Consider yet another variation on Examples 16.1 through 16.4, where everything remains as…

Consider yet another variation on Examples 16.1 through 16.4, where everything remains as specified except the probabilities. The probabilities are specified below. Determine a value for π(y = g|H) = α such that (1) the monitor is useless and not controllable, (2) useful and allows zero risk premium, (3) useful and implies observing y = g is “good” news and (4) useful and implies observing y = g is “bad” news. Verify each of your claims by solving for the associated optimal contract.

((Example 16.1: Assume the manager’s input can be H or L, with H preferred by the firm. Output can be either x1 or x2 (with x1

As a benchmark, if output itself is the only contractible variable it is routine to verify the optimal pay-for-performance arrangement has respective (low and high output) payments of I ∗ x1 = 0 and I ∗ x2 = 7, 305.66. The firm’s cost is C(H) = 3, 652.83 and the manager’s risk premium is 652.83. (This should be familiar.)

Next (continuing to reprise the earlier example) we introduce an additional performance measure. This measure will report either y = g or y = b. The probabilities are specified in Table 16.1, where you should note that absent the additional measure we are back to the benchmark setting of π(x1|H) = π(x2|H) = .50 and π(x1|L) = 1.

Also notice the new, improved optimal pay-for-performance arrangement displayed in Table 16.1.1 The additional measure is useful, as we have I ∗ x1g > I∗ x1b . Following our earlier definition in Chapter 14, it is also informative, as the conditional likelihood measure of expression (14.10) is a nontrivial function of measure y (when x1 obtains). Overall, the firm’s cost is reduced to C(H) = 3, 148.70, which implies a risk premium for the manager of 148.70

Now check out the new measure’s controllability by the manager. Notice, glancing at the probabilities in Table 16.1, that we have π(g|H) = .75, π(b|H) = .25, π(g|L) = .20 and π(b|L) = .80. The respective unconditional likelihood ratios, using expression (16.1), are LRg = 20/75

((Example 16.4: Finally, consider yet another variation, that in Table 16.4. Here the new measure is not useful, is not informative; but is controllable by the manager.

The competing tests are once again in conflict. The optimal contract ignores the additional information. The new measure fails the informativeness test, but is surely controllable by the manager.

What are we to conclude? Informativeness, as assessed by the conditional likelihood ratio is the gold standard here. And, as evidenced by the four examples, there is no logical connection between controllability and informativeness. In Example 16.1 they are both present, and in Example 16.2 they are both absent. In Example 16.3 we have informativeness and absence of controllability, while in Example 16.4 we have controllability and absence of informativeness. There simply is no logical connection between informativeness and controllability.

The slippage, or error if you want to be less polite, is in how the presence of other information is handled. Controllability asks whether the manager can affect the statistical description of the new measure, regardless of what other information is present. While intuitive and, yes, simple, this ignores what is already being learned from the other information as well as any possible interactions between the existing and the new measures. Informativeness, on the other hand, stresses whether the manager can affect the statistical description of the new measure, conditional on what other information is present.3

Both tests boil down to a nonconstant, well-chosen likelihood ratio. The difference is informativeness looks for new information; it is conditional on the existing information. In this sense, informativeness could be interpreted as conditional controllability, controllability that is conditioned on what is already being learned from other information sources. This, however, is just another way of saying the fundamentals come down to whether the conditional likelihood ratio is nonconstant.4

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