Assume there is positive utility in period 1 from consumption, and negative utility in period 2 (a reduced-form expression for poor health, say). Because the large size has some production economies, it is cheaper, which is reected in higher immediate consumption utility. The Table below shows numerical utilities. (If she chooses to eat 1 serving from the large pack in period 1, then she gets utility of +3 in period 1, and 2 in period 2.) Consider a -J quasi-hyperbolic framework. For simplicity assume 6 = 1 to focus attention on the [9 term. Analyze the optimal consumption decisions of three types of agents: Exponential ( = l); nai've quasi-hyperbolic (,6 = 0.5); and sophisticated quasi- hyperbolic ( = 0.5). Consum ntion Decision in Period 1 in Period 2 lservin- Large lserving 2 servin _s For each type of agent, answer: a. What will they expect to do, at time 0, if they buy either the large and small packages? b. Given your answer in (a), which package will the period-O \"self" purchase, for each of the three types? c. After they buy their optimal package, how much will they consume in period 1? (1. Which of the type's (if any) plans embedded in (a) are actually violated in (c)? e. Suppose agents could purchase external commitment, in which they could only consume 1 of the 2 servings in the large pack in period 1, at a price of P > 0 (think of this as buying prepackaged dietary portions of food). Which agents would commit at time zero to pay P, and how much would they be willing to pay