# Intertemporal choice with possibility of death

Here is the setup: Suppose that there is an individual who lives up to two periods. He lives with absolute certainty during period $$1$$, and during this period his sub-utility function is given by: $$u(c_1) = \log c_1.$$ The individual is alive in period $$2$$ with probability $$p$$, and if he is alive, his state sub-utility function is given by $$u(c_2) =\log c_2$$ and if he dies, he receives the fixed utility level $$B$$. Suppose he has access to a perfect capital market and discounts the future at rate $$\beta$$. His endowment is given by $$(\omega_1, \omega_2)$$.

I have a few questions about how to approach this problem. The first question is: do we assume he can die with debt/excess savings at the end of period $$1$$? If I assume that $$p=0$$ (dies with certainty), I get that he takes out a loan $$s= \frac{\omega_2}{1+r}$$ in the first period. Does this make sense, i.e., do we assume that even if he dies he gets an endowment of $$\omega_2$$ to pay off his debts? This result would correspond to $$c_2 =0$$.

The second question is about the fixed utility level. Should this be interpreted as, if he dies, he gets utility $$u(c_1) + B$$? Or $$u(c_1) + \beta B$$? Or that we assume his total lifetime utility from consumption is $$u(c_1) = B$$?

Thanks!

• There is a lot of information missing from the problem. In terms of $\beta B$ vs $B$, it doesn't make much of a difference, just change the interpretation based on how you set up the problem. – lunar_props Dec 1 '19 at 3:46
• Yeah I agree I'm not sure why I asked that. So I guess we should assume that his person receives the endowment or something to pay off any debt – David Bowman Dec 1 '19 at 16:43
• You model an agent receiving utility from dying? Seems dumb. – 123 Dec 1 '19 at 20:24

It is safe to assume that if the consumer passes away, his creditors get his endowment. This actually happens in real life. $$B$$ should obviously be zero. As far as the second period is concerned you can represent it as a von Neuman-Morgenstern utility from the lottery:
$$\beta u(c_2)$$ with probability $$p$$ and $$0$$ with probability $$1-p$$.
I.e. $$U(c_1, c_2) = log(c_1) + p\beta log(c_2).$$