You and your spouse plan your lifetime household consumption collectively. This is complicated because you have different discount factors. Your wife is patient, her discount factor is .8; you are not so patient, your discount factor is .5. But you are a utilitarian household so you resolve your conflicts by maximizing the total household utility.
Leeat Yariv and Matt Jackson show in this cool paper that your household necessarily violates a basic postulate of rationality: your household preferences are not time consistent. For example, consider how you rank the following two streams of household consumption:
- (0,10,0,0, …)
- (0,0,15,0,0, …)
Each of you evaluates the first plan by computing the present value of 10 units of consumption one period from now. Total household utility for the first plan is the sum of your two utilities, i.e. For the second plan you each discount the total consumption of 15 two periods from now. Total utility for the second plan is Your utilitiarian household prefers the second plan.
But now consider what happens when you actually reach date 1 and you re-consider your plan. Now the total utilities are for the first plan (since it is date 1 and you will each consume the 10 immediately if you choose the first plan) and for the second plan. Your household preference has reversed.
Indeed your household exhibits a present bias: present consumption looms large in your household preferences, so much so that you cannot forego consumption that, earlier on, you were planning to delay in exchange for a greater later reward.
Jackson and Yariv show that this example is perfectly general. If a group of individuals is trying to aggregate their conflicting time preferences, and if that group insists on a rule that respects unanimous preferences and is not dictatorial, then it must be time inconsistent.