Geometric Discounting in Discrete, Infinite-Horizon Choice Problems Asen Kochov∗ August 12, 2013

The rate of time preference is traditionally defined as the marginal rate of substitution between current and future consumption. This definition is not applicable when outcomes are indivisible. Such is the case in all discrete-choice dynamic problems which arise, for example, in modeling housing or occupational decisions. Assuming an infinite horizon and a standard time-additive utility representation, this note shows that the discount factor can be uniquely recovered from the underlying preference order, provided that the decision-maker is sufficiently patient. Under the same conditions, the utility index over outcomes is cardinally unique. Finally, an algorithm for approximating the discount factor is provided which, at each stage, uses only finite-horizon data. Let X be a finite set of outcomes. Time is discrete and varies over the set T := {0, 1, 2, ...}. A preference order  is a complete and transitive binary relation on the space of all sequences X T . A time-additive utility representation (u, β) for  takes the form X β t u(xt ), (0.1) U (x0 , x1 , ...) := (1 − β) t∈T

where β ∈ (0, 1), u : X → R. An axiomatic characterization of all preferences which admit the representation in (0.1) is beyond the scope of this paper. Some important behavioral implications, such as stationarity and independence across time, are wellunderstood from the classical work of Koopmans [2]. We focus on the uniqueness ∗

Department of Economics, University of Rochester, Rochester, NY 14627. [email protected].

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of the representation which, when X is finite, depends critically on the following assumption. (P) There exist outcomes x, y ∈ X such that (x, x, ...)  (y, y, ...) and (y, x, x, x, ...)  (x, y, y, y, ...). If the preference order  has a time-additive representation (u, β), the assumption is satisfied if and only if the discount factor β is greater than 21 . It turns out that this lower bound on the rate of time preference is sufficient for the uniqueness of the representation. Proposition 1 Suppose the preference order  on X T has a time-additive representation (u, β) and satisfies axiom (P). Then, the discount factor β is unique and the instantaneous utility function u : X → R is unique up to positive affine transformations. Before we prove Proposition 1, it is useful to comment on the neccesity of assumption (P). For simplicity, take X to be {0, 1} and let ≥lex be the usual lexicographic order on {0, 1}T . Recall that that a preference order  is stationary if for all z ∈ X and (xt )t , (yt )t ∈ X T , (x0 , x1 , ...)  (y0 , y1 , ...) if and only if (z, x0 , x1 , ...)  (z, y0 , y1 , ...). An order on {0, 1}T is monotone if (xt )t  (yt )t whenever xt ≥ yt for all t ∈ T and xt > yt for some t ∈ T . It is straight-forward to check that the lexicographic order is the only nontrivial stationary and monotone ranking onP {0, 1}T which violates (P). In particular, all time-additive representations (1 − β) t β t xt with β < 21 induce the lexicographic order on {0, 1}T . It is an open question if assumption (P) remains necessary if there are more than two outcomes. Since the bound it imposes on the rate of time preference is reasonable in many economic contexts, we do not pursue the question. The proof of Proposition 1 is based on a number-theoretic result due De Vries and Komornik [1]. Fix an order  and let (u, β), (v, δ) be two time-additive representations. By axiom (P), we know that both β, δ are greater or equal to 21 . Let x∗ , x∗ ∈ X be such that (x∗ , x∗ , ...), (x∗ , x∗ , ...) ∈ X T are the best and respectively the worst sequences in X ∞ . By axiom (P) again, the two sequences cannot be indifferent. Rescaling if necessary, it is w.l.o.g. to assume that u(x∗ ) = v(x∗ ) = 1 and u(x∗ ) = v(x∗ ) = 0. Abusing notation, identify the order  on {x∗ , x∗ }T with an order on {0, 1}T . Say that a sequence in {0, 1}T is infinite if it contains infinitely many 1’s. A sequence (a0 , a1 , ...) is lexicographically decreasing 2

if (a0 , a1 , ...) ≥lex (at , at+1 , ...) for every t ∈ T . Let D be the class of all infinite, lexicograpically decreasing sequences in {0, 1}T . It follows from Proposition 2.3 in [1] P that there is a bijection λ 7→ (aλ0 , aλ1 , ...) from [ 12 , 1) onto D such that t∈T λt+1 aλt = 1. Since, by assumption, both (u, β) and (v, δ) induce the same order  on D, we have that X X β t+1 at = 1 ⇔ (0, a0 , a1 , ...) ∼ (1, 0, 0, ...) ⇔ δ t+1 at = 1, ∀(at )t ∈ D. t∈T

t∈T

Conclude that β = δ. Turn to uniqueness of the indices u, v : X → R. Having made the normalization above, it remains to show that u = v. Since β ≥ 12 , it is well-known that every number c ∈ P [0, 1] has a β-expansion, that is, a sequence (at )t ∈ {0, 1}T such that c = (1 − β) t∈T β t at . It follows that, for every z ∈ X, there is a sequence (xt )t ∈ {x∗ , x∗ }T such that (z, z, ...) ∼ (x0 , x1 , ...). In terms of the two representations (u, β) and (v, β), the latter implies that u(z) = v(z) for all z ∈ X, as desired. an algorithm for computing For a given λ ∈ [ 21 , 1), De Vries and Komornik P[1] provide t+1 at = 1. It is not difficult to the unique sequence (at )t ∈ D such that t∈T λ adapt the algorithm to find the discount factor of a given preference order with an unknown representation (u, β). Once again, set u(x∗ ) = 0, u(x∗ ) = 1 and think of  as a ranking on {0, 1}T . We define two sequences (an )n ∈ {0, 1}T and (βn )n ∈ [ 21 , 1]T through the following procedure. Set a0 = 1, β0 = 1. If ak , βk have been defined for all k < n, set ( 1, if (1, 0, 0, ...)  (0, a0 , a1 , ..., an−1 , 1, 0, 0, 0, ...) an = 0, else βn = arg max λ

n X

n

λ

t+1

at

t=0

X 1 s.t. λ ∈ [ , 1], λt+1 at ≤ 1. 2 t=0

It follows from [1, Prop 2.1] that (an )n is the unique sequence in D such that P∞ t+1 at = 1. We now show that (βn )n converges to the true β. By construction, t=0 β the sequence is decreasing and βn ≥ β for every n. If δ denotes the limit of (βn )n , we have 1≥

n X

βnt+1 at

≥

t=0

Taking limits as n → ∞ gives

n X

δ

t+1

t=0

P∞

t=0

at ≥

n X

β t+1 at ,

∀n.

t=0

δ t+1 at = 1 and therefore δ = β, as desired. 3

References [1] M. De Vries and V. Komornik. Unique expansions of real numbers. Advances in Mathematics, 221(2):390–427, 2009. [2] T. Koopmans. Representation of preference orderings over time. In C. McGuire and R. Radner, editors, Decision and organization, pages 79–100. Amsterdam: North Holland, 1972.

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