# Comparative precautionary saving under higher-order risk and recursive utility

## Abstract

Measuring and comparing the precautionary saving motive rest almost exclusively on the expected utility framework, and only focus on income risk or coefficients of the Arrow–Pratt type. We generalize the standard approach by characterizing comparative precautionary saving under recursive utility for increases in income risk and increases in risk on the saving return, including higher-order risk effects. We express the comparisons in terms of precautionary premia. In addition, we define a new class of preference coefficients, and derive the associated conditions to predict a stronger precautionary motive. The coefficients provide a detailed picture of the preferences sustaining precautionary saving and could be useful in applications.

## Keywords

Precautionary saving Prudence Recursive utility Higher-order risk Precautionary premium Preference coefficient## JEL Classification

D91 D81## Notes

### Acknowledgements

We are grateful for valuable comments made by Sebastian Ebert and two anonymous reviewers, as well as Arthur Charpentier, Louis Eeckhoudt, Rachel Huang, Olivier l’Haridon, Liqun Liu, Debrah Meloso, Richard Peter, Art Snow, and participants in the 2016 International Conference on Economic and Financial Risks in Niort, the CEAR/MRIC Behavioral Insurance Workshop in Munich, and the 2017 EGRIE seminar. This research was partially supported by the European Union’s Seventh Framework Programme FP7/2007-2011, Grant Agreement no. 290693 (FOODSECURE), and by the William J. Fulbright Foundation.

## References

- Attanasio, O.P., and G. Weber. 2010. Consumption and saving: Models of intertemporal allocation and their implications for public policy.
*Journal of Economic Literature*48: 693–751.CrossRefGoogle Scholar - Bansal, R., and A. Yaron. 2004. Risks for the long run: A potential resolution of asset pricing puzzles.
*Journal of Finance*59: 1481–1509.CrossRefGoogle Scholar - Barro, R.J. 2009. Rare disasters asset prices, and welfare costs.
*American Economic Review*99: 243–264.CrossRefGoogle Scholar - Binsbergen, J.V., J. Fernández-Villaverde, R. Koijen, and J. Rubio-Ramírez. 2012. The term structure of interest rates in a DSGE model with recursive preferences.
*Journal of Monetary Economics*59: 634–648.CrossRefGoogle Scholar - Bostian, A.A. and Heinzel, C. 2018. Consumption smoothing and precautionary saving under recursive preferences. Working paper.Google Scholar
- Briys, E., L. Eeckhoudt, and H. Loubergé. 1989. Endogenous risks and the risk premium.
*Theory and Decision*26: 37–46.CrossRefGoogle Scholar - Capéraà, P., and L. Eeckhoudt. 1975. Delayed risks and risk premiums.
*Journal of Financial Economics*2: 309–320.CrossRefGoogle Scholar - Chiu, W.H. 2005. Skewness preference, risk aversion, and the precedence relations on stochastic changes.
*Management Science*51: 1816–1828.CrossRefGoogle Scholar - Choi, H., S. Lugauer, and N.C. Mark. 2017. Precautionary saving of Chinese and U.S. households.
*Journal of Money, Credit and Banking*49: 635–661.CrossRefGoogle Scholar - Denuit, M.M., and L. Eeckhoudt. 2010a. Stronger measures of higher-order risk attitudes.
*Journal of Economic Theory*145: 2027–2036.CrossRefGoogle Scholar - Denuit, M.M., and L. Eeckhoudt. 2010b. A general index of absolute risk attitude.
*Management Science*56: 712–715.CrossRefGoogle Scholar - Denuit, M.M., and L. Eeckhoudt. 2013. Risk attitudes and the value of risk transformations.
*International Journal of Economic Theory*9: 245–254.CrossRefGoogle Scholar - Drèze, J.H., and F. Modigliani. 1972. Consumption decisions under uncertainty.
*Journal of Economic Theory*5: 308–335.CrossRefGoogle Scholar - Eeckhoudt, L., and H. Schlesinger. 2008. Changes in risk and the demand for saving.
*Journal of Monetary Economics*55: 1329–1336.CrossRefGoogle Scholar - Eeckhoudt, L., and H. Schlesinger. 2009. On the utility premium of Friedman and Savage.
*Economics Letters*105: 46–48.CrossRefGoogle Scholar - Ekern, S. 1980. Increasing Nth degree risk.
*Economics Letters*6: 329–333.CrossRefGoogle Scholar - Epstein, L.G., E. Farhi, and T. Strzalecki. 2014. How much would you pay to resolve long-run risk?
*American Economic Review*104: 2680–2697.CrossRefGoogle Scholar - Epstein, L.G., and S.E. Zin. 1989. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework.
*Econometrica*57: 937–969.CrossRefGoogle Scholar - Epstein, L.G., and S.E. Zin. 1991. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: An empirical analysis.
*Journal of Political Economy*99: 263–286.CrossRefGoogle Scholar - Gollier, C. 2001.
*The Economics of Risk and Time*. Cambridge, MA: MIT Press.Google Scholar - Gollier, C. 2013.
*Pricing the Planet’s Future: The Economics of Discounting in an Uncertain World*. Princeton, NJ: Princeton University Press.Google Scholar - Gomes, F., and A. Michaelides. 2005. Optimal life-cycle asset allocation: Understanding the empirical evidence.
*Journal of Finance*60: 869–904.CrossRefGoogle Scholar - Gourinchas, P.-O., and J.A. Parker. 2001. The empirical importance of precautionary saving.
*American Economic Review*91: 406–412.CrossRefGoogle Scholar - Hall, R.E. 1988. Intertemporal substitution in consumption.
*Journal of Political Economy*96: 339–357.CrossRefGoogle Scholar - Jindapon, P., and W.S. Neilson. 2007. Higher-order generalizations of Arrow-Pratt and Ross risk aversion: A comparative statics approach.
*Journal of Economic Theory*136: 719–728.CrossRefGoogle Scholar - Kihlstrom, R.E., D. Romer, and S. Williams. 1981. Risk aversion with random initial wealth.
*Econometrica*49: 911–920.CrossRefGoogle Scholar - Kimball, M.S. 1990. Precautionary saving in the small and in the large.
*Econometrica*58: 53–73.CrossRefGoogle Scholar - Kimball, M., and P. Weil. 2009. Precautionary saving and consumption smoothing across time and possibilities.
*Journal of Money, Credit and Banking*41: 245–284.CrossRefGoogle Scholar - Kreps, D.M., and E.L. Porteus. 1978. Temporal resolution of uncertainty and dynamic choice theory.
*Econometrica*46: 185–200.CrossRefGoogle Scholar - Langlais, E. 1995. A measure of the sensitivity of saving to interest rate uncertainty with non-expected preferences.
*Economics Letters*48: 325–330.CrossRefGoogle Scholar - Li, J. 2009. Comparative higher-degree Ross risk aversion.
*Insurance: Mathematics and Economics*45: 333–336.Google Scholar - Liu, L. 2014. Precautionary saving in the large: Nth degree deteriorations in future income.
*Journal of Mathematical Economics*52: 169–172.CrossRefGoogle Scholar - Liu, L., and J. Meyer. 2013a. Normalized measures of concavity and Ross’s strongly more risk averse order.
*Journal of Risk and Uncertainty*47: 185–198.CrossRefGoogle Scholar - Liu, L., and J. Meyer. 2013b. Substituting one risk increase for another: A method for measuring risk aversion.
*Journal of Economic Theory*148: 2706–2718.CrossRefGoogle Scholar - Lugilde, A., R. Bande, and D. Riveiro. 2017. Precautionary saving: A review of the theory and the evidence. MPRA Paper No. 77511.Google Scholar
- Martin, I.W. 2013. Consumption-based asset pricing with higher cumulants.
*Review of Economic Studies*80: 745–773.CrossRefGoogle Scholar - Menezes, C.E., C. Geiss, and J. Tressler. 1980. Increasing downside risk.
*American Economic Review*70: 921–932.Google Scholar - Menezes, C.F., and X.H. Wang. 2005. Increasing outer risk.
*Journal of Mathematical Economics*41: 875–886.CrossRefGoogle Scholar - Mirrlees, S.J., S. Adam, T. Besley, R. Blundell, S. Bond, R. Chote, M. Gammie, P. Johnson, G. Myles, and J. Poterba. 2010.
*Dimensions of tax design: The Mirrlees review*. Oxford: Oxford University Press.Google Scholar - Ross, S.A. 1981. Some stronger measures of risk aversion in the small and the large with applications.
*Econometrica*49: 621–638.CrossRefGoogle Scholar - Rothschild, M., and J.E. Stiglitz. 1970. Increasing risk: I. a definition.
*Journal of Economic Theory*2: 225–243.CrossRefGoogle Scholar - Selden, L. 1978. “Certain x Uncertain” consumption pairs: The ordinal certainty equivalent hypothesis.
*Econometrica*46: 1045–1060.CrossRefGoogle Scholar - Tallarini, T.D. 2000. Risk-sensitive real business cycles.
*Journal of Monetary Economics*45: 507–532.CrossRefGoogle Scholar - Weil, P. 1989. The equity premium puzzle and the risk-free rate puzzle.
*Journal of Monetary Economics*24: 401–421.CrossRefGoogle Scholar - Weil, P. 1990. Nonexpected utility in macroeconomics.
*Quarterly Journal of Economics*105: 29–42.CrossRefGoogle Scholar