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Conditional Impatience and Concavity of Consumption Functions

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  • Alexis Akira Toda

Abstract

Concave consumption functions imply a marginal propensity to consume that falls with wealth. I characterize the utility functions that guarantee this property in finite-horizon optimal saving problems with stochastic discounting, returns, income, and borrowing limits. Under conditional impatience---the conditional expected discounted gross return does not exceed one---consumption functions are always concave if and only if inverse absolute prudence, $-u''/u'''$, is concave. When no conditional-impatience restriction is imposed, hyperbolic absolute risk aversion (HARA) is necessary and sufficient for uniform concavity. Thus conditional impatience permits declining marginal propensities to consume for a preference class strictly larger than HARA.

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  • Alexis Akira Toda, 2026. "Conditional Impatience and Concavity of Consumption Functions," Papers 2608.29488, arXiv.org, revised Sep 2026.
    • Handle: RePEc:arx:papers:2608.29488
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    References listed on IDEAS

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    1. Toda, Alexis Akira, 2021. "Necessity of hyperbolic absolute risk aversion for the concavity of consumption functions," Journal of Mathematical Economics, Elsevier, vol. 94(C).
    2. Stachurski, John & Toda, Alexis Akira, 2019. "An impossibility theorem for wealth in heterogeneous-agent models with limited heterogeneity," Journal of Economic Theory, Elsevier, vol. 182(C), pages 1-24.
    3. Park, Myung-Ho, 2006. "An analytical solution to the inverse consumption function with liquidity constraints," Economics Letters, Elsevier, vol. 92(3), pages 389-394, September.
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