Attitudes Towards Income Risk in the Presence of Quantity Constraints

The Geneva Risk and Insurance Review, 2013, 38, (183–209) © 2013 The International Association for the Study of Insurance Economics 1554-964X/13 www.palgrave-journals.com/grir/ Attitudes Towards Income Risk in the Presence of Quantity Constraints* Fred Schroyen Department of Economics, Norwegian School of Economics, Helleveien 30, Bergen N-5045, Norway. E-mail: . Considering a consumer with standard preferences, I trace out how quantity constraints on markets impact on relative risk aversion and prudence. I first show how this impact decomposes into a local curvature effect and an endogenously changing risk aversion/ prudence effect. Next, I calibrate both effects on relative risk aversion and prudence, using estimates on household demand for durables and labour supply. The calibrations show that commitments to durable goods have large effects on attitudes towards risk. And while small wedges between realised and desired levels of labour supply have only moderate effects, becoming full time unemployed on a 60 per cent unemployment benefit significantly raises risk aversion and prudence. The Geneva Risk and Insurance Review (2013) 38, 183–209. doi:10.1057/grir.2013.3 Keywords: household demand; income risk aversion; prudence; quantity constraints; labour supply; durables Article submitted 19 February 2013; accepted 23 September 2013; published online November 2013 Introduction How are a consumer’s attitudes towards income risk affected when her trading opportunities get restricted because of quantity constraints, such as having to work full-time while wishing a part-time job (or vice versa), or being stuck with a small car, when in need for a large one? I consider two types of attitude towards risk: (i) risk aversion and (ii) prudence or downside risk aversion. The Arrow-Pratt coefficient of risk aversion measures a consumer’s willingness to pay for disposing of * This paper is a substantial revision of an earlier paper titled “Income risk aversion with quantity constraints”. I am especially grateful to Jacques H Drèze for very stimulating discussions and useful suggestions, and to the Editor (Achim Wambach) and two anonymous referees for their detailed comments. Also thanks to Louis Eeckhoudt for fruitful discussions. The comments by Kåre Petter Hagen, Julien Hardelin, Karl Rolf Pedersen, Agnar Sandmo, and Dirk Schindler on an earlier version are gratefully acknowledged. Part of the paper was written during a very hospitable research stay at CORE (Louvain-la-Neuve, Belgium). The Geneva Risk and Insurance Review 184 a zero mean risk. Likewise, Kimball’s coefficient of prudence measures the certain reduction in income required to bring the marginal utility of consumption in line with the expected marginal utility of consumption when a zero mean risk is added.1 Intuitively, one would expect that quantity constraints make a consumer both more risk averse and more prudent, since they reduce the opportunity set and thus allow for smaller adjustments of the consumption bundle after the income risk has realised. For example, this is true when the consumer’s utility function over consumption (c) and leisure (l) is u(c, l) = v(c)+l, with v′, v′″>0 and v″<0. Because preferences are quasilinear in leisure, all exogenous income risk is absorbed by leisure. Since also the utility function is linear in leisure, the consumer is risk neutral with respect to this income risk and exhibits zero prudence. But if she faces a binding quantity constraint on her labour supply, the exogenous income risk is absorbed by the consumption of other goods, whose marginal utility is strictly falling and convex. Hence, the constraint turns the consumer into a strictly risk-averse and prudent person with respect to income risk. This example points to the central role of the income elasticity of the constrained good in determining whether risk attitudes are affected by constraints. At the same time, it shuts down a number of channels that are important to determine the size and sign of the effects. In this paper, I characterise these effects making use of the virtual price and income approach introduced by Neary and Roberts.2 They show how constrained consumer behaviour under certainty may be analysed using standard tools by redefining the price vector and income as those that support the optimal bundle under quantity constraints. I show that under a quantity constraint, an income risk translates into a virtual price and a virtual income risk, both of which are positively correlated (under normality). I also show that the positive net evaluation of the price risk (due to substitution possibilities) is more than offset by the negative net evaluation of the correlation between both risks (due to the income effect). The result is a mark-up on the risk premium compared with an unconstrained situation. At least for weakly binding constraints, this mark-up is positive. A similar mark-up applies for the prudence premium, though this time its sign is a priori undetermined. 1 Since the solution to the consumer's decision problem under uncertainty will satisfy a condition on the expected marginal utility of consumption, the optimal response to changes in the zero mean risk will depend on the sign and size of the prudence coefficient. Hence, this coefficient measures “the propensity to prepare or forearm oneself in the face of uncertainty” (Kimball, 1990, p. 54). The notion of prudence was originally defined in a temporal context (savings decision). It is predated by the concept of downside risk aversion, defined by Menezes et al. (1980), which is often used in the atemporal context. A downside risk-averse decision maker dislikes a mean-preserving spread that is followed by a mean-preserving contraction on the right of the spread that restores the variance to its original value. Menezes et al. show that for an EU-decision maker, this is equivalent to u′″>0. For brevity, I use prudence throughout in the paper. 2 Neary and Roberts (1980). Fred Schroyen Attitudes Towards Income Risk in the Presence of Quantity Constraints 185 The size of these mark-ups depends on the behavioural elasticities and the extent to which the quantity constraint is binding. To assess their empirical importance, I calibrate them for two settings: commitments to durable goods (for the U.S.) and hour constraints on the labour market (for the U.S., the U.K. and Sweden). For durable goods, I find that small frictions in the adjustment to their desired level have positive effects on relative risk aversion and negative effects on relative prudence. I also find that these effects are very non-linear, both in the level of the constraint and in income. The effects of restrictions on labour supply are more modest, although “large” restrictions such as being full-time unemployed do boost both risk attitudes. I believe there are several reasons that call for a better understanding of the effects of constraints on attitudes towards risk. First, the level (...truncated)