Intertemporal utility with heterogeneous goods and constant elasticity of substitution
Introduction
The intertemporal evaluation and efficient management of heterogeneous goods is a key challenge for economics. This is particularly relevant when assessing long-lived public goods, such as infrastructure or atmospheric carbon. Such evaluation and management often crucially depends on the substitution possibilities between private and public goods. For example, the evaluation of climate policy strongly depends on the degree of substitutability between private consumption and atmospheric carbon as well as on the intertemporal degree of substitutability (e.g. Sterner and Persson, 2008, Traeger, 2011, Drupp and Hänsel, 2020). Parsimony in modeling the effect of substitution possibilities on the economic evaluation and efficient intertemporal allocation suggests assuming a constant elasticity of substitution (CES) between private and public goods, and a constant intertemporal elasticity of substitution for the private good (CIES).
We characterize intertemporal utility functions over two heterogeneous goods, say a private and a public good, when preferences have these two properties1 : (i) The elasticity of substitution between the two goods is constant (but not necessarily identical) at each point in time (CES). (ii) The intertemporal elasticity of substitution for consumption of one of the goods, say the private good, is constant over time (CIES). We show that a standard, stationary intertemporal utility function satisfies these two properties simultaneously if and only if either all elasticities are equal, that is at each point in time the CES is equal to the CIES for the private good, or if the instantaneous utility function is Cobb–Douglas. As a consequence, the intertemporal elasticity of substitution for the public good must also be constant. Thus, except for the case where the instantaneous utility function is Cobb–Douglas, the only intertemporal utility function satisfying (i) and (ii) is a utility function with identical elasticity of substitution for any pair of goods at any two points in time. We call this the intertemporal constant elasticity of substitution (ICES) utility function. The other possibility is that the instantaneous utility function is Cobb–Douglas, i.e. that the elasticity of substitution between the two goods at each point in time equals one. In this case, the constant intertemporal elasticities of substitution for each of the two goods, and between the two goods, may differ. In other words, if the CES between the two goods is fixed to one, there is flexibility in specifying the constant intertemporal elasticities of substitution. We also characterize the families of standard intertemporal utility functions that have either of the two properties (i) or (ii), but not the respective other one.
Our findings are in line with results on constant elasticities of substitution in production technologies with finitely many inputs. McFadden (1963) shows that a linear homogeneous production function must have a nested CES-Cobb–Douglas structure with identical elasticities if one imposes constant Hicks elasticities of substitution between the inputs. The present paper differs from the literature on production functions with constant elasticities of substitution (summarized in Blackorby and Russell, 1989) in several respects: We study preference orderings, i.e. we do not impose linear homogeneity, and we study an infinite time horizon, i.e. infinitely many variables, but focus on two different goods. As a result, the assumption of constant elasticities of substitution allows for either a nested Cobb–Douglas-CIES, or the ICES utility function.
Section snippets
Characterizing intertemporal preferences with constant elasticities of substitution
We study a decision problem where the objects of choice are intertemporal allocations of two goods in discrete time, and with . For concreteness, we refer to the good as ‘private consumption’, and to the good as ‘public good’. We write , , and , . Throughout we assume that the decision maker’s preferences over allocations can be represented by an ordinal, continuous intertemporal utility function ,
Conclusion
We have characterized the intertemporal constant elasticity of substitution (ICES) utility function over heterogeneous goods as the only specification (up to monotone transformations, and except for the case where instantaneous utility is Cobb–Douglas) of a stationary intertemporal utility function that is consistent with the two properties of (i) a constant elasticity of substitution between goods at each point in time and (ii) a constant intertemporal elasticity of substitution for at least
Acknowledgments
Financial support by the German Federal Ministry of Education and Research under Grant 01LC1826A is gratefully acknowledged.
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