Return to Example 15.2, but now assume the second variable has a variance of s 2 = 5, 000….
Return to Example 15.2, but now assume the second variable has a variance of σ 2 = 5, 000. Determine an optimal (linear) contract. Explain the difference between this contract and that in the original setting.
((Example 15.2 To illustrate, stay with the setting in Example 15.1 and assume the new information variable is structured precisely as the original one. So we have x = a1 + .7a2 + ε and y = a1 + .7a2 + ε along with σ 2 = σ 2 = 10, 000. Think of this as a random sample of size two. In any event, solving program (15.12) provides β ∗ 1 = β ∗ 2 = .10, ω ∗ = −30 and a cost to the firm of C(H) = 70. This cost is less than the original example’s cost of C(H) = 80. To provide the intuition, notice that for any contract described by (15.9), the manager’s risk premium will be
We know from our earlier work that the manager will allocate all of his input to the first task. His certainty equivalent if H is supplied is therefore
Motivating H requires CEH ≥ CEL, or β1 + β2 ≥ cH H−L = 60 500−200 = .20. This reflects the fact here that the sum of the piece rates is the source of the incentive. With this observation, it is clear we want to minimize the manager’s risk premium while simultaneously ensuring the two piece rates sum to .20. With the additional information, this means we want to minimize 500[β 2 1 + β 2 2 ] subject to β1 + β2 ≥ .20. And we thus have β1 = β2 = .10. From here, setting CEH = 0 provides ω = −30. But the central observation is the fact the second variable allows us to diversify the noise in the original performance measure, thereby creating a (modest) portfolio of performance measures that, in total, lowers the performance assessment noise and thus the manager’s risk premium.
Example 15.2, then, is a case where an additional performance measure is useful. And this returns us to the informativeness criterion. Is variable y in the example informative in the presence of measure x? Definitely. With independence between the two error terms (ε and ε), we readily conclude the conditional likelihood ratio (expression (14.10) to be precise) in this case is
With independent error terms, knowledge of measure x has no effect on our assessment of variable y’s likelihood ratio. The new information is an independent assessment, unaffected by what we have learned from the first measure. And this likelihood ratio surely suggests variable y informs us about the manager’s behavior.10
Notice, however, that the control problem is well isolated here. It centers on the choice between L and H, because we assume the firm wants the input used exclusively on the first task and the performance measures more heavily weight that first task. Given this, the additional performance measure speaks precisely to the control problem of motivating choice of H over choice of L.