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 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.

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 quasi-linear 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.Footnote 1 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.

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 of relative risk aversion and prudence governs the response of an expected utility (EU) decision maker when adjusting the fraction of initial wealth to be invested in a risky asset, or the level of savings when the rate of return gets more risky.Footnote 2 In times where underemployment equilibria prevail, the investment responses of consumer/workers to increased uncertainty on financial markets cannot be expected to be identical when labour markets clear at full employment levels. Moreover, if the anticipation of future quantity constraints impacts on the degrees of risk aversion and prudence, this will trigger changes in the optimal amount of savings of which the macroeconomic effects may confirm the expectation. A better understanding of these impacts is therefore relevant for studying the nature of underemployment equilibria.

Second, in recent years several studies have measured the empirical variation in attitudes towards risk and explained this variation in terms of socioeconomic characteristics of the decision maker. See, for example, Barsky et al.,Footnote 3 Dohmen et al.,Footnote 4 Guiso and Paiella,Footnote 5 and Aarbu and SchroyenFootnote 6 for risk aversion, and Deck and SchlesingerFootnote 7 for prudence. A salient feature of these studies is that observable characteristics leave a large part of the variation in risk attitudes unexplained. I will argue that one source of unobserved heterogeneity may stem from frictions that people experience in adjusting themselves to desired levels of quantities of goods and services. For example, in the calibration part of the paper, I show that small deviations from the optimal level of durable goods may produce large variation in relative risk aversion around its sample mean. Another example are labour market frictions. Dohmen et al.Footnote 8 analyse the responses of 22,000 respondents in the 2004 wave of the German Socioeconomic Panel to the hypothetical question how much of a €100,000 windfall gain one chooses to invest in a hypothetical asset which in two years’ time either doubles the invested amount or returns only half of it, both with 50 per cent probability. The answer (on average about €15,000) is regressed on gender, age, height, parental education and a large set of controls, including household income, household wealth and employment status. The unemployment dummy has a coefficient of €−4,938.2 (st error €2,222.5, p<5 per cent—cf. their Table A.1 (column 9)). Since income is controlled for, this result indicates that being unemployed in itself significantly decreases the willingness to take financial risk and begs the question through which mechanism such restriction on labour supply affects risk attitudes. In the calibration part, I show that despite the low income elasticity of labour supply, the prospect of full-time unemployment may more than double both relative risk aversion and relative prudence—even under constant relative risk aversion (CRRA).

The paper is related to a recent literature that investigates how commitments affect risk taking behaviour or the willingness to take risk and the normative implications this may have for contract design. For example, Chetty and SzeidlFootnote 9 show that when an EU decision maker finds it optimal to follow a sluggish adjustment policy w.r.t. housing or some other lumpy good, his indirect utility function becomes more concave in some income regions, but convex in others (compared with full flexibility). They explore how this may help to reconcile a number of empirical puzzles, such as the simultaneous purchasing of insurance and lottery tickets, or the presence of substantial aversion towards moderate gambles without implying unrealistically high aversion towards large gambles.Footnote 10

Finally, the paper is related to the literature on timeless vs temporal risky prospects. This literature asks whether an agent who can choose a lottery and take some action after observing the outcome of the lottery (a timeless prospect) has a larger willingness to bear risks than an agent who has to commit to an action before observing the lottery outcome (a temporal prospect). MachinaFootnote 11 has shown through an example that this is not always the case. GollierFootnote 12 has recently generalised this result and identified a set of sufficient conditions for the flexible context to lead to a higher risk tolerance. He then examines how rigidities may induce a household to more risk-prone behaviour in portfolio allocation and/or savings decisions. In the present paper, the focus is on timeless risks. I examine the effect of one particular set of constraints—quantity constraints on purchased levels of goods and services—on the willingness to accept small income risks, and decompose it in terms of consumer preferences. To the best of my knowledge, this paper is also the first to examine the consequences of constraints for the decision maker’s rate of prudence and thus her willingness to change precautionary behaviour when background risk increases.

The next section briefly reminds of the consumer’s decision problem and its properties, and formulates the coefficients of risk aversion and prudence with respect to income risk in terms of the direct utility function. In the subsequent section, I introduce quantity constraints and derive their effect on relative risk aversion using the virtual price approach. In the latter section, I derive the effects for relative prudence. Using estimation results for the demand for durables (U.S.) and the supply of labour (U.K., U.S. and Sweden), I calibrate in the penultimate section the effects of constraints on risk attitudes. The final section concludes.

Income risk aversion and prudence without quantity constraints

A consumer cares about n commodities whose quantities are given by the bundle q, belonging to the consumption set X⊂R+n. She has well-defined preferences over simple lotteries on X, which can be represented by a vNM utility function u(·), which is smooth, monotone and strongly concave (implying that she prefers the certain bundle to the uncertain consumption prospect Let the price vector be certain and given by p∈R+ n . The consumer’s income is random and given by with Thus it has expectation m and variance

The consumer is informed about the income draw before she makes her consumption decision. In other words, she faces a timeless income prospect: the decision on q is made after the uncertainty is resolved. With an income draw her problem is then to solve

Let the unique solution be given by the bundle yielding a utility level The function v(p,·) is the vNM utility function representing the consumer’s preferences for timeless income lotteries.Footnote 13 Then this solution satisfies the first-order conditionsFootnote 14

The local properties of are well known but repeated here for future reference. Defining K as the matrix of Slutsky substitution effects and q m as the vector of income effects, we have:

Expression (2(ii)) is the Slutsky decomposition. A similar decomposition of the price effect on the marginal utility of income, v m , is

The first right-hand side term is a real income effect that can be neutralised by an appropriate change in income. The second right-hand side term is a substitution effect: the change in the marginal utility of income when the consumer is compensated so as to remain at the same utility level.

Differentiating both sides of (1) with respect to , and making use of the adding-up property (2(i)) gives v mm =q′ m u qq q m , which is negative by the concavity of u(·). Since v m =q′ m u q , the Arrow-Pratt coefficient of relative risk aversion is given by

The rhs expression may be added to Hanoch’s list of alternative representations of relative risk aversion (Hanoch,Footnote 15 Theorem 1). It is twice the relative risk premium per unit of variance for infinitesimal proportional timeless income risk.

When the consumer faces an uninsurable income risk but can take actions to mitigate it, Kimball’sFootnote 16 coefficient of relative prudence, RP, measures the sensitivity of these actions to the risk.Footnote 17 For a consumer who cares about many goods, one expects this coefficient, defined as to depend on the set of third (cross) derivatives of the utility function u(·). This is indeed the case. In the appendix, I show that

where is the effect on the quadratic form q′ m u qq q m because of a perturbation in the Hessian u qq following dq.Footnote 18 Expression (5) thus reveals that the coefficient of relative prudence for income risk can be expressed as the ratio of a cubic form in the (three dimensional) array of third derivatives of u(·) to a quadratic form in the Hessian of u(·).

Proposition 1

  • When a consumer has a utility function u(·) defined over n commodities, the coefficients of relative risk aversion and prudence for timeless income risks are given by (4) and (5), respectively.

For future reference, it is useful to establish how a compensated increase in the price of good i affects these attitudes towards risk. In the appendix, I prove the following result:

Theorem 1

  • The m-compensated effects of p i on RR and RP are

The term (∂2q i /∂m2)(m2/q i ) measures the curvature of the Engel curve for good i; (∂3q i /∂m3)(m3/q i ) is the third-order counterpart. If (6) [(7)] is zero for all i, then relative risk aversion [prudence] is constant along the indifference curve (but can vary along an Engel curve).Footnote 19 Clearly, this will be the case with homothetic preferences (when all Engel curves are straight lines through the origin), but in general, attitudes towards risk will not be constant along the indifference curve. The important implication is that, even when these attitudes are constant w.r.t. income (e.g. because of a CRRA assumption), a change in the relative price structure will affect both of them in a non-trivial way.

Effects of quantity constraints on relative risk aversion

Virtual income and price risk

Suppose now that , , where z is the quantity of a single good or service which is no longer at the consumer’s discretion because it is fixed at z̄.Footnote 20 Her problem when observing the income draw then turns into

Let the solution be given by and the indirect utility function defined as Repeating the procedure of the previous section, the coefficient of relative risk aversion for income risk is given by

In order to relate to RR(p, m), I will use the “virtual price” approach of Neary and Roberts.2 This consists in defining a virtual price , and adjusting the consumer’s income to the virtual income level such that the consumer’s notional demand for the z-good coincides with the imposed quantity z̄. That is,

This is illustrated by Figure 1 for the case of two normal goods. The Engel curve is given by EE. Panel (a) sketches a situation of a consumer who faces the timeless income prospect (m L , m H ; 1/2, 1/2) with expectation m but cannot consume more than z̄ units of the second good. Both under the high-income state (m H ) and the low-income state (m L ), she feels rationed Her choice of can be rationalised by a large virtual price π zH and a virtual income Likewise, (π zL , μ L ) supports the constrained choice . With z being a normal good, the virtual price and income are positively correlated. Panel (b) depicts a situation of forced consumption (). Again, the supporting virtual price and income are positively correlated. The figures also show the virtual price (π z ) and income (μ) that support the choice when receiving the mean income m—if unconstrained, z* is demanded.

Figure 1
figure 1

Income risk {m L , m H } translates into a virtual price risk {π zL , π zH } and a virtual income risk {μ L , μ H }: (a) rationing; (b) forced consumption.

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Implicitly differentiating (9) and using the Slutsky equation (2(ii)) shows that

where is the income effect for z and k zz is the own compensated price effect, that is, Intuitively, the consumer would like to respond to a marginal income shock by However, because the quantity constraint prevents this, the virtual price of that good has to rise by and virtual income by such that she is exactly compensated for the virtual price rise. The constraint translates the income risk into a virtual income and price risk.

Figure 1 clearly shows that with a binding constraint, utility in each state (and therefore EU) is lower than when unconstrained, that is, than the utilities corresponding to the indifference curves going through the intersection of the Engel curve and the budget lines m L and m H (not drawn). The fact that the consumer gets worse off is going to affect her relative risk attitudes to the extent that RR and RP depend on real income. For example, under increasing relative risk aversion (IRRA), this income effect of rationing will lower RR. Later in the empirical section, I am going to assume away this “standard” effect by imposing CRRA. As I will show next, what remains are ordinal effects (unrelated to RR and RP) and substitution effects on RR and RP (defined in Theorem 1).

A decomposition result

The marginal utility of income in the constrained situation is given by

where the equality sign follows from Roy’s identity. Differentiating one more time with respect to yields

where the second line follows from (3).Footnote 21 Let (π z , μ) be the virtual price and income combination corresponding to the mean income m. Then use of (12) into (13) and the definition (8) of leads to:

Proposition 2

  • When facing the quantity constraint z̄, the degree of relative risk aversion can be decomposed into a virtual degree of relative risk aversion and a positive ordinal term:

The term RR(p x , π z , μ) on the right-hand side of (14) can be coined the virtual coefficient of relative risk aversion. The second rhs term is a mark-up that is strictly positive (since k zz <0) and entirely ordinal. Multiplying and dividing the second term by and defining the income and compensated price elasticity of z as and and the virtual budget share as (14) may also be written as

The mark-up term w z (η z 2/ɛ zz ) is ordinal because all of its components are derived from the demand function z(·) which is in principle observable from market behaviour—it is invariant to any positive monotone transformation of the cardinal utility function u(x, z), and thus reflects the preference order (under certainty).

It remains to show that (14) amounts to twice the risk premium (as a fraction of m) per unit of variance in case the consumer is facing a small proportional risk around the expected income m. In the appendix, I show that (12) allows me to approximate the normalised variances of and , and their covariance, as

Taking expectations of a second-order Taylor expansion for around (p x , π z , μ) gives

I show in the appendix that the valuations of the variance of the price risk and of the covariance are

Thus the consumer is correlation loving (averse) to the extent that RR exceeds η z . Income risk aversion itself makes the consumer price risk averse, but then a positive income elasticity and a negative Hicksian price elasticity makes her price risk loving. The last result is intuitive: the possibility to substitute in response to price shocks is desirable. Using (16)–(18) and rearranging then results in

To complete the argument, I mimic the utility of the certainty equivalent income, as v(px, π z CE, μCE) (cf. (11)) and take a first-order Taylor expansion around (p x , π z , μ):

where the second line follows from the definition of the virtual certainty equivalent income:

Equating the rhs of (21) and (22) and rearranging then gives

The underlined term is exactly as derived in (15), and hence may be interpreted as the coefficient of relative risk aversion when facing the constraint z̄

Discussion

There are two things worth noticing about First, all terms in the round bracket expressions (15) or (23) are evaluated at the virtual price and income that support the consumer’s decision when facing the mean income m and the constraint z̄: (p x , π z , μ). The relative virtual price will exceed its nominal counterpart in case of rationing (—the case shown in Figure 1(a)) and fall short of it in case of forced consumption (—Figure 1(b)). For the same reason will virtual income exceed or fall short of nominal income. The extent to which the quantity constraint is binding will thus be reflected by the discrepancy between (p x , π z , μ) and (p x , p z , m), and in this way influence the degree of relative risk aversion. Second, the entire round bracket term is multiplied with a factor m/μ. Again, this factor is larger than 1 in case of forced consumption and below one in case of rationing. The reason for this scale effect is our interest in the risk premium as a fraction of nominal income, not of virtual income.

A special case is a weakly binding constraint: when the quantity z̄ exactly coincides with the quantity that the consumer wishes to consume at (p x , p z , m). I denote this level by z*, that is, z*=z(p x , p z , m). In this case, π z =p z and μ=m, and

This expression illustrates the graphical effect that the constraint makes the maximum value function more concave in the neighbourhood of income for which the desired quantity z* coincides with the imposed level z̄. I therefore call the second rhs term the local curvature effect.Footnote 22

Once the constraint becomes strictly binding, virtual price and income start to deviate from their nominal counterparts. The difference can be decomposed as

Thus the difference consists of the local curvature effect and a second effect that I will term the endogenously changing risk aversion effect. Since , all three rhs terms depend on π z and μ which will be affected by ζ. Consider the effect on the first term:

Thus implies a change in the relative price structure that has a substitution effect on RR (1st rhs term). At the end of the previous section, I argued that in general, this substitution effect will not vanish. Second, except for CRRA preferences when ∂RR(·)/∂μ≡0, there is a real income effect on RR (2nd rhs term): a rationed consumer (, π z >p z ) will get better off when z̄ is increased, and under IRRA turn more risk averse. In addition, there is the effect on the mark-up (clearly depending on higher order behavioural effects), and finally that on the scaling factor . It is only the last effect that may be signed unambiguously (positive). Thus I will conclude that both the sign and size of the endogenously changing risk aversion effect is an empirical issue, and I will document it in the section on calibration. Before doing so, I briefly discuss the effect on relative prudence.

Relative prudence under a quantity constraint

In the appendix, it is shown that

where τ is a dimensionless term collecting all “second-order” ordinal responses:

Then taking the ratio and evaluating at mean income m (and therefore at (π z ,μ)), gives:

Proposition 3

  • When facing a quantity constraint, the coefficient of relative prudence is given by

where all terms in the large round bracket are evaluated at (p x , π z , μ), the terms of trade that mimic the decision when facing mean income m and being constrained at z̄.

Thus a positive sign of (26) indicates that a constrained consumer disproves of the income shock being replaced by a shock that is the result of a mean preserving spread on the left followed by a mean preserving contraction on the right to keep the variance fixed.Footnote 23

As with relative risk aversion, I decompose the effect of a constraint as

The square bracket term is a local curvature effect (but now of the marginal utility of income function), while the next term is an endogenously changing prudence effect. The former effect is no longer unambiguously positive. This can be seen by evaluating the rhs of (26) at (p x , p z , m) and subtracting RP*=RP(p, m):

The important thing to notice is that this local curvature effect for prudence, as that for relative risk aversion, vanishes when the income effect is zero (since then η z =τ=0).

Again, there are four channels that make up the endogenously changing prudence effect: compensated price effects on RR and RP, real income effects on RR and RP, effects on the ordinal mark-up terms w z (η z 2/ɛ zz ) and τ, and an effect on the scaling factor m/μ. I now turn to the calibration exercises.

Calibrations of the effects on RR and RP.

I will look at two settings. The first is one where the consumer cares about non-durables and durables. The second is a model of labour supply. I am then interested in calibrating the effects of constraints on durable goods or on labour supply on the coefficients of relative risk aversion and prudence.Footnote 24

Durable goods

In a recent paper, PakošFootnote 25 estimates the Euler equation and the intraperiod first-order condition corresponding to an intertemporal utility maximisation problem where the period utility index is a generalised CES function defined over the consumption of non-durables (x) and the service flow of durables, assumed to be proportional to the stock of durables (z). In my notation, the period-CES index is given by

with α,σ,θ>0. It can be shown that the elasticity of intraperiod substitution is given by σ+ɛ zz (θ−1). If θ=1, preferences reduce to standard homothetic CES preferences and both income elasticities are equal to 1. If θ<1, durables have a (non-constant) income elasticity larger than 1.

Using aggregate quarterly data for the U.S. (1955:1-2001:4) compiled by Yogo,Footnote 26,Footnote 27 Pakoš obtains superconsistent estimates for σ and θ by estimating the intratemporal first-order condition as a co-integrating regression.Footnote 28 This regression picks up long-run co-movements in durables and non-durables, and thus reflect behaviour that is unconstrained by adjustment restrictions and costs. The remaining parameters are estimated from the Euler equation, conditional upon the consistent estimates for σ and θ.Footnote 29 Since is just a utility index representing the preference order over bundles under certainty, it does not necessarily represent preferences over uncertain consumption prospects. Thus I take a concave iso-elastic transformation by defining the vNM index as , and set γ=2. This choice of γ implies a value of RR* of about 2, a value frequently found in the literature (the results of a sensitivity analysis w.r.t. γ is discussed below).Footnote 30

Table 1, panel (a) gives the parameter estimates and sample means (x*, z*) for non-durables and the stock of durables. Panel (b) gives the relative price (p z *) and real income (m*) that support (x*, z*) as an optimal solution, as well as the corresponding budget share, the income and own Hicksian price elasticity for durables and the second-order behavioural elasticity τ*.Footnote 31 Panel (c) gives the attitudes towards risk when unconstrained, and with a weakly binding quantity constraint on durables (i.e. ). It also gives the local curvature effects.

Table 1 Summary of the durable goods model and effects on RR and RP of weakly binding quantity constraints on durable flows
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The reason RR* is slightly lower than 2 is the non-homotheticity of the CES index (θ<1). For the same reason RP* is not exactly 3. The local curvature effects on risk attitudes (panel (c)) are very strong: the mere impossibility the adjust durables in response to small income shocks almost doubles relative risk aversion, and almost reduces relative prudence to a sixth. This reduction is due to the significant size of τ*. To gauge the endogenously changing risk aversion and prudence effects, I have plotted in Figure 2(a), where I vary z̄ from 80 per cent to 120 per cent of the notional level z*. Being constrained at 90 per cent of the optimal flow of durable services further increases relative risk aversion to about 5. On the other hand, being rationed at 90 per cent again increases relative prudence, but still to only a third of its unconstrained value. Figure 2(a) also shows that is about 10 ((5−3)/(1.1−0.9)). The empirical implication is that a sample of individuals, all identical except for some small heterogeneity in their degree of adjustment to the desired level of durables may display a considerable degree of heterogeneity in relative risk aversion. The U-shape for in panel (a) also suggests that an EU saver who has recently optimised w.r.t. z (and therefore is stuck at a z̄ close to z*) reacts more fiercely to an increase in risk in the rate of return on savings (by saving less) than another one, stuck at far more suboptimal levels of z.Footnote 32

Figure 2
figure 2

RR*=2 and RP*=3: (a) RR(p, m*|φz*) and RP(p, m*|φz*); (b) RR(p, φm*|z*) and RP(p, φm*|z*).

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In Figure 2(b), I keep fixed z̄ at z* and change mean disposable income from 80 per cent to 120 per cent of the value that supports z* (i.e. m*=3.395). When unconstrained, relative risk aversion is almost constant w.r.t. income (the bold dashed line is slightly falling), but the figure shows that it becomes strongly increasing in income when constrained at z*. Relative prudence, on the other hand, is non-monotone in income—again, if unconstrained this risk attitude would be almost constant around 3.

In the appendix, I have drawn similar figures for γ=1, which corresponds to RR*=1.03 and RP*=2.02, and for γ=4.2, which corresponds to RR*=4.04 and RP*=5.06 (see Figures A1 and A2).Footnote 33 While the qualitative properties of the curves are preserved, the main difference is that with a low baseline level for RR* and RP*, relative prudence turns negative when constrained—the consumer turns imprudent or downside risk loving. With γ=4.2, the local curvature effect for relative prudence is still negative, but much smaller. A back of the envelope calculation shows that as γ becomes large, LCE RP approaches twice the value of LCE RR , and thus positive.Footnote 34

I believe these results warrant the conclusion that risk attitudes are very sensitive to the presence of consumption commitments, and that small heterogeneity in the possibility to adjust may result in large heterogeneity in relative risk aversion, and in market behaviour governed by RR and RP. I next turn to labour supply restrictions.

Labour supply

The second calibration is based on estimates of the labour supply equation

where a is the real wage rate and m denotes non-labour income. This equation has been estimated by Blundell et al.Footnote 35 for U.K. married or cohabitating women, by de Linde LeonardFootnote 36 for U.S. male heads of household, and by Flood and MaCurdyFootnote 37 for Swedish married men.

SternFootnote 38 showed that the above labour supply equation integrates to the following utility index:

where Ei is the standard exponential integral, and a(c, L) is the reservation wage when consuming c and supplying L hours.Footnote 39 I define the vNM utility function as The indirect utility function is then given by

Again I assume γ=2, and calibrate the constant ν such that the “unconstrained” coefficient of relative risk aversion RR*=−(v mm /v m )m* attains the value 2 at (a*, m*), the sample mean values for m and a. This also ensures that RP*=3. The sample means for the wage rate, hours worked and non-labour income, as well as the parameter estimates and the income and compensated wage elasticities are given in Table 2.Footnote 40

Table 2 Sample means, parameters and implied elasticities for labour supply studies
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Using the information in Table 2, I have calculated the effects of labour supply constraints on relative risk aversion and prudence. The results are given in Table 3. Unlike for durables, constraints on labour supply hardly affect attitudes towards risk. The local curvature effects are strongest for U.S. male workers: 0.17 for relative risk aversion and 0.24 for relative prudence. The reason these effects are small are the low-income elasticities. Indeed, for U.K. females without children the income effect is absent. In Figure 3, I have plotted RR(·|φL*) (lower curves) and RP(·|φL*) (upper curves) for φ∈[0.8, 1.2]. Both risk attitudes are decreasing in the constraint on labour supply.Footnote 41 A constraint to supply about 10 per cent more than the optimal amount of hours typically makes the worker as risk averse and prudent as without any constraint. On the other hand, an involuntary 20 per cent reduction in labour supply has non-negligible positive effect on both risk attitudes, especially for male U.S. workers.

Table 3 Effects of labour supply constraints on relative risk aversion and prudence
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Figure 3
figure 3

γ=1(RR*=1.03, RP*=2.02): (a) RR(p, m*|φz*) and RP(p, m*|φz*); (b) RR(p, φm*|z*) and RP(p, φm*|z*).

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I therefore conclude that labour supply constraints have small effects on relative risk aversion and prudence, at least when these constraints are not too binding. But this does not imply that risk attitudes are immune to unemployment status. For example, using the estimates of de Linde Leonard,Footnote 42 I find that for an average worker hit by full-time unemployment and receiving an unemployment benefit at 60 per cent of past earnings during 26 weeks (the maximum benefit period in most U.S. states), relative risk aversion increases to 4.26 and relative prudence to 6.30.Footnote 43

Conclusion

I have traced out how quantity constraints on a good or service affect the consumer’s attitude towards timeless income risk. Using the virtual price approach, I have decomposed the effect into a local curvature effect and an endogenously changing risk aversion/prudence effect.

Calibration of the model shows that restrictions on service flow of durables—for example, because of adjustment costs—have large effects on both relative risk aversion and prudence. Moreover, for relative prudence the effect turned out to be non-monotone in the degree that the constraint is binding, and in the income level. A natural next question is how durables are decided upon when consumers realise the consequences of costly adjustment for risk attitudes. Under simplifying assumptions, Shore and SinaiFootnote 44 provide a first answer to this (difficult) question.

Constraints on the supply of labour were shown to be more moderate, the main reason being the small income effect of notional labour supply. Still, assuming a base rate relative risk aversion of 2, this attitude towards risk raises with more than 0.4 when labour market frictions force a U.S. worker to 80 per cent of his notional labour supply, and with almost 2.3 when fully unemployed on a 60 per cent unemployment benefit.

Eeckhoudt and Schlesinger,Footnote 45 p. 1335) show that an increase in the skewness of the interest rate distribution triggers more saving if relative temperance (RT) exceeds 3. An obvious extension of the paper would consider the effects of constraints on RT, and verify whether, for example, costly adjustment of durables makes RT trespass this critical value.

Quantity constraints may also apply to savings or portfolio decisions, in the form of borrowing constraints or constraints on taking short positions. However, such decisions are made before the uncertainty (about future income and/or rates of return) is resolved. Future research should focus on the effects of such constraints on the attitudes towards temporal risks.