Correction to exam-solution covered in lecture 13

The solution steps I followed were to

1. Identify optimal response for agent as a function of w and b (with p1 and p2 in general form)
2. Calculate optimal effort using the solution and adding in that p1=0.5 and p2=0.5 - This gave the answer to question 1 in the exam question
3. Calculate the optimal b from the firm's profit maximization Lagrangian. In this problem, as in the first, it makes sense to do this "generally" by employing p1 and p2. I did not do this consistently in the solution I wrote on the blackboard in class. Instead, I sometimes wrote in the optimal effort given that  p1=0.5, at other times using p1 and p2 in symbolic form. The error was pointed out by a student after class: This will give us the right answer to question 2, but this solution is only valid for p1=p2=0.5. In the second part of the exam, when examining how the optimal b is altered with changes in the value of p1, the equation I derived in problem 2 would not give the right answer.

To make sure you have the right expressions, I ran the exam through Mathematica - the full set of expressions and solutions for different values of p1 is given in the below screenshot. The general solution follows the intuition I gave when I solved the optimal b for values p1=p2=0.5: Give the agent the full marginal product of his or her effort, and suck out the "extra utility" this gives the agent (relative to the outside option) with the fixed (and negative) part of the wage. 

Here are what I (hope) are the correct solutions ;)
Let me know in the comments if anything is unclear or confusing: