A monotone model of intertemporal choice
Summary (2 min read)
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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