Q256. Consider the collective action model described in Lecture 17.4: Collective Action and Common Pool Resource Problems where $x_j$ is the cost to me to "pitch in" and do my part in some collective effort. Each member of the collective reaps benefits from the contributions of those who decide to pitch in. In particular, they receive some fraction $\beta$ of all the contributions. Their net benefit is thus, this amount minus the effort they contribute. In other words,

\begin{align} Payoff_j = -x_j + \beta \sum x_i \end{align}

Suppose you are in a class of 21 students and everyone is expected to prepare for class in a manner that costs 1 unit of life. In the class itself, things go much better when people are prepared and we estimate that the benefit a student derives from the class is equal to 0.2 units of life for each person who comes prepared.

(a) What is your net payoff if you do the reading half the class rest of the class comes prepared too?
(b) What is the benefit to a shirker under the same conditions?
(c) How many people do you need to think are going to do the reading to make it worth it to do the reading?