Intertemporal choice with savoring of yesterday

Abstract

The problem of intertemporal choice arises when outcomes are received in different moments of time. This paper presents an axiomatic model of intertemporal choice when consumption in the previous moment of time contributes to utility evaluation of consumption in the current moment. This model generalizes classic discounted utility theory (also known as constant or exponential discounting) in two ways. First, in every moment of time, a decision maker derives utility not only from current consumption but also from “residual” consumption in the previous moment of time. Second, these utilities are discounted with weights that are essentially a quasi-hyperbolic discounting function. The paper presents an application of the proposed model to the problem of optimal consumption and savings given a fixed income (wealth). When a decision maker derives satisfaction from both instantaneous consumption as well as a share of consumption in the previous moment of time, optimal consumption path is cyclic—periods of relatively high consumption are interchanged with periods of relatively low consumption. These cycles decay over time. Asymptotically, the consumption path exhibits conventional properties (constant, increasing or decreasing over time when a gross interest rate multiplied by discount factor is correspondingly equal to, greater than, or less than one).

Appendices

Appendix

Proof of Proposition 1

It is relatively straightforward to show that utility function (2) satisfies axioms 1–4. We shall prove only the sufficiency of these axioms. If all moments of time are null, then proposition 1 holds trivially by setting u(x) = 0 for any x (discount factors could be arbitrary). If only one moment of time t is nonnull, then there is a continuous utility function that represents preferences over outcomes in this nonnull moment of time (Debreu 1954, Theorem I, p.162). Proposition 1 is then satisfied by setting utility function (2) equal to this utility function and letting β_t be strictly positive. If two moments of time are nonnull, then by setting s = t and w = y in Axiom 4 we obtain a hexagon or Thomsen-Blaschke condition (Wakker 1984, p.112): whenever $a_{t} {\varvec{x}}^{{\varvec{\alpha}}}$≽$b_{t} {\varvec{y}}^{{\varvec{\alpha}}}$, $a_{t} {\varvec{y}}^{{\varvec{\alpha}}}$≽$c_{t} {\varvec{x}}^{{\varvec{\alpha}}}$, and $b_{t} {\varvec{z}}^{{\varvec{\alpha}}}$≽$a_{t} {\varvec{y}}^{{\varvec{\alpha}}}$ then $a_{t} {\varvec{z}}^{{\varvec{\alpha}}}$≽$c_{t} {\varvec{y}}^{{\varvec{\alpha}}}$. Additively separable utility representation (2) is then due to Hauptzatz über Sechseckgewebe (Blaschke and Bol, 1938, p. 10; Debreu 1960) and Theorem 15 in Krantz et al. (1971, Section 6.11.2). If more than two moments of time are nonnull, then preference relation ≽ satisfies ordinal independence ($a_{t} {\varvec{x}}^{{\varvec{\alpha}}}$≽$a_{t} {\varvec{y}}^{{\varvec{\alpha}}}$ implies $b_{t} {\varvec{x}}^{{\varvec{\alpha}}}$≽$b_{t} {\varvec{y}}^{{\varvec{\alpha}}}$) due to Lemma 2 in Blavatskyy (2013). Additively separable utility representation (2) is then due to Theorem 3 in Debreu (1960) and Theorem 15 in Krantz et al. (1971, Section 6.11.2).

Q.E.D.

3.1 Proof of Proposition 2

A preference relation ≽ satisfies axioms 1–4 if and only if it admits representation (2) due to Proposition 1. It is relatively straightforward to show that utility function (1) satisfies axiom 6. It remains to show that when preferences are represented by utility function (2) and Axiom 6 holds then they are indeed represented by utility function (1).

Let us consider two streams such that $\left( {x_{0} , x_{1} , x_{2} ,x_{3} ,x_{4} \ldots , ,x_{T} } \right)\sim \left( {x_{0} , x_{1} ,y_{2} ,x_{3} ,x_{4} \ldots , ,x_{T} } \right)$. If preferences are represented by utility function (2) then this preference indifference implies

$$ \beta_{2} u\left( {x_{2}^{\alpha } } \right)\, + \,\beta_{3} u\left( {x_{3}^{\alpha } } \right)\, = \,\beta_{2} u\left( {y_{2}^{\alpha } } \right)\, + \,\beta_{3} u\left( {y_{3}^{\alpha } } \right) $$

which can be rearranged as

$$ \beta_{2} \left[ {u\left( {x_{2}^{\alpha } } \right) - u\left( {y_{2}^{\alpha } } \right)} \right] = \beta_{3} \left[ {u\left( {y_{3}^{\alpha } } \right) - u\left( {x_{3}^{\alpha } } \right)} \right] $$

(4)

If Axiom 6 holds, then we must also have $\left( {x_{1} , x_{2} ,x_{3} , \ldots , ,x_{T} ,x_{0} } \right)\sim \left( {x_{1} ,y_{2} ,x_{3} , \ldots , ,x_{T} ,x_{0} } \right)$. If preferences are represented by utility function (2), then this preference indifference implies

$$ \beta_{1} u\left( {x_{2}^{\alpha } } \right)\, + \,\beta_{2} u\left( {x_{3}^{\alpha } } \right)\, = \,\beta_{1} u\left( {y_{2}^{\alpha } } \right)\, + \,\beta_{2} u\left( {y_{3}^{\alpha } } \right) $$

which can be rearranged as

$$ \beta_{1} \left[ {u\left( {x_{2}^{\alpha } } \right) - u\left( {y_{2}^{\alpha } } \right)} \right]\, = \,\beta_{2} \left[ {u\left( {y_{3}^{\alpha } } \right) - u\left( {x_{3}^{\alpha } } \right)} \right] $$

Dividing this equation by Eq. (4) yields result $\beta_{3} = \beta_{2}^{2} /\beta_{1}$. Iterating the same argument for subsequent moments of time we obtain that $\beta_{t} = \beta_{2}^{t - 1} /\beta_{1}^{t - 2}$ for all t ∊ {1, …, T-1}. Finally, since utility function is unique up to a positive affine transformation, we can divide by $\beta_{0}$ to obtain the conventional normalization that utility in the current moment of time is not discounted.

Q.E.D.