Abstract
This paper analyzes self-control problems in intertemporal consumption of a sin good where current consumption imposes a negative externality on one’s future selves due to the existence of present-biased preferences. We introduce the negative externality dubbed ‘internality’, which is augmented by the cumulative distribution function of the sum of previous consumption. We show that the shot-run self with self-control problem over consumes the sin good compared to that of the long-run self without present-biased preferences. Further, we investigate the effectiveness of self-bargaining with the presence of an external control devise as possible punishment. The short-run self faces three possible options: do nothing, accept the long-run self’s offer, or reject the offer. In the latter case an external control devices is implemented. It is shown that successful self-bargaining depends on the efficiency of the control device.
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Notes
There are many variants of paternalism: patronizing paternalism (Burrows 1993), libertarian paternalism (Sunstein and Thaler 2003), permissible paternalism (Goodin 1991), benign paternalism (Choi et al. 2003), and asymmetric paternalism (Camerer et al. 2003). For a survey of the literature see Ott (2007). “State Paternalism and Rules of Reason”, presented at the 64th International Atlantic Economic Conference in Savannah, Georgia, October 2007.
This is the standard form as proposed by Laibson (1997).
Given the cost of implementing the external control device e, we define two critical types of LR-2 in terms of his/her reduction of consumption denoted by ξ. There is LR-2 self who is indifferent between doing nothing and making a rejectable offer, this is denoted as the ‘credible-threat threshold’ \(\ifmmode\expandafter\bar\else\expandafter\=\fi{\Delta }^{{LR - 2}} \) where
$$\underline \Delta ^{{\text{LR}} - 2} \equiv \alpha \ln x_2^{{\text{SR}} * } - \alpha \ln \left( {x_2^{{\text{SR}} * } - \xi } \right) + c\delta \left[ {F\left( {x_1 + x_2^{{\text{SR*}}} - \xi } \right) - F\left( {x_1 + x_2^{{\text{SR}} * } } \right)} \right].$$Also, we define the ‘acceptable-offer threshold’, \(\bar \Delta ^{{\text{LR}} - 2} \) in which LR-t self is indifferent between making an acceptable offer and making a rejectable offer:
$$\overline \Delta ^{{\text{LR}} - 2} \equiv \alpha \ln x_2^A - \alpha \ln \left( {x_2^{{\text{SR}} * } - \xi } \right) + c\delta \left[ {F\left( {x_1 + x_2^{{\text{SR}} * } - \xi } \right) - F\left( {x_1 + x_2^A } \right)} \right].$$Given the efficiency of the external control device, we define two critical types of SR-2 self in terms of the reduction in consumption level. There is SR-2 self who is indifferent between accepting the offer and rejecting the offer is denoted by the ‘acceptance threshold’, \(\overline \Delta ^{SR - 2} \) where
$$\overline \Delta ^{SR - 2} \equiv \alpha \ln x_2^A - \alpha \ln \left( {x_2^{SR * } - \xi } \right) + \left( {x_2^{SR * } - x_2^A } \right) + c\beta \delta \left[ {F\left( {x_1 + x_2^{SR * } - \xi } \right) - F\left( {x_1 + x_2^A } \right)} \right].$$Also, we define the ‘reject threshold’ as \(\underline \Delta ^{SR - 2} \) in which SR-2 self is indifferent between to doing nothing and rejecting the offer:
$$\underline \Delta ^{SR - 2} \equiv \alpha \ln x_2^{SR * } - \alpha \ln \left( {x_2^{SR * } - \xi } \right) + c\beta \delta \left[ {F\left( {x_1 + x_2^{SR * } - \xi } \right) - F\left( {x_1 + x_2^{SR * } } \right)} \right].$$
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Bae, S.H., Ott, A.F. The public economics of self control. J Econ Finance 32, 356–367 (2008). https://doi.org/10.1007/s12197-008-9047-1
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DOI: https://doi.org/10.1007/s12197-008-9047-1