A popular metaphor is that of a balanced scorecard, that a variety of measures should be used in…

A popular metaphor is that of a balanced scorecard, that a variety of measures should be used in evaluating a manager and that they should be treated in balanced fashion. Taking this literally, return to the multitask setting in Chapter 15, and especially the general setup in Examples 15.4 and 15.5. The firm seeks a balanced supply of input H to two underlying tasks. Two performance measures, specified in slightly different fashion than in Chapter 15, are available: x = a1 + αa2 + ε and y = γa1 + a2 + ε. As before, ε and ε are independent, normal random variables, and each has a variance of 10,000. Remaining details can be found in Examples 15.4 and 15.5.

((Example 15.4 Continue with the setting of Example 15.1. Recall that, with α = .7 and thus a performance measure of x = a1 + .7a2 + ε (and σ 2 = 10, 000), the cost to the firm of high input was C(H) = 80. We also know a balanced allocation of this input is infeasible here, as the single performance measure more heavily weights the first task. Suppose a second performance measure of y = a1 + .6a2 + ε along with σ 2 = 10, 000 is available. So γ = .6. If the manager is to equally value the two tasks, the balance requirement in (15.13) must hold, or

Solving program (15.12), but with this balance requirement appended provides an optimal solution of β ∗ 1 = .80, β ∗ 2 = −.60, ω ∗ = 460 and a cost to the firm of C(H) = 560. 14 Be certain you verify this solution, and understand how it motivates the desired behavior. First notice the manager has a balanced view of the two tasks, as β ∗ 1 + β ∗ 2 = .80 − .60 = .20 = .7β ∗ 1 + .6β ∗ 2 = .7(.80) − .6(.60). Second, the manager is willing to supply high input as doing so has no effect on his risk premium but does increase his expected compensation by his increased personal cost of cH − cL = 60. In particular, a balanced supply of H provides expected compensation of

Likewise, a balanced supply of L provides expected compensation of15

ω ∗ + .2L = ω ∗ + .2(200) = ω ∗ + 40

Third, accepting and behaving as instructed (and motivated), provides the manager a certainty equivalent of

But why has the firm’s cost increased so dramatically relative to the unbalanced case? This is the balance requirement at work. Both evaluation measures underweight the second task, and neutralizing this requires relatively large, counterbalancing incentive weights on the two measures, which leads to a dramatic increase in the manager’s compensation risk.

Example 15.5 Contrast Example 15.4 with a setting where everything remains as before except the second performance measure is biased in favor of the second task. γ = 1.2 will suffice. You should now find an optimal incentive function of β ∗ 1 = .1038, β ∗ 2 = .0962, ω ∗ = −29.986 and a cost to the firm of C(H) = 70.01. Here it is relatively easy to weight the two measures so as to provide the manager with balanced incentives, as one measure favors the first task and the other favors the second.