cτ )] is concave whenever v < 0. More
N
generally, when the utility function is the limit of the functions (TW
(0))(c), (W3) ensures
concavity is inherited by U .
Let C be the space of continuous functions on S. Let ϕ ∈ C with ϕ > 0. Define the
ϕ-weighted norm by ||f ||ϕ = sup |f (x)/ϕ(x)|. The space Cϕ = {f ∈ C : ||f ||ϕ < ∞} is then
a Banach space under the ϕ-norm || · ||ϕ .
Weighted Contraction Mapping Theorem. Suppose T : Cϕ → C such that:
(1) T is non-decreasing (f ≤ g implies T f ≤ T g).
(2) T (0) ∈ Cϕ .
(3) T (ξ + Aϕ) ≤ T ξ + Aθϕ for some constant θ < 1 and all A > 0.
Then T has a unique fixed point.
Proof. The proof is inspired by Blackwell (1965). Let f, g ∈ Cϕ and consider ||f − g||ϕ .
Then −||f − g||ϕ ϕ ≤ f − g ≤ ||f − g||ϕ ϕ. Rearranging, we find f ≤ g + ||f − g||ϕ ϕ and
g ≤ f + ||f − g||ϕ ϕ. Using properties (1) and (3), we obtain T f ≤ T f + θ||f − g||ϕ ϕ and
T g ≤ T g + θ||f − g||ϕ ϕ. Together, these yield ||T f − T g||ϕ ≤ θ||f − g||ϕ . This shows T is a
strict contraction from Cϕ to C.
To show T maps into Cϕ , set g = 0 to obtain ||T f − T (0)||ϕ ≤ θ||f ||ϕ . By (2) T (0) ∈ Cϕ
which means T f ∈ Cϕ with ||T f ||ϕ ≤ ||T (0)||ϕ + θ||f ||ϕ .
As T is a strict contraction on Cϕ , it has a unique fixed point. Before attempting to construct the utility function, we must decide what domain is appropriate. Obviously, the utility function will live on a subset of S+ . The domain ultimately
chosen may depend on the problem at hand. One of the motivations for studying recursive
utility is to admit non-degenerate equilibria. This demands we use a subset that is appropriate for equilibrium problems, a linear space. If we are focusing on capital accumulation
problems we may further restrict the domain. Streufert (1990) exploits that fact to sharpen
the utility existence theorem.
For β ≥ 1, define the β-norm by |c|β = supt ||ct ||/β t } where || · || is the Euclidean norm
6
JOHN H. BOYD III
on Rn . Then define the β-weighted `∞ space by `∞ (β) = {c ∈ S : |c|β < ∞}. The space
`∞ (β) is a Banach space under the norm | · |β . We refer to the associated topology as the
β-topology. Since a sequence that converges in β-norm must converge in each coordinate,
the β-topology is stronger than the product topology on `∞ (β).
Continuous Existence Theorem. Suppose W : X × Y → Y obeys (W1) and (W2), ϕ
is continuous on some A ⊂ S with πA ⊂ X and SA ⊂ A. Suppose further W (πc, 0)
is ϕ-bounded and δ||ϕ ◦ S||ϕ < 1. Then there exists a unique U ∈ Cϕ (A) such that
N
W (πc, U (Sc)) = U (c). Moreover, (TW
0)(c) → U (c) in Cϕ .
Proof. The conditions on A insure everything makes sense. Since W is increasing in y,
the Koopmans operator TW is increasing. Now
|W (c1 , 0)|
|TW (0)|
=
<∞
ϕ(c)
ϕ(c)
because W (πc, 0) is ϕ-bounded. Moreover,
TW (ξ + Aϕ) = W (c1 , ξ(Sc) + Aϕ(Sc))
≤ W (c1 , ξ(Sc)) + Aδϕ(Sc)
≤ TW ξ + Aδ||ϕ ◦ S||ϕ ϕ(c)
by the Lipschitz condition (W2). Applying the Weighted Contraction Mapping Theorem
with θ = δ||ϕ ◦ S||ϕ < 1 shows that TW has a unique fixed point U .
N
0)(c)||ϕ ≤ δ N ||U (S N c)||ϕ ≤ ||U ||ϕ (δ||ϕ ◦ S||ϕ )N . As the rightNow consider ||U (c) − (TW
N
hand side converges to zero, (TW
0)(c) → U (c). A couple of applications will help clarify how the theorem may be used. The general
strategy is to pick either W (x, 0) or a function bounding it for the weighting function ϕ.
Consider the TAS aggregator W (x, y) = x1−σ + δy for 0 < σ < 1. Choose β with δβ 1−σ < 1
and set A = `∞ (β)+ . Here W (x, 0) = x1−σ . This is not positive, so we add one and
compose with the β-norm to get a weighting function. That is, ϕ(c) = 1 + |c|1−σ
β . Then
≤
W (c1 , 0) = c1−σ
≤ |c|1−σ
< ϕc, so W (πc, 0) is ϕ-bounded. Also, ϕ(Sc) = 1 + |Sc|1−σ
1
β
β
1 + (β|c|β )1−σ ≤ β 1−σ |c|β , which implies δ||ϕ ◦ S||ϕ < 1.
The EH aggregator W (x, y) = (−1 + y)e−v(x) provides a second example. Here X = R+
and Y = R− . Suppose v is increasing with v(0) > 0. Then W is increasing and obeys
a Lipschitz condition with δ = e−v(0) < 1. Since |W (x, 0)| = e−v(x) ≤ 1, we set ϕ = 1.
DISCRETE-TIME RECURSIVE UTILITY
7
The existence theorem then shows that the corresponding utility function is continuous and
bounded on A = S+ .
This approach has several limitations. When W (x, 0) is not bounded below it becomes
impossible to construct an appropriate ϕ. This difficulty can be handled by first constructing
the utility function on a restricted space of sequences that are bounded away from zero (so
utility can be bounded below), and then using a limiting argument to remove the lower
bound.
For 0 < γ ≤ β < ∞, define γ |c| = inf ||ct ||/γ t−1 and `∞ (β, γ) = {c ∈ S : 0 < γ |c| and
|c|β < ∞}. This is the set of paths with growth factors of at least γ and at most β.
To see how utility can be defined on such a space, consider the partial sums of U (c) =
P∞
∞
t−1
≤ ct ≤ |c|β β t−1 . Then
t=1 log ct . If c ∈ ` (β, γ), γ |c|γ
T
X
[(t − 1) log γ + log γ |c|] ≤
t=1
T
X
δ t−1 log ct
t=1
≤
T
X
δ t−1 [(t − 1) log β + log |c|β ].
t=1
Here the utility partial sums converge because they are squeezed between the partial sums
of convergent series. The limit is not uniform, so we cannot conclude it is continuous.
A slightly different approach gives us upper semicontinuity. Consider the set X = {c ∈
P
`∞ (β)+ : |c|β < A}. On this set, Tt=1 δ t−1 (log ct − (t − 1) log β − log A) has non-positive
terms. As the infimum of upper semicontinuous functions, the limit is upper semicontinuous.
It differs from the utility function by a constant, so utility is also upper semicontinuous X.
Boyd’s “partial summation” method adapts this approach to recursive utility.
Before proceeding, we have to consider the consequences of admitting −∞ as a possible
value for utility. The obvious solution to Koopmans’s equation may not be the only one.
In fact, U (c) = −∞ may satisfy the recursion, as it does in the logarithmic case. However,
it does not match up with the solution we derived on `∞ (β, γ)+ . We will rule out such
solutions as unreasonable.
The general strategy is to first derive utility on some well-behaved sequences in `∞ (β, γ)+ ,
and then use recursive substitution to extend utility to `∞ (β)+ .
Assumption.
(W10 ) W : X × Y → Y is increasing in both arguments, upper semicontinuous on X × Y
continuous for x > 0 and y > −∞, and obeys W (x, −∞) = W (0, y) = −∞ for all
8
JOHN H. BOYD III
x ∈ X and y ∈ Y.
Upper Semicontinuous Existence Theorem. Suppose W obeys (W10 ) and satisfies
the Lipschitz condition (W2) whenever it is finite. Suppose further there are increasing
functions g and h with g(||x||) ≤ W (x, 0) ≤ h(||x||). Set ϕ(c) = max{h(|c|β ), −g(γ |c|)}. If
ϕ > 0 with δ||ϕ ◦ S||ϕ < 1 for some β > γ > 0 with 1 ≤ β, then there exists a unique U
that is ϕ-bounded on `∞ (β, γ)+ , obeys Koopmans’s equation W (πc, U (Sc)) = U (c), and is
β-upper semicontinuous on `∞ (β)+ .
Proof. We first construct the function on `∞ (β, γ)+ . Temporarily give A = `∞ (β, γ)+ the
discrete topology. All functions are continuous there. Since W (c, 0) is clearly ϕ-bounded,
the Continuous Existence Theorem applies and yields a unique ϕ-bounded recursive utility
function Ψ defined on `∞ (β, γ)+ .
Next let z be an arbitrary element of `∞ (β, γ)+ and define the “partial sums” on all of
`∞ (β)+ by replacing the utility of the tail of c with the utility of the tail of z. Formally,
N
ΨN (c; z) = [(TW
Ψ)(S N z)](c)
= W (c1 , W (c2 , . . . , W (cn , Ψ(S N z)) · · · ))
Now for z, z0 ∈ `∞ (β, γ)+ ,
|ΨN (c; z) − ΨN (c; z0 )| ≤ δ N |Ψ(S N z) − Ψ(S N z0 )|
≤ δ N M [ϕ(S N z) + ϕ(S N z0 )]
≤ M 0 (δ||ϕ ◦ S||ϕ )N
for some M, M 0 . The first step uses the Lipschitz bound (W2), the second uses the ϕboundedness of Ψ on `∞ (β, γ)+ , and the third uses the fact that ϕ(S N z) ≤ (||ϕ ◦ S||ϕ )N ϕ(z).
It follows that if limN ΨN (c; z) exists, it must be independent of the choice of z. Note
that if c ∈ `∞ (β, γ)+ , ΨN (c; c) = Ψ(c), so limN ΨN (c; z) exists and is equal to Ψ(c) for
c ∈ `∞ (β, γ)+ .
We next show U (c) = limN ΨN (c; z) exists and is β-upper semicontinuous on all of `∞ (β)+ .
Consider the ball B about zero of radius κ. Set zt = κβ t−1 . For c ∈ B, ct ≤ zt . It follows that
ΨN (c; z) is a decreasing sequence. Its limit U (c) not only exists, but is upper semicontinuous
on B as the infimum of a sequence of upper semicontinuous (in c) functions. Since B was
any ball, U is upper semicontinuous on all of `∞ (β)+ .
DISCRETE-TIME RECURSIVE UTILITY
9
The function U is also recursive. If πc = 0 or if U (c) = −∞, our hypotheses imply
W (πc, U (Sc)) = −∞ = U (c). Otherwise, we may write:
W (πc, U (Sc)) = W (πc, lim(Sc); Sz))
N
= lim W (πc, ΨN (Sc; Sz))
N
= lim ΨN +1 (c; z)
N
= U (c)
which demonstrates Koopmans’s equation for c ∈ `∞ (β)+ .
This leaves uniqueness. Let Φ be a β-upper semicontinuous recursive utility function
that is ϕ-bounded on `∞ (β, γ)+ . Since Ψ is the unique such function on `∞ (β, γ)+ , Φ is an
extension of Ψ. Let zt = |c|β β t−1 so that c ≤ z. Thus Φ(c) ≤ ΨN (c; z). Taking the limit
shows Φ(c) ≤ U (c).
Now if ct = 0 for some t, U (c) = −∞ = Φ(c) and we are done. So suppose ct > 0 for
all t. Now set zt = γ t−1 and consider cn = (c1 , . . . , cn , zn+1 , zn+2 , . . . ). By construction,
Φ(cn ) = Ψn (c; z). Since γ < β, |cn − c|β → 0. As Φ is upper semicontinuous, Φ(c) ≥
limn Ψn (c; z) = U (c). Thus U = Φ, proving uniqueness. The sort of situation this applies to is W (x, y) = (1 − σ)−1 x1−σ + δy when σ > 1. Then
g(x) = −x1−σ and h = 0 is a good choice. Choose γ large enough that γ 1−σ δ < 1 (note that
δ > 1 is ok here) and β > γ arbitrary. Then set h = 1 and use the fact that g(γ |Sc| ≥ γ γ |c|
to show δ||ϕ ◦ S||ϕ < 1.
Rincón-Zapatero and Rodríguez-Palmero (2003b) also use a two-stage approach to derive
the recursive utility function. The biggest difference is that they use a different type of
contraction mapping theorem that allows them to handle a wider variety of aggregators.
Streufert (1990) followed a different path to existence of recursive utility. He focused on
the case where consumption sequences of interest are bounded above. Let ωt > 0 for every
t and consider [0, ω] = {c ∈ S : 0 ≤ ct ≤ ωt for every t}. A utility function U is upper
convergent over [0, ω] if for every c ∈ [0, ω]:
lim U (c1 , . . . , cT , S T ω) = U (c)
T →∞
while a utility function is lower convergent if
lim U (c1 , . . . , cT , 0) = U (c).
T →∞
10
JOHN H. BOYD III
A utility function is biconvergent over [0, ω] if it is both upper and lower convergent over
[0, ω]. A function U1 : [0, ω] → [0, ∞) equivalent to U is a general solution to Koopmans’s
equation if there is a sequence of subutility functions such that
Ut (S t−1 c) = W (ct , Ut+1 (S t c)).
Such a solution is admissible if for all c ∈ [0, ω], U (0) ≤ U1 (c) ≤ U (ω). Streufert showed
that if U is biconvergent, it is the only admissible solution to Koopmans’s equation. He was
also able to prove a converse under a mild additional hypothesis on the connectedness of the
image of U .
3. Existence of Optimal Paths
The basic method of showing optimal paths exist is to apply the Weierstrass Theorem,
which states that an upper semicontinuous function has a maximum on any compact set.
The existence theorems for recursive utility establish upper semicontinuity. We need only
show that the feasible set is compact.
As usual in normed spaces, β-bounded sets in `∞ (β) are not compact. However, if α < β,
any α-bounded set is pre-compact in `∞ (β). If it is product-closed, it is compact.5 Thus
the key to showing existence of optimal paths will be to show that all feasible paths grow
at most by a growth factor that is below β.
Lemma 1. Suppose 0 < α < β and there is an A > 0 such that ||ct || ≤ Aαt whenever
c ∈ X. Then if X is closed in the product topology, X is also compact.
Proof. Let cn be a sequence in X. Since X is α-bounded, we can extract a subsequence
that converges in the product topology. We also denote the subsequence by cn and its
product-limit by c.
Let > 0. Choose T so that Aαt /β t < /2 for all t > T . Now choose N so that
n
m
||cnt − cm
t || < for all n, m > N and t = 1, . . . , T . Then |c − c |β < for n, m > N . As
{cn } is a Cauchy sequence in the β-topology, it has a limit in `∞ (β). Since β-convergence
implies product convergence, this limit coincides with the product-topology limit c. As X
is β-closed, c ∈ X. 5
Since all of the β-topologies are stronger than the product topology, β-closed implies product closed. The
converse does not hold, but on α-bounded sets we only need the weaker condition.
DISCRETE-TIME RECURSIVE UTILITY
11
Proposition 1. Suppose U is β-upper semicontinuous on a product-closed and α-bounded
set X with α < β. Then the problem of maximizing U (c) for c ∈ X has a solution.
Proof. By the preceding lemma, X is β-compact. The Weierstrass Theorem then applies
to yield a maximum.
4. One-sector Model with Recursive Utility
The traditional one-sector growth model (Ramsey model) has one all-purpose good available
at each point in time. This good is used both for consumption and as an input to production
in the next period. Production proceeds under conditions of diminishing returns to scale,
and is described by a production function. The planner starts with an initial capital stock,
and maximizes utility over all feasible consumption paths.
Let ct denote consumption in time period t and let kt denote the capital stock accumulated
during period t, used for production in period t + 1. The initial capital stock is k0 . Consider
∞
the sequences of consumption levels, c = {ct }∞
t=1 , and capital stocks, k = {kt }t=1 . Both c
and k are elements of S.
Let f be a non-decreasing continuous production function such that f (0) ≥ 0. In each
time period, income yt = f (kt−1 ) is freely divided between consumption ct and capital kt .
Any income that is not accumulated as capital may be consumed.6 A pair of sequences (c, k)
is feasible from k if ct , kt ≥ 0 and 0 ≤ kt + ct ≤ f (kt−1 ) for t = 1, 2, . . . . The feasible set
is Y(k0 ) = {(c, k) : (c, k) is feasible from k}. The sets of feasible capital and consumption
programs are F(k0 ) = {k : (c, k) ∈ Y(k0 ) for some c} and B(k0 ) = {c : (c, k) ∈ Y(k0 ) for
some k}, respectively. Feasible consumption paths obey 0 ≤ ct ≤ f (kt−1 ) − kt , and B(k0 ) is
referred to as the budget set.
Proposition 2. Both B(k0 ) and F(k0 ) are compact in the product topology.
Proof. Define the t-th iterate of f , f t , inductively by f 1 (x) = f (x), f t (x) = f (f t−1 (x)). As
Q
t
f is increasing, ct , kt ≤ f t (k0 ), and so both B(k0 ) and F(k0 ) are contained in ∞
t=1 [0, f (k0 )].
This set is compact by Tychonoff’s Theorem.
ν
Take feasible kν with kν → k. Then 0 ≤ ktν ≤ f (kt−1
). Taking the limit shows 0 ≤ kt ≤
f (kt−1 ). Thus F(k0 ) is closed. As a closed subset of a compact set, F(k0 ) is compact.
Now suppose cν ∈ B(k0 ) with cν → c. Consider the associated kν ∈ F(k0 ). Take
a convergent subsequence with limit k. Retaining notation, we denote it kν . Then 0 ≤
6
We can interpret this as 100 percent depreciation. An alternative interpretation is that f denotes output
net of depreciation and investment is reversible.
12
JOHN H. BOYD III
ν
). Taking the limit in the feasibility constraints, we find c ∈ B(k0 ). As a
ktν + cνt ≤ f (kt−1
closed subset of a compact set, B(k0 ) is also compact. Ramsey Model Existence Theorem. Suppose U is β-upper semicontinuous and f (k) ≤
a + bk with b ≥ 1 and b, a ≥ 0. If b < β then an optimal path exists.
Proof. All that is really left is to show B(k0 ) is α-bounded for some α < β. Choose α with
b < α < β. Now ct ≤ f t (k0 ) and f t (k0 ) ≤ a+aα+· · ·+aαt−1 +αt k0 = a(αt −1)/(α−1)+αt k0
by induction. Since α > 1, ct ≤ [k0 + a/(α − 1)]αt , which shows B(k0 ) is α-bounded. 4.1. Dynamic Programming
We again consider the Ramsey problem, the problem of maximizing utility given an initial
capital stock k and production function f . This time, instead of the direct method, we
approach the optimal growth problem via dynamic programming, using the value function
J(k) = sup{U (c) : c ∈ B(k)}. The value function always exists, although it may be either
+∞ or −∞. If it is a continuous function, we will be able to use it to find optimal paths.
We establish Bellman’s equation using the Principle of Optimality. The classic statement of
the Principle of Optimality is:7
The Principle of Optimality. An optimal policy has the property that whatever the
initial state and initial decision are, the remaining decisions must constitute an optimal
policy with regard to the state resulting from the first decision.
The Optimality Principle says once on the optimal path, it is optimal to stay there.
Optimal choices are time consistent. Koopmans’s Stationarity and Limited Independence
Axioms are the key to establishing the Principle of Optimality in the context of the recursive
one-sector model.
The idea of time consistency underlies the following proof, although it is a bit obscured
due to the use of the supremum rather than a maximum. If the maximum does exist, the
epsilons can be dispensed with, making clear how the Principle of Optimality is employed.
Bellman’s equation is the analytical implementation of the Principle of Optimality. The
basic methodology of dynamic programming is to solve the optimization problem by finding
the solution to Bellman’s equation. The value function stores all the relevant information
necessary to solve the original problem.8
7
Bellman (1957, p. 83).
There are many treatments of dynamic programming. We recommend Stokey and Lucas (1989) for an
exhaustive treatment of dynamic programming in deterministic and stochastic economic models. Streufert’s
8
DISCRETE-TIME RECURSIVE UTILITY
13
Bellman’s Equation. In the one-sector model,
J(k) = sup {W (c, J(f (k) − c)) : 0 ≤ c ≤ f (k)} .
Proof. Let > 0, and take a feasible path c with U (c) > J(k) − . The path c0 =
0
{ct+1 }∞
t=1 is feasible from f (k) − c1 , and so U (c ) ≤ J(f (k) − c1 ). Thus J(k) − < U (c) =
W (c1 , U (c0 )) ≤ W (c1 , J(f (k) − c1 )). It follows that J(k) − ≤ sup {W (c, J(f (k) − c))}.
Because was arbitrary, J(k) ≤ sup{W (c, J(f (k) − c))}.
For step two, fix > 0. Take any c ∈ [0, f (k)] and choose c feasible from f (k) − c with
U (c) ≥ J(f (k)−c)−/δ. Letting c∗ = (c, c), we obtain U (c∗ ) = W (c, U (c)) ≥ W (c, J(f (k)−
c)) − by (W2). As c∗ is feasible from initial stocks k, + J(k) ≥ sup{W (c, J(f (k) − c))}.
Since was also arbitrary, J(k) ≥ sup{W (c, J(f (k)−c))}. Combining this with the previous
paragraph yields Bellman’s equation. Similar results hold in multisector models. When the optimal path is interior, we can use
the Envelope Theorem to show the value function is differentiable and that its derivative is
the derivative of U with respect to time 1 consumption evaluated at the optimal path. Let
Ut denote ∂U/∂ct .
Regular Paths. An optimal path of capital accumulation k is regular if 0 < kt < f (kt−1 )
for all t = 1, 2, . . . .
Differentiability of the Value Function. The value function J is non-decreasing
and concave. If U is differentiable with respect to consumption in period 1, and optimal
paths are regular, then J is differentiable and obeys J 0 (k) = U1 (c) where c is any optimal
path from k.
Proof. The value function is increasing because the feasible set grows when the initial stock
increases. Concavity follows since U is concave and the production function is concave.
Differentiability is established as follows.9 Let h > 0, h = (h, 0, . . . ) and let c be an
optimal path with initial endowment k so that J(k) = U (c). Clearly, J(k + h) ≥ U (c + h)
(1998) chapter surveys deterministic and stochastic dynamic programming using his biconvergence technique.
More recent literature on dynamic programming in the context of recursive utility includes Durán (2000)
and Rincón-Zapatero and Rodríguez-Palmero (2003). Alvarez and Stokey (1998) and Le Van and Morhaim
(2002) consider the problem of unbounded returns in a TAS context.
9
This method is adapted from Mirman and Zilcha (1975), which is simpler in this context than adapting
Benveniste and Scheinkman (1982).
14
JOHN H. BOYD III
and thus J(k+h)−J(k) ≥ U (c+h)−U (c). Dividing by h and taking the limit shows that the
right-hand derivative D+ J(k) satisfies D + J(k) ≥ U1 (c).10 Since c is regular, c1 = f (k) − k1
is non-zero. We may then repeat this with −c1 < h < 0, to show D− J(k) ≤ U1 (c) ≤ D+ J(k).
As J is concave, D+ J(k) ≤ D− J(k), thus J 0 (k) = U1 (c). Corollary 1. Under the above conditions, J 0 (k) = W1 (c1 , U (Sc)) where c is any optimal
path from k.
For regular paths, the Euler equations are easy to derive.11
Euler Equations. Suppose k is a regular optimal path from initial stock k and that Ut
exists at every time period. Then Ut (c) = f 0 (kt )Ut+1 (c).
Proof. Consider the path kt () = kt for t 6= s, ks () = ks + . The associated consumption
path is ct () = ct for t 6= s, s + 1, cs () = cs − , cs+1 () = f (ks + ) − f (ks ) + cs . For ||
small, this will be feasible and we can define g() = U (c()). This attains a maximum when
= 0, so g 0 (0) = 0. Now g 0 (0) = Us (c) − f 0 (ks )Us+1 (c), which establishes the result. The Euler equations can also be written Ut /Ut+1 = f 0 (kt ). In the regular case, we can
simplify the Euler equations by using the aggregator. The chain rule tells us that
Ut (c) = W2 (c1 , U (Sc))W2 (c2 , U (S 2 c)) · · · W2 (ct−1 , U (S t−1 c))W1 (ct , U (S t c))
Plugging this into the marginal rate of substitution Ut /Ut+1 , we find that most of the W2
terms cancel, leaving us with
W1 (ct , U (S t c))
Ut (c)
=
.
Ut+1 (c)
W2 (ct , U (S t c))W1 (ct+1 , U (S t+1 c)
This motivates the definition of the marginal rate of impatience by
1 + R(x, y, u) =
so that
W1 (x, W (y, u))
,
W2 (x, W (y, u))W1 (y, u)
Ut (c)
= R(ct , ct+1 , U (S t+2 c)).
Ut+1 (c)
In the TAS case, R is independent of future utility and reduces to δ −1 u0 (x)/u0 (y) − 1.
10
11
We use D+ and D− to denote the right- and left-hand derivatives.
The case of non-regular paths is more complex, see Boyd (1990) for details.
DISCRETE-TIME RECURSIVE UTILITY
15
Along stationary paths, we define the rate of impatience by ρ(c) = R(c, c, Φ(c)) where
Φ(c) = U (c, c, . . . ). Thus ρ(c) = −1 + 1/W2 (c, Φ(c)). In the TAS case, it coincides with the
discount rate as ρ(c) = −1 + δ −1 , so δ = 1/(1 + ρ(c)).
The Euler equations for the TAS case, W (x, y) = u(x) + δy have the usual form u0 (ct ) =
δf 0 (kt )u0 (ct+1 ). The Epstein-Hynes aggregator W (x, y) = (−1 + y) exp(−v(x)) yields W1 =
−v 0 (x)(−1 + y) exp(−v(x)) and W2 = exp(−v(x)). After some simplification,
v 0 (ct ) −1 + U (S t c)
= f 0 (kt )
v 0 (ct+1 ) U (S t c)
The difference is particularly noticeable on stationary paths. Suppose ct = c and kt = k.
Then the TAS case yields δf 0 (k) = 1 while the EH case gives 1 − 1/U (c) = f 0 (k). Overall
utility (or wealth) has no effect in the TAS case, but plays a key role in the EH case.
If we specialize further to the case of a constant interest rate r, so f (k) = 1+r, stationarity
for TAS utility requires δ(1+r) = 1. In the EH case, 1−1/U = 1+r or r = −1/U (recall that
utility is negative here). Different interest rates lead to different steady state consumption
levels for EH utility. For TAS utility, interest rates other than δ −1 − 1 do not have stationary
solutions. Either the optimal path shrinks toward zero (if r is small) or grows without bound
(if r is large).
We can go a bit further. We can find Φ(c) by solving Koopmans’s equation W (c, Φ(c)) =
Φ(c). Using the expression for W , we find (−1 + Φ) exp(−v(c)) = Φ, so Φ(c) = [1 − ev(c) ]−1 .
Thus exp v(c) = r along stationary paths. Only if r is outside the range of exp v(c) do we
get optimal paths converging to zero or growing without bound.
5. Optimal Paths in the One-Sector Model
In this section, we focus on one-sector capital accumulation models with differentiable concave production function and differentiable strictly concave recursive utility U . We will
require that the aggregator W obey either (W1) or (W10 ). It will also obey (W2) with
Lipschitz bound δ < 1. We require Ut (c) > 0 exist whenever U (c) is finite and that f 0 > 0
on R+ +. Further, assume that the feasible set F(k) is α-bounded and that U is β-upper
semicontinuous for some β > α.12 These conditions ensure that optimal paths exist. The
concavity of the production function implies that the feasible set is convex. The strict
concavity of U then yields a unique optimal path.
12
If W has the time additive form W (x, y) = u(x) + δy, this implies 0 < δ < 1 and that u0 > 0 on R+ +.
16
JOHN H. BOYD III
5.1. The Inada Conditions
The Euler equations are one of our main tools for investigating the properties of optimal
paths. We would like to use them to characterize optimal paths. However, we have only
derived the standard Euler equations as necessary conditions when optimal paths are regular.
There are two ways to work around this. One is to allow for boundary points by using a
Kuhn-Tucker inequality when kt = 0 or ct = 0. Modified Euler equations of this sort are
presented in Boyd (1990). The other, and simpler, method is to guarantee interiority by
imposing the Inada conditions. The Inada conditions come in two parts: the Inada utility
condition is Ut+ (c) = +∞ when ct = 0; the Inada production conditions are f 0 (0+) = +∞
and f 0 (∞) sup W2 < 1. When U is time additive separable, the Inada utility condition
becomes u0 (0+) = +∞, and the production condition is δf 0 (∞) < 1. We have:
Lemma 2. Suppose the Inada condition for utility is satisfied and f (0) = 0 and f 0 (k) > 0
for k ≥ 0. Whenever k > 0 and J(k) > −∞, any optimal path is regular, it obeys ct , kt > 0
for all t.
Proof. Let c be optimal and suppose cs = 0. If cs+1 > 0, then f (ks ) − ks+1 = cs+1 > 0, so
f (ks ) > ks+1 and hence ks > 0. Take ∆ > 0 small enough that ks > ∆ and f (ks −∆) > ks+1 .
We try an arbitrage between times s and s + 1 that accelerates consumption. Increase
consumption by ∆ at time s by taking the path c0 defined by c0t = ct for t 6= s, s + 1,
c0s = cs + ∆, and c0s+1 = f (ks − ∆) − ks+1 = cs+1 + f (ks − ∆) − f (ks ), which is feasible. Now
0 ≥ U (c0 )−U (c). Dividing by ∆, and letting ∆ → 0+ , we find 0 ≥ Us (c)−Us+1 (c)f 0 (ks ). But
the right-hand side is +∞. This contradiction shows that cs+1 = 0 also. Once consumption
reaches 0, it must stay there.
Now let s be the earliest time with cs = 0. If s = 1, ct = 0 for all t. Thus J(k) = U (0).
Of course, this is impossible if U (0) = −∞, so we may assume U (0) is finite. But then, the
path c∗ = (f (k), 0, 0, . . . ) is feasible, and yields utility W (k, U (0)) > W (0, U (0)) = U (0).
This is also impossible as c is optimal. Thus s > 1.
Note that all of the capital must be used up at s. No more consumption will take place,
and we would be made better off by consuming any leftover capital. Try an arbitrage between
s and s − 1 that delays consumption. Let ∆ > 0 and define c0 by c0t = ct for t 6= s − 1, s,
c0s−1 = cs−1 − ∆, and c0s = f (∆). This is feasible for small ∆. Again, 0 ≥ U (c0 ) − U (c). We
again divide by ∆, and let ∆ → 0+ . This yields 0 ≥ −Us−1 (c) + Us (c)f 0 (0). The right-hand
side is +∞ by the Inada condition on U . This contradiction shows that there is no s with
cs = 0. DISCRETE-TIME RECURSIVE UTILITY
17
5.2. Monotonicity
The basic monotonicity and turnpike results for the recursive one-sector model were established by Beals and Koopmans (1969), and under slightly weaker conditions by Magill and
Nishimura (1984).13 We prove optimal paths are monotonic and that they cannot cross.
Convergence to a steady state (or infinity) then follows.
First let c(k) and k(k) denote the optimal paths of consumption and capital stocks,
respectively.
Monotonicity Theorem. Suppose ∂R/∂c1 6= 0. For any initial stock k, kt (k) is a strictly
increasing function of k and the optimal path k(k) is strictly monotonic.
Proof. The strict concavity means k(k) is single-valued, hence continuous. Let k < k 0 ,
and let k = k(k), k0 = k(k 0 ) be optimal. Suppose k1 = k10 . By the Principle of Optimality,
kt = kt0 for t = 2, 3, . . . . Further, c1 = f (k) − k1 < f (k 0 ) − k10 = c01 and c2 = f (k1 ) − k2 = c02 .
Thus c0t = ct for t = 2, 3, . . . . The Euler equations yield R(c1 , c2 , . . . ) = f 0 (k1 ) = f 0 (k10 ) =
1 + R(c01 , c02 , . . . ). Since c0t = ct for t = 2, 3, . . . , R(c1 , c2 , . . . ) = R(c01 , c2 , . . . ). But this is
impossible since R is decreasing in c1 and c1 < c01 . Thus k1 6= k10 .
Now suppose k1 > k10 . Since k1 (0) = 0 < k10 < k1 (k), and k1 (k) is continuous, there is a k 00
with 0 < k 00 < k and k1 (k 00 ) = k10 . This is impossible by the preceding argument. Therefore
k1 is strictly increasing. Since kt (k) is the t-th iterate of k1 , it too is strictly increasing.
Further, k(k) is strictly monotonic by the usual argument. Since ∂R/∂c1 is continuous, it must be either always positive or always negative. Many
people consider the requirement that ∂R/∂c1 < 0 as most natural. R is the marginal rate
of substitution between consumption today and consumption tomorrow. As we increase
today’s consumption, we expect the rate of substitution to fall. In the additive case, it
must fall as ∂R/∂c1 = u00 (c1 )/δu0 (c2 ). In the more general recursive case it is equivalent to
requiring
∂[W1 (c1 , u)/W2 (c1 , u)]/∂c1 < 0.
This says that the indifference curves in (c1 , u)-space are convex to the origin.
Monotonicity immediately implies that optimal paths converge either to 0, or to a steady
state, or to +∞.
13
Boyer (1975) and Iwai (1972) examined recursive utility models with one sector. They utilized dynamic
programming ideas and conjectured the presence of multiple steady states in some cases.
18
JOHN H. BOYD III
Initial stocks can be divided into three disjoint sets. Let I0 = {k : k = 0 or f 0 (k) =
1 + ρ(f (k) − k)}, I+ = {k : f 0 (k) > 1 + ρ(f (k) − k)}, and I− = {k : f 0 (k) < 1 + ρ(f (k) − k)}.
For k ∈ I0 , the Euler equations and transversality condition are clearly satisfied by the
stationary path kt = k. Thus every element of I0 is a steady state. The Euler equations
also show that all steady states are in I0 . Accumulation is definitely possible in I+ since
f 0 (k) > 1 + ρ(f (k) − k) > 1. Define Ψ(k) = Φ(f (k) − k) where Φ(c) is the utility of the
constant path ct = c.
Recursive Non-Optimality Lemma. Suppose k ∈ I+ (k ∈ I− ) and kt ≤ k (kt ≥ k) for
t < n with kt = k for t ≥ n. Then U (c) ≤ Ψ(k), and k is not optimal.
Proof. First suppose k ∈ I+ . That U (c) ≤ Ψ(k) is trivial for n = 1. We proceed by
induction. Suppose U (c) ≤ Ψ(k) when n = m ≥ 1 and consider a path k with kt ≤ k and
kt = k for t ≥ m + 1. If km = k, U (c) ≤ Ψ(k) by the induction hypothesis, so we may
suppose km < k.
0
First consider the path k0 defined by kt0 = k for t 6= m and km
= k + ∆. Obviously
f 0 (k) > 1, so this path will be feasible from k for ∆ > 0 small enough. Taking a Taylor
expansion shows
U (c0 ) − U (c) = −Um (c)∆ + Um+1 (c)∆f 0 + o(∆)∆
= (W2 )m−1 [−W1 + W2 W1 f 0 ]∆ + o(∆)∆
= W1 (W2 )m−1 [W2 f 0 − 1]∆ + o(∆)∆
where all derivatives are evaluated at k. Now
1 + ρ(f (k) − k) = 1/W2 (k, Ψ(k)) < f 0 (k)
as k ∈ I+ . So W2 f 0 > 1 and ∆ may be chosen small enough that U (c0 ) > Ψ(k). Note that
remaining at k cannot be optimal.
Now take λ, 0 < λ < 1 with λ(k + ∆) + (1 − λ)km = k. (Here λ = (k − km )/(k − km + ∆).)
Then k00 = λk0 +(1−λ)k satisfies the hypotheses of the lemma for n = m, so U (c00 ) ≤ Ψ(k) by
the induction hypothesis. Now Ψ(k) ≥ U (c00 ) ≥ λU (c0 )+(1−λ)U (c) > λΨ(k)+(1−λ)U (c).
Thus Ψ(k) > U (c). The inequality holds for all n by induction. Further, since the stationary
path kt = k is feasible and not optimal, k cannot be optimal.
The case of k ∈ I− is similar. DISCRETE-TIME RECURSIVE UTILITY
19
Using the Recursive Non-Optimality Lemma, we can prove a turnpike result. Since both
I and I− are open, they are the countable union of open intervals. The endpoints of these
intervals must be in I0 .14 Now label the endpoints k̄i such that k̄i < k̄i+1 . We allow +∞
+
as the largest k̄i . If k ∈
/ (k̄i , k̄i+1 ), the optimal path cannot cross the steady states at the
endpoints, so kt ∈ (k̄i , k̄i+1 ). Further, since kt is monotonic, it must converge to some k̄.
Taking the limit in the Euler equations shows f 0 (k̄) = 1 + ρ(f (k̄) − k̄). The optimal path
converges to one of the endpoints. Similarly, if k is greater than all of the steady states it
converges either to the largest steady state, or to ∞. The next theorem shows that kt → k̄i+1
when k ∈ (k̄i , k̄i+1 ) ⊂ I+ and kt → k̄i when k ∈ (k̄i , k̄i+1 ) ⊂ I− .
Recursive Turnpike Theorem. Suppose ∂R/∂c1 < 0. If k ∈ (k̄i , k̄i+1 ) ⊂ I+ , the
optimal path obeys kt ↑ k̄i+1 ; if k ∈ (k̄i , k̄i+1 ) ⊂ I− , it obeys kt ↓ k̄i+1 ; and if k ∈ I0 , kt = k
is the optimal path.
Proof. When k ∈ I0 , the path kt = k satisfies the Euler equations and transversality
condition. Thus it is optimal.
Consider the case where k ∈ I+ . We know that kt is strictly monotonic. Suppose kt is
decreasing. Take a sequence of feasible paths kν such that kν → k in the product topology
with ktν ≤ k for all t and ktν = k for large t. (This is possible since f 0 > 1 and f (k) > k on
[ktν , k].) Then U (kν ) ≤ Ψ(k) by the Recursive Non-Optimality Lemma. Since U is product
continuous on the feasible set, U (k) ≤ Ψ(k), contradicting the fact that k is optimal. Thus
kt is increasing. By the Monotonicity Theorem, kt < k̄i+1 . Taking the limit in the Euler
equations shows the limit point is in I0 . It must be k̄i+1 .
The case k ∈ I− is similar, except that the optimal path may simply be truncated to
obtain the desired kν . 6. Homogeneous Recursive Utility and Sustained Growth
Many applications of TAS utility use a homogeneous or logarithmic period utility function.
This yields a utility function that is homothetic. Rader (1981) showed that these are the
only homothetic TAS utility functions. Such functions are of interest because they can yield
balanced growth paths (given appropriate technology). As a result, they are widely used in
macroeconomic models of economic growth.
Interestingly, these are not the only homothetic recursive utility functions. Dolmas (1995,
1996) was able to characterize homothetic recursive utility both axiomatically and via the
14
Note that I0 may contain points other than these endpoints.
20
JOHN H. BOYD III
aggregator. As usual, if a utility function is homothetic, it is equivalent to a utility function
that is homogeneous of degree 1.
Proposition 3. Let U be a recursive utility function that is homogeneous of degree γ. Its
aggregator obeys W (u(λc), λγ y) = λγ W (u(c), y).
Proof. This follows from the fact that
λγ W (u(c1 ), U (Sc)) = λγ U (c)
= U (λc) = W (u(λc1 ), U (λSc))
= W (u(λc1 ), λγ U (Sc)).
A converse can be derived whenever the existence theorems apply. E.g., if the ContinN
(0)(c), which is easily seen to be
uous Existence Theorem applies, U (c) is the limit of TW
homogeneous of degree γ. It follows that U is homogeneous of degree γ.
Dolmas (1996) gave the following example of a class of non-additive homogeneous recursive
utility functions. Let u be homogeneous of degree γ and set W (x, y) = u(x)w(y/u(x)). It is
easy to see that W (λx, λγ y) = λγ W (x, y), so the resulting utility function is homogeneous
of degree γ, provided it exists.
Now specialize to the one-good case with w(y) = [1 + δy]ρ /ρ where 0 < δ, ρ < 1 and
u(x) = xγ . Then W2 = δ[1 + δy]ρ−1 ≤ δ and W (x, 0) = xγ , so the Continuous Existence
Theorem applies on `∞ (β)+ whenever β γ δ < 1. It is easy to see this is not equivalent to an
additive representation by examining the marginal rates of substitution, which depend on
future utility as well as the consumption levels ct and ct+1 .
Now consider a path that grows at a constant rate, so ct+1 = βct . Since W (λc, λγ u),
W1 (λc, λγ u) = λγ−1 W1 (c, u) and W2 (λc, λγ u) = W2 (c, u). Then the marginal rate of substitution between consumption in adjacent periods is
W1 (ct , U (S t c))
W1 (ct , U (S t c))
=
W2 (ct , U (S t c))W1 (ct+1 , U (S t+1 c))
W2 (ct , U (S t c))W1 (λct , λγ U (S t c))
1
= γ−1
λ W2 (ct , U (S t c))
Now W2 (ct , U (S t c)) = W2 (c1 , U (Sc)), which implies that R is constant along such paths,
as noted by Farmer and Lahiri (2004). This constancy allows the possibility of balanced
growth paths.
DISCRETE-TIME RECURSIVE UTILITY
21
Farmer and Lahiri also note that W2 (c1 , U (S t c)) is independent of the starting level of
consumption since replacing ct by χct (so U (Sc) is replaced by χγ U (c) leaves W2 unchanged
due to the homogeneity property established above.
In models with a maximum sustainable stock, recursive preferences allow for heterogeneity
in discounting while permitting a non-degenerate long-run capital distribution.15 Farmer and
Lahiri conclude that recursive preferences add little flexibility in the case of balanced growth.
Either the existing wealth distribution is maintained, or it becomes degenerate in the long
run. They propose a generalization of recursive utility that allows for more heterogeneity in
discounting while maintaining balanced growth.
It should be noted that TAS utility already allows more flexibility in long-run behavior
under sustained growth. Boyd (2000) noted that the growth rate could affect whether it
was possible for agents with differing discount factors to both hold capital in the long run,
and that the growth rate could also affect which one ended up with all of the capital in
the degenerate case. These results suggest that the advantages of recursive utility occur
primarily in models with a maximum sustainable stock.
References
Fernando Alvarez and Nancy L. Stokey (1998), Dynamic Programming with Homogeneous Functions,
J. Econ. Theory 82, 167–189.
Richard Beals and Tjalling C. Koopmans (1969), Maximizing stationary utility in a constant technology,
SIAM J. Appl. Math. 17, 1001–1015.
Robert A. Becker (1980), On the long-run steady state in a simple dynamic model of equilibrium with
heterogeneous households, Quart. J. Econ. 95, 375–382.
Robert A. Becker and John H. Boyd III (1992), Recursive Utility and Optimal Capital Accumulation, II:
Sensitivity and Duality, Econ. Theory 2, 547–563.
Robert A. Becker and John H. Boyd III, “Capital Theory, Equilibrium Analysis and Recursive Utility,”
Blackwell, Oxford, 1997.
Robert A. Becker, John H. Boyd III and Bom Yong Sung (1989), Recursive Utility and Optimal Capital
Accumulation, I: Existence, J. Econ. Theory 47, 76–100.
Richard Bellman, “Dynamic Programming,” Princeton University Press, Princeton, NJ, 1957.
Lawrence M. Benveniste and Jose A. Scheinkman (1982), Duality theory for dynamic optimization models
of economics: the continuous time case, J. Econ. Theory 27, 1–19.
David Blackwell (1965), Discounted dynamic programming, Annals Math. Stats. 36, 226–235.
John H. Boyd III (1990), Recursive Utility and the Ramsey Problem, J. Econ. Theory 50, 326–345.
John H. Boyd III, “Sustained Growth with Heterogeneous Households,” Working Paper, Florida Interna15
Becker (1980) finds that the TAS case does lead to a degenerate capital distribution, where only the most
patient household owns any capital stock in the steady state.
22
JOHN H. BOYD III
tional University, 2000.
Marcel Boyer (1975), An optimal growth model with stationary non-additive utilities, Can. J. Econ. 8,
216–237.
Rose-Anne Dana and Cuong Le Van (1990), On the structure of Pareto-optima in an infinite horizon economy
where agents have recursive preferences, J. Opt. Theory Appl. 64, 269–292.
Rose-Anne Dana and Cuong Le Van (1991a), Equilibria of a stationary economy with recursive preferences,
J. Opt. Theory Appl. 71, 289–313.
Rose-Anne Dana and Cuong Le Van (1991b), Optimal growth and Pareto optimality, J. Math. Econ. 20,
155–180.
James F. Dolmas (1995), Time-additive representations of preferences when consumption grows without
bound, Econ. Letters 47, 317–326.
James F. Dolmas (1996), Balanced-growth-consistent recursive utility, J. Econ. Dyn. Control 20, 657–680.
Jorge Durán (2000), On Dynamic Programming with Unbounded Returns, Econ. Theory 15, 339–352.
Larry G. Epstein (1983), Stationary cardinal utility and optimal growth under uncertainty, J. Econ. Theory
31, 133–152.
Larry G. Epstein (1987a), The global stability of efficient intertemporal allocations, Econometrica 55, 329–
355.
Larry G. Epstein (1987b), A simple dynamic general equilibrium model, J. Econ. Theory 41, 68–95.
Larry G. Epstein and J. Allan Hynes (1983), The rate of time preference and dynamic economic analysis,
J. Polit. Econ. 91, 611–635.
Larry G. Epstein and Stanley E. Zin (1989), Substitution, risk aversion, and the temporal behavior of
consumption and asset returns: a theoretical framework, Econometrica 57, 937–969.
Roger E. A. Farmer and Amartya Lahiri, “Recursive Preferences and Balanced Growth,” Working Paper,
UCLA, 2004.
Irving Fisher, “The Rate of Interest,” Macmillan Co., New York, 1907.
Katsuhito Iwai (1972), Optimal economic growth and stationary ordinal utility—a Fisherian approach,
J. Econ. Theory 5, 121–151.
Tjalling C. Koopmans (1960), Stationary ordinal utility and impatience, Econometrica 28, 287–309.
Tjalling C. Koopmans (1972a), Representation of preference orderings over time, in “Decision and Organization” (C. B. McGuire and Roy Radner, eds.), North-Holland, Amsterdam.
Tjalling C. Koopmans (1972b), Representation of preference orderings with independent components of
consumption, in “Decision and Organization” (C. B. McGuire and Roy Radner, eds.), North-Holland,
Amsterdam.
Tjalling C. Koopmans, Peter A. Diamond, and Richard E. Williamson (1964), Stationary utility and time
perspective, Econometrica 32, 82–100.
Cuong Le Van and L. Morhaim (2002), Optimal Growth Models with Bounded or Unbounded Returns: A
Unifying Approach, J. Econ. Theory 105, 158–187.
Robert E. Lucas, Jr. and Nancy L. Stokey (1984), Optimal growth with many consumers, J. Econ. Theory
32, 139–171.
Michael J. P. Magill and Kazuo Nishimura (1984), Impatience and accumulation, J. Math. Anal. Appl. 98,
270–281.
DISCRETE-TIME RECURSIVE UTILITY
23
Leonard J. Mirman and Itzhak Zilcha (1975), On optimal growth under uncertainty, J. Econ. Theory 11,
329–339.
Trout Rader (1981), Utility over time: the homothetic case, J. Econ. Theory 25, 219–236.
Frank P. Ramsey (1928), A mathematical theory of saving, Econ. J. 38, 543–559.
Juan P. Rincón-Zapatero and Carlos Rodríguez-Palmero (2003a), Existence and Uniqueness of Solutions to
the Bellman Equation in the Unbounded Case, Econometrica 71, 1519–1555.
Juan P. Rincón-Zapatero and Carlos Rodríguez-Palmero, “On the Koopmans Equation with Unbounded
Aggregators,” Mimeo, University of Valladolid, 2003b.
Nancy L. Stokey and Robert E. Lucas, Jr., with Edward C. Prescott, “Recursive Methods in Economic
Dynamics,” Harvard University Press, Cambridge, MA, 1989.
Peter Streufert (1990), Stationary Recursive Utility and Dynamic Programming under the Assumption of
Biconvergence, Rev. Econ. Studies 57, 79–97.
Peter Streufert (1998), Recursive Utility and Dynamic Programming, in “Handbook of Utility Theory” (S.
Barberá, P. J. Hammond and C. Seidl, eds.), Kluwer Academic, Dordrecht, 93–121.
H. Uzawa (1968), Time preference, the consumption function, and optimum asset holdings, in “Value,
Capital and Growth: Papers in Honour of Sir John Hicks” (J. N. Wolfe, ed.), Edinburgh University
Press, Edinburgh.
Department of Economics, Florida International University, Miami, FL 33199