Journal ArticleDOI

# A monotone model of intertemporal choice

01 Oct 2016-Economic Theory (Springer Berlin Heidelberg)-Vol. 62, Iss: 4, pp 785-812

AbstractExisting models of intertemporal choice such as discounted utility (also known as constant or exponential discounting), quasi-hyperbolic discounting and generalized hyperbolic discounting are not monotone: A decision maker with a concave utility function generally prefers receiving \$1 m today plus \$1 m tomorrow over receiving \$2 m today. This paper proposes a new model of intertemporal choice. In this model, a decision maker cannot increase his/her satisfaction when a larger payoff is split into two smaller payoffs, one of which is slightly delayed in time. The model can rationalize several behavioral regularities such as a greater impatience for immediate outcomes. An application of the model to intertemporal consumption/saving reveals that consumers may exhibit dynamic inconsistency. Initially, they commit to saving for future consumption, but, as time passes, they prefer to renegotiate such a contract for an advance payment. Behavioral characterization (axiomatization) of the model is presented. The model allows for intertemporal wealth, complementarity and substitution effects (utility is not separable across time periods).

Topics: Intertemporal choice (68%), Exponential discounting (66%), Discounted utility (65%), Hyperbolic discounting (63%)

### 1. Introduction

• Section two presents their model of intertemporal choice and discusses its properties.
• Section three applies this model to the problem of intertemporal consumption/savings.
• Behavioural characterization of the model is presented in section four.
• Section five concludes with a general discussion.

### State of the world

• It is relatively straightforward to rearrange utility formula (2) into formula (3).
• ( EQUATION -5 -To illustrate model (3) let us return to the first example from the introduction.
• Table 2 shows the values of the leftmost-hand and the rightmost-hand side of inequality ( 16) for several values of parameter β∊(0,1) and the weighting function (17) proposed by Tversky and Kahneman (1992,p.309) with γ=0.61 (a median parameter elicited in the experiment of Tversky and Kahneman, 1992, p. 312) . ( Thaler (1981) provides another example of dynamically inconsistent preferences.
• Thus, a discount factor inferred from equation ( 18) may be almost constant over time if the authors use an inverse S-shaped weighting function w(.) and a concave utility function u(.).
• As in the previous example, these discount factors increase over time-it appears as if a decision maker is more impatient for payoffs that are closer to the present moment.

### 3.1. Case 1: consumption of all disposable income at the current moment of time and no savings

• Impatient consumers with w(β)≤1/R would never voluntarily hold any savings (and even try to accumulate a credit card debt if it were possible).
• Such decision makers prefer to consume all income immediately and "starve" in the subsequent periods.
• To prevent such behavior a social planner can either increase an interest rate (so that condition w(β)≤1/R is rarely satisfied) or introduce restrictions on the intertemporal movement of income (e.g., a system of social security).
• The latter option appears to be more effective.
• Table 3 shows that a large increase in the interest rate is required for any substantial decrease in the upper bound on β when weighting function w(.) is inverse S-shaped.

### Weighting function w(.)

• The upper bound imposed by condition (25) on a subjective discount factor β when T=50 for various values of the interest rate R and two weighting functions w(.).
• Consider the case when income increases by 10% from one moment of time to another and a decision maker has a weighting function (17) with γ=0.61.
• Without the possibility of commitment, the decision maker chooses not to save at the penultimate moment of time iff β≤0.9895.
• Condition (26) is not only sufficient but also necessary when the utility function is differentiable.

### Thus, a decision maker optimally chooses to consume nothing in period

• This is a standard result-a decision maker, who maximizes discounted present value, consumes all income either at the present moment t=0 (when β≤1/R) or at the last moment t=T (when β>1/R).
• 1/R. A decision maker with a linear utility function and a weighting function that is convex in the neighborhood of one (such as an inverse S-shaped function often found in empirical studies) consumes all income either at the present moment t=0 (when w(β)≤1/R) or at the last moment t=T (when w(β)>1/R).
• Condition ( 32) is stronger for a concave utility function (with a declining marginal utility) than for a linear utility.
• This fits well with a stylized fact that few people over-save but many people over-consume.

### 3.3. Optimal consumption path with a possibility of a debt (negative consumption)

• Optimal consumption, however, starts to decline at the later moments of time.
• Impatient decision makers (with a low β) have a higher level of constant consumption at the initial moments and a higher level of debt at the last moment.

### 4. Behavioral Characterization (Axiomatization) of Rank-Dependent Discounted Utility

• In choice under uncertainty, a separable utility representation is traditionally derived from an axiom known as tradeoff consistency (Wakker 1984; 1989) or Reidemeister closure condition in geometry (Blaschke and Bol, 1938) .
• Blavatskyy (2013b) recently showed that this condition can be weakened to an axiom known as cardinal independence or standard sequence invariance (e.g., Krantz et al., 1971, Section 6.11.2) . (.

### Axiom 4 (Cardinal

• The proof follows immediately from proposition 1 in Blavatskyy (2013b) when axiom 3 is used and proposition 3 in Blavatskyy (2013b)-when axioms 3a and 3b are used.
• Representation (37) for step programs can be extended to all other programs.
• A standard method is to enclose any bounded program f∊ℱ by step programs that period-wise dominate (approximation from above) or are period-wise dominated (approximation from below) by program f∊ℱ.

### 5. General discussion

• Consider the following example of "subadditive" time preferences (Scholten and Read , 2010, anomaly 1, p. 928).
• Let β denote a monthly discount factor and let us normalize the utility function so that u(\$0)=0.
• According to model (13) a decision maker reveals the above mentioned choice pattern if inequality (40) holds.

### 𝑤(𝛽

• Such decision makers would prefer plan I) over plan J) but, at the same time, they would prefer plan L) over plan K), in violation of the assumption of payoff independence.
• A necessary condition for (47) to hold is inequality (48), which defines a concave utility function u(.).
• The separability of utility in intertemporal choice may not be normatively appealing because payoffs are not mutually exclusive.

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RESEARCH REPOSITORY
This is the author’s final version of the work, as accepted for publication
following peer review but without the publisher’s layout or pagination.
The definitive version is available at:
http://dx.doi.org/10.1007/s00199-015-0931-6
Blavatskyy, P.R. (2016) A monotone model of intertemporal choice.
Economic Theory, 62 (4). pp. 785-812.
http://researchrepository.murdoch.edu.au/id/eprint/29017/
It is posted here for your personal use. No further distribution is permitted.

ECONOMIC THEORY
A Monotone Model of Intertemporal Choice
--Manuscript Draft--
Manuscript Number: ECTH-D-14-00353R2
Full Title: A Monotone Model of Intertemporal Choice
Article Type: Research Article
Corresponding Author: Pavlo Blavatskyy, Ph.D.
AUSTRIA
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Information:
Corresponding Author's Institution:
Corresponding Author's Secondary
Institution:
First Author: Pavlo Blavatskyy, Ph.D.
First Author Secondary Information:
Order of Authors: Pavlo Blavatskyy, Ph.D.
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Funding Information:
Abstract: Existing models of intertemporal choice such as discounted utility (also known as
constant or exponential discounting), quasi-hyperbolic discounting and generalized
hyperbolic discounting are not monotone: a decision maker with a concave utility
function generally prefers receiving \$1m today plus \$1m tomorrow over receiving \$2m
today. This paper proposes a new model of intertemporal choice. In this model, a
decision maker cannot increase his/her satisfaction when a larger payoff is split into
two smaller payoffs, one of which is slightly delayed in time. The model can rationalize
several behavioral regularities such as a greater impatience for immediate outcomes.
An application of the model to intertemporal consumption/saving reveals that
consumers may exhibit dynamic inconsistency. Initially, they commit to saving for
future consumption but, as time passes, they prefer to renegotiate such a contract for
an advance payment. Behavioral characterization (axiomatization) of the model is
presented. The model allows for intertemporal wealth, complementarity and
substitution effects (utility is not separable across time periods).

Summary of main changes in the revised version
1) I restructured the manuscript to focus on the main issue: existing models of intertemporal
choice violate intertemporal monotonicity and this paper proposes a new model to remedy
this problem. I dropped all discussion of separability as this is a secondary point (even though
interesting in itself) which diverts reader’s attention from the main issue.
2) I dropped my criticism of discontinuity and pointed out that violations of monotonicity may
arise in a discrete time framework (and they are more severe in a continuous time
framework).
3) Since the referee did not criticize the macroeconomic application but found axiomatization
“not more appealing than those used in decision under risk” I moved the section with
macroeconomic application to section 3 and axiomatizationto section 4 (I am also happy to
delegate axiomatization to the appendix, if required).
With best regards, Pavlo

- 1 -
A Monotone Model of Intertemporal Choice
Pavlo R. Blavatskyy
School of Management and Governance
Murdoch University, 90 South Street,
Murdoch, WA 6150 AUSTRALIA
Ph: (08)9360 2838 Fax: (08)9360 6966
Email: P.Blavatskyy@murdoch.edu.au
Abstract Existing models of intertemporal choice such as discounted utility (also known as constant
or exponential discounting), quasi-hyperbolic discounting and generalized hyperbolic discounting are
not monotone: a decision maker with a concave utility function generally prefers receiving \$1m
today plus \$1m tomorrow over receiving \$2m today. This paper proposes a new model of
intertemporal choice. In this model, a decision maker cannot increase his/her satisfaction when a
larger payoff is split into two smaller payoffs, one of which is slightly delayed in time. The model can
rationalize several behavioral regularities such as a greater impatience for immediate outcomes. An
application of the model to intertemporal consumption/saving reveals that consumers may exhibit
dynamic inconsistency. Initially, they commit to saving for future consumption but, as time passes,
they prefer to renegotiate such a contract for an advance payment. Behavioral characterization
(axiomatization) of the model is presented. The model allows for intertemporal wealth,
complementarity and substitution effects (utility is not separable across time periods).
Keywords: Intertemporal Choice; Discounted Utility; Time Preference; Dynamic Inconsistency;
Expected Utility Theory; Rank-Dependent Utility
JEL Classification Codes: D01; D03; D81; D90
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- 2 -
A Monotone Model of Intertemporal Choice
1. Introduction
Intertemporal choice involves payoffs to be received at different points in time. Samuelson
(1937) proposed a classical model of intertemporal choice that is known as discounted utility or
constant (exponential) discounting. The model is parsimonious and analytically convenient but its
descriptive validity has been questioned. For instance, Thaler (1981, p. 202) argued that some people
may prefer one apple today over two apples tomorrow but, at the same time, they may prefer two
apples in one year plus one day over one apple in one year. Discounted utility cannot account for
such a switching choice pattern. The descriptive limitations of discounted utility motivated the
development of alternative models such as quasi-hyperbolic discounting (Phelps and Pollak, 1968)
and generalized hyperbolic discounting (Loewenstein and Prelec, 1992). These models replaced a
constant (exponential) discount factor in discounted utility with a more general discount function.
Discounted utility and its subsequent generalizations such as quasi-hyperbolic and generalized
hyperbolic discounting may produce rather counterintuitive results which are seldom discussed in
the literature. These models may violate intertemporal monotonicity when utility function is
concave (as usually assumed in economics). For example, consider a decision maker who receives
one million dollars now as well as one million dollars at a later moment of time t (with a convention
that t=0 denotes the present moment).
1
According to the above mentioned models, this decision
maker behaves as if maximizing utility
(1)
󰇛

󰇜
󰇛
󰇜
󰇛

󰇜
where u(.) is utility function and D(.) is a discount function. According to the same models, receiving
two million now yields utility u(\$2m). For a decision maker with a concave utility function u(.) and
enough patience (i.e. with a discount function D(t) sufficiently close to one) utility (1) is greater than
u(\$2m) due to Jensen’s inequality. Moreover, in a continuous time framework, for any decision
maker with a concave utility function u(.) it is always possible to find a moment of time t sufficiently
close to the present moment such that utility (1) is greater than u(\$2m) due to the property


󰇛
󰇜
(cf. Figure 1 in Loewenstein and Prelec, 1992, p. 581). In other words, the above
mentioned models predict that a decision maker with a concave utility function prefers receiving
one million now plus one million at a later moment of time t over receiving two million immediately.
More generally, according to the existing models of intertemporal choice, the desirability of
any payoff may increase if this payoff is split into two smaller payoffs one of which is slightly delayed
in time. Such an implication is clearly testable in a controlled laboratory experiment. Yet, readers
would probably agree with my tentative conjecture that very few people are likely to reveal such a
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This example is also given in Blavatskyy (2015, p. 143).
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