Discount rates are robust to background consumption and consumption smoothing if the utility function is jointly elicited. However, this result holds only if background consumption and consumption smoothing are time-invariant and independent of reward size. What happens to estimates of individual discount rates if the level of background consumption and the extent of consumption smoothing are not constant over time and rewards? I analyze the sensitivity of individual discount rates when the level of background consumption and
the extent of consumption smoothing vary over time, and I discuss how individual discount rates can be elicited if consumption smoothing varies with rewards.
5.1 Time-Variant Background Consumption and Consumption Smoothing Unlike the results in the previous section, discount rates are highly sensitive to time-variant background consumption and consumption smoothing. I illustrate this sensitivity by keeping background consumption and consumption smoothing of the sooner reward constant and vary both assumptions for the later reward. I assume that the utility function is constant over time and identified from the same decision task as before, which implies that r = 0.77.
When the level of background consumption and the extent of consumption smoothing can vary over time, the individual discount rate is identified by
30+λXS−1
In Figure 7 (a), I vary background consumption on the later date assuming that λS = λL = 1 and background consumption on the sooner date is equal to 118 kroner. As background consumption on the later date ωL increases, the discount rate decreases. For example, an increase in ωL from 118 kroner to 150 kroner implies that the elicited discount rate falls from 9.5% to 4.8%. When the level of background consumption on the later date is 200 kroner or more per day, the elicited discount rate is negative.19
In Figure 7 (b), I vary consumption smoothing on the later date and assume that λS = 1.
Background consumption is kept constant at 118 kroner. As the degree of consumption smoothing λLincreases, the elicited discount rate also increases. For example, the individual discount rate increases from 9.5% when λL = 1 to more than 50% when λL = 2.20
19When the utility function is concave, higher background consumption leads to lower marginal utility of income, and the implied individual discount rate falls when background consumption on the later date increases.
20When the utility function is concave, consumption smoothing leads to higher marginal utility of income, and the implied discount rate increases when λ increases.
Figure 7: Individual Discount Rates and Time-Variant Consumption Smoothing and Background Consumption
(a) (b)
The sensitivity of individual discount rates to time-variant consumption smoothing and background consumption has interesting consequences. For example, an expected increase in background consumption can be one explanation for seemingly present biased preferences.
Suppose a person is indifferent between 3,000 kroner now and 4,000 kroner in one year and is indifferent between 3,000 kroner in one year and 3,750 kroner in two years.21 Under the assumption of time-invariant background consumption and consumption smoothing, the first indifference point would imply a higher discount rate than the second, which is consistent with quasi-hyperbolic discounting. However, the two indifference points are also consistent with exponential discounting and time-varying background consumption. A subject may expect an increase in future background consumption, which implies that the discount rate inferred from the first indifference point would decrease and could become the same as the discount rate inferred from the second indifference point. In this scenario, present bias is not the result of quasi-hyperbolic discounting but the result of an expected increase in background consumption on later dates.
This example illustrates that time-varying background consumption and consumption smoothing are relevant to consider in future research. How one best identifies variation in the level of background consumption and the extent of consumption smoothing is an open
21See, for example, Thaler (1981), where the subjects reveal similar preferences over delayed rewards with and without a front end delay to the sooner reward.
question. One suggestion is to elicit risk preferences at different points in time and infer variation in background consumption or consumption smoothing from variation in absolute risk aversion. However, this method requires that the underlying utility function is the same for all subjects and constant over time, and one cannot separate effects from background consumption and consumption smoothing.
Andersen et al. (2008b), Andersen et al. (2014) and Andreoni and Sprenger (2012) iden-tify the atemporal utility function at a single point in time. Other studies have investigated to what extent risk aversion changes with the timing of lotteries. For example, Noussair and Wu (2006) find that 38.6% of the subjects in their sample are more risk averse over lotteries that are paid out in the present compared to similar lotteries that are paid out in the future. The observed decrease in risk aversion could be influenced by an increase in background consumption or consumption smoothing (or both) over time. Abdellaoui et al. (2011) also find that risk aversion is sensitivity to the timing of lottery rewards, but attribute this behavior to changes in probability weighting. Andersen et al. (2008a) elicit risk preferences from binary choices between lotteries, and repeat the same decision task with the same subjects after 17 months. They do not assume that rewards are integrated with background consumption, and find no general tendencey for risk attitudes to increase or decrease. It will be an interesting avenue for future research to further investigate time-varying risk aversion, and analyze to what extent risk aversion varies with expected future income.
5.2 Consumption Smoothing that Varies with Rewards
It is likely that the extent of consumption smoothing depends on reward size. For example, small rewards may be consumed straight away, whereas large rewards may be consumed over longer periods of time. Suppose that a subject converts the rewards into days he can buy an additional coffee, and that a cup of coffee costs 30 kroner. When the subject is indifferent between 300 kroner today or 360 kroner in a year from now, the sooner reward is converted into an extra coffee on 10 subsequent days and the later reward is converted
into an extra coffee on 12 subsequent days. This indifference point is given by
Added utility from consuming z over xzS days
=
Added utility from consuming z over xzL days
. (17)
where z is the price of coffee, xS is the sooner reward at t = S and xL is the later reward at t = L. If background consumption ωt is constant over time, then equation (17) simplifies to
S+XxSz −1
such that identification of the discount rate becomes independent of the utility function.
One can use this insight and consider an alternative experimental design that elicits individual discount rates over annuities. Instead of offering a choice between two rewards xS and xL at different points in time, one can offer a choice between two annuities with different start dates and durations. By varying the duration of the annuities one can identify the indifference point and the implied individual discount rate without identifying the utility function.