11. task assignment with an old friend Multitask issues are also discernible in a modest extension of our original single task story in Chapters 13 and 14. To see this suppose we have two tasks, each of which will produce output of 10,000 or 20,000, which we will code as "1" or "2" for each task. So the possible joint outcomes are 11, 12, 21, or 22, where 12 denotes outcome 1 from the first and outcome 2 from the second task,etc. As usual, the manager can supply input H or input L, but now to two tasks.

The possible input combinations are therefore HH,HL,LH or LL, where, for example, HL refers to input H supplied to the first task and input L supplied to the second. Input H carries a personal cost of 3,000, while the personal cost of input L and the manager’s outside certainty equivalent are normalized to 0. The personal cost of HH, then, is 6,000, of HL is 3,000, etc. We also model the manager’s preferences in the usual fashion of constant risk aversion, and here assume a risk aversion measure of ρ = .0001. The manager will be assigned two tasks, but will not see the outcome of the first task before providing input to the second. Two types of tasks are under consideration. Their respective probabilities are detailed below.

outcome 1 2 task type one

π(x|H) .4 .6

π(x|L) 1 0 task type two

π(x|H) .4 .6

π(x|L) .7 .3

(a) Initially suppose this is a single task setting. Determine an optimal contract to provide supply of input H to a type one task. Do the same for a type two task.

(b) Now suppose two of task type one are assigned the manager.

Given the possible combinations of inputs supplied to either task, we have the probability structure noted below. Suppose the following pay-for-performance arrangement is offered: -601.30 if "11" is observed, 6,410.97 if "12" or "21" are observed, and 1,0492.43 if outcome "22" is observed. Determine the manager’s certainty equivalent for each combination of inputs. Is this con tract individually rational and incentive compatible? Explain.
Is the noted contract the optimal contract?
11 12 21 22 π(x|HH) .16 .24 .24 .36 π(x|HL) .40 0 .60 0 π(x|LH) .40 .60 0 0 π(x|LL) 1 0 0 0

(c) Determine an optimal contract for the case where two of task type two are assigned the manager.

(d) In both cases above, it turns out assigning a pair of type one or of type two tasks to the manager is more efficient than contracting on each task separately. Verify this claim. What is the explanation?

(e) Does it remain efficient to assign the pair of tasks rather than contract for each task separately if the firm wants one of each type task performed? Explain.