13. task assignment Ralph owns a production function. Output can be either x1 or x2, with x1 < x2. The manager’s input can be L or H, with H desired.

Ralph is risk neutral. The probabilities are:

x1 x2

π(x|H) .1 .9

π(x|L) .8 .2 The risk averse manager is modeled in the usual fashion, with personal cost of high input cH = 5, 000 and personal cost of low input (cL)

and outside opportunity certainty equivalent (M) normalized to zero.

Under constant risk aversion, his risk aversion measure is ρ = .0001.
The only observable for contracting purposes is the manager’s output.

(a) What is the best way to motivate supply of input H by the manager? How much would Ralph pay to be able to observe the manager’s input?

(b) Call the above task one. A second task, task two, requires the same personal cost, and so on. The only difference is the probability structure:
x1 x2 π(x|H) .1 .9 π(x|L) .7 .3 Suppose only this task is present. What is the best way to motivate supply of input H? How much would Ralph pay to be able to observe the manager’s input?

(c) Now suppose both tasks are present, and Ralph wants supply of input H to both. The output of each task is separately observed.
Also, the manager does not see the outcome of the first task before providing input to the second; so the input supply options are H to both, L to both, L and H or H and L. Will the above two incentive schemes motivate supply of input H to both tasks?
Verify your claim, and give the intuition.

(d) What is the best way to motivate supply of input H to both tasks?

(e) How much would Ralph pay to observe the input supplied to task one?