I Live for Two Time Periods: Today and Tomorrow $2 \bar{u}$

Essay

I live for two time periods: today and tomorrow. At the beginning of today, I must choose to smoke cigarettes or not smoke cigarettes. On the first day that I smoke cigarettes, I receive a utility of $2 \bar{u}$ and pay a cost of $c$ . I only receive a utility $\bar{u}$ from smoking on the second day and still pay cost $c$ . But, I also receive a fraction of utility from yesterday's smoking $\delta \in$ $(0,1)$ . If I choose not to smoke on a given day, then my utility is 0 . Let $\bar{u}=10$ , and $c=22$ . I am risk neutral.
a. If I do not choose to start smoking today, will I choose to smoke tomorrow? No. My utility from today is 0 and my utility tomorrow will be -12 if I choose to smoke and 0 if I choose not to smoke.
b. Suppose $\delta=0$ , that is, I carry no utility from smoking today into tomorrow. If I choose to smoke today, will I quit tomorrow?
No. $U($ smoke, smoke $)=u($ today $)+u($ tomorrow $)=(2 * 10-22)+$ $(20+10-22)=6 \cdot U($ smoke, no smoke $)=(2 * 10-22)+0=-2$ .
d. What is the minimum level of $\delta$ that I will smoke today and tomorrow.
$(20-22)+\delta 20+10-22 \geq 0$ . This is always true if $\delta \geq \frac{7}{10}$ .