Group A: At the end of the academic year, if their students outperform the median student in the school district, they will earn a bonus of $8k. Otherwise, they don't get any bonus.

Group B: They each receive a $4k upfront payment at the beginning of the academic year. At the end of the academic year, if their students outperform the median student in the school district, they will earn an additional bonus of $4k. Otherwise, they need to return the $4k to the experimenter.

This study finds that Group B teachers' students obtain significantly higher scores than Group A's students. They interpret it as a result of loss aversion.

Let's build a model to analyze the situation. Suppose the teachers have reference dependence utility for money and have disutility for work, which could be represented by

2

u(m,e;r)=v(mr)10e^2.

m denotes the amount of money; r is their reference point of money; e [0, 1] is their effect level; v(x) = x for x 0 and v(x) = 2x for x < 0, where "2" is the loss aversion parameter. Higher e leads to better performance of the students. Let's assume that, if they spend effort e, the probability of beating the median student is p = e, and the probability of failing

is 1 p = 1 e.

1. Suppose Group A teachers have reference point 0, what's the lottery they are facing if they spend effort e? Suppose Group B teachers have a reference point $4k, what's the lottery they are facing if they spend effort e? (Note: use the numbers 8 and 4 instead of 8000 and 4000.)
2. Based on your answer from part (a), write down Group A teachers' maximization problem.
3. What's Group A teachers' optimal effort level? (Note again: in your calculations, use the numbers 8 and 4 instead of 8000 and 4000.)
4. Based on your answer from part (a), write down Group B teachers' maximization problem.
5. What's Group B teachers' optimal effort level? Does this model explain the higher performance of Group B? Aside from the math results, please provide an intuitive explanation of why the model explains/doesn't explain Group B's success.