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Social time preference revisited


The article reconsiders the implications of the choice of pure social time preference for intergenerational equity in the presence of a time-consistent utilitarian social welfare criterion. The analytic framework is a setting with overlapping generations, lifetime uncertainty, population growth and technical progress. The analysis identifies upper and lower bounds for the feasible range of social discount rates and draws a corresponding distinction between “gerontocratic” and “Stalinist” optimal plans. The paper corrects a number of inaccurate propositions in a related earlier contribution by Marini and Scaramozzino (2000) to this journal.

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  1. It is noted en passant that the corresponding condition “"9"” in M/S contains an error. However, this error is not the source of the inconsistencies in the authors’ reasoning discussed later on.

  2. See, e.g., Mirrlees (1967). See Barro and Sala-i-Martin (1995), 59–92, for a recent textbook exposition.


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Helpful comments by an anonymous referee and an associate editor are gratefully acknowledged.

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Correspondence to Dirk Willenbockel.

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Responsible editor: Alessandro Cigno



Derivation of the indirect utility function U(c) when felicity is iso-elastic

It is assumed that the felicity function u(.) in Eq. (1) takes the iso-elastic form

$$u{\left( {\overline{c} } \right)} = \frac{{\overline{c} ^{{1 - \theta }} }} {{1 - \theta }}\;,\theta > 0\;,\theta \ne 1\;;\;u{\left( {\overline{c} } \right)} = \ln {\left( {\overline{c} } \right)}\;{\text{if}}\;\theta = 1.$$

Evans (2005) provides empirical support for the assumption of a constant elasticity of marginal utility θ=1/σ. Integration of Eq. (8) from th to t yields

$$\overline{c} {\left( {t - h,t} \right)} = \overline{c} {\left( {t,t} \right)}e^{{\sigma {\left( {\delta - \rho } \right)}h}} $$

Inserting (A-2) into the intratemporal consumption constraint, one obtains

$$c{\left( t \right)} = \frac{{C{\left( t \right)}}} {{L{\left( t \right)}}} = \beta {\int_0^\infty {e^{{{\left[ {\sigma {\left( {\delta - \rho } \right)} - \beta } \right]}h}} } }{\text{d}}h = \frac{{\beta c{\left( {t,t} \right)}}} {{\beta - \sigma {\left( {\delta - \rho } \right)}}}\;,$$

provided that β>σ(δρ). If this inequality—which is equivalent to condition (19) in the main text and implies an upper limit for the feasible range of δ—does not hold, the intratemporal consumption allocation rule (8) is not consistent with the intratemporal resource constraint (5) and an optimal plan does not exist for the chosen social discount rate. Combining (A-2) and (A-3), one finds

$$\overline{c} {\left( {t - h,t} \right)} = c{\left( t \right)}\frac{{\beta - \sigma {\left( {\delta - \rho } \right)}}} {\beta }e^{{\sigma {\left( {\delta - \rho } \right)}h}} \;,$$

which is equivalent to (20) in the main text. As U(c) in Eq. (9) is by definition equal to the inner integral of Eq. (3), its functional form is found by inserting (A-4) into

$$U{\left[ {c{\left( t \right)}} \right]} = {\int_0^\infty {\frac{{\overline{c} {\left( {t - h,t} \right)}^{{1 - \theta }} }} {{1 - \theta }}} }e^{{{\left( {\delta - n - \rho - \lambda } \right)}h}} {\text{d}}h\;,$$

and, hence,

$$U{\left[ {c{\left( t \right)}} \right]} = \Phi \frac{{c{\left( t \right)}^{{1 - \theta }} }} {{1 - \theta }}\;,\Phi \equiv {\left( {\beta - \sigma {\left( {\delta - \rho } \right)}} \right)}^{{ - \theta }} \beta ^{{\theta - 1}} .$$

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Willenbockel, D. Social time preference revisited. J Popul Econ 21, 609–622 (2008).

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  • Intergenerational equity
  • Overlapping generations
  • Social discount rate

JEL Classification

  • H43
  • E13
  • O41