Imagine a simple three-period model of the decision to undertake an unhealthy activity - drinking alcohol heavily. The agent has the opportunity to drink in the first two periods, but no such opportunity to drink in the third period, so utility in period 3 depends only on actions in period 2. The decision to drink is denoted by x, in any time period t. The agent gets positive utility from drinking today but that leads to negative utility tomorrow. The utility payoff from drinking today is 4. However, if you drink you receive negative utility of 5 the following day. ie. u1(x1) = 4x1, u2(x1, x2) = 5x1 + 4x2, and so on. If you don't drink, there are no consequences, today or tomorrow.

a) Assume that the agent's preferences can be represented by exponential discounting. Write down the total utility U(x1, x2) of someone over those three periods from the perspective of the third period with respect to x1 and x2 [Hint: start by writing down U(x1, x2) as a sequence of u1(x1), u1(x1, x2) and u3(x2) with appropriate discounting].