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A monotone model of intertemporal choice

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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 $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).

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Notes

  1. This example is also given in Blavatskyy (2015, p. 143).

  2. It seems that an inverse S-shaped weighting function rather than a concave utility function is the main driving force behind an apparent “hyperbolic” discounting. For example, consider discount factors implicitly defined by Eq. (25) with a weighting function (24) with parameter \(\gamma =0.56\) and a linear utility function. In this case, a quarterly discount factor would be 0.7115 for payoffs in one quarter, 0.6559 for payoffs in 1 year, and 0.7855 for payoffs in 3 years.

  3. Alternatively, we can make an additional assumption that utility function is differentiable and investigate when the first derivative of the objective function in (32) is strictly positive for all \(C_{T-1}\in [0, Y_{T-1}]\).

  4. When negative consumption is not possible (i.e., \(C_{T-1}\) is restricted to be between zero and \(Y_{T-1})\), Eq. (33) is a solution to problem (20) only when constrains (22) and (29) are violated, i.e., when \(\frac{1}{R}\le w\left( \beta \right) \le \frac{1}{1+\frac{R-1}{e^{Y_{T-1} Rd}}}\).

  5. When \(t=T-1\), recursive Eq. (35) becomes \(C_{T-1} =\frac{1}{d}\ln \left( {e^{C_T d}+\frac{R-1}{1-w\left( \beta \right) }w\left( {\beta ^{2}} \right) \left[ {e^{-C_T d}-1} \right] } \right) \).

  6. If undesirable payoffs occur, then it is possible that \({{\mathbf {f}}}({{\mathbf {t}}})\succcurlyeq {{\mathbf {f}}} ({{\mathbf {s}}})\) even though a moment of time \(t\in S\) precedes a moment of time \(s\in S\). In this case, we need to assume explicitly that all step programs are comonotonic.

  7. Rank-dependent utility with a linear utility function is also known as Yaari’s dual modal (Yaari 1987).

  8. Typical parameterizations of rank-dependent utility also cannot resolve the classical St. Petersburg paradox (Blavatskyy 2005).

  9. A similar example is also given in Rubinstein (2003, experiment 1, section 3.1, p. 1211).

  10. For example, inequality (40) is satisfied when \(\beta =0.8\), utility function is \(u(x)=x^{0.4}\), and weighting function is (17) with parameter \(\gamma =0.3\).

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  1. School of Management and Governance, Murdoch University, 90 South Street, Murdoch, WA, 6150, Australia

    Pavlo R. Blavatskyy

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Correspondence to Pavlo R. Blavatskyy.

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Blavatskyy, P.R. A monotone model of intertemporal choice. Econ Theory 62, 785–812 (2016). https://doi.org/10.1007/s00199-015-0931-6

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