# We have repeatedly stressed the connection between shadow prices in an incentive design program…

We have repeatedly stressed the connection between shadow prices in an incentive design program and what we call control “hot spots.” Determine the shadow prices in Examples 18.1 and 18.2. What are the control hot spots in these two settings?

((Example 18.1 Assume a two period setting patterned after Example 15.1. The manager is described by a risk aversion measure of ρ = .1 and a personal cost of cH = 60. The inputs are H = 500 and L = 200; and noise in the performance measures is specified by σ 2 1 = σ 2 2 = 10, 000. Also let parameter θ = 1 in the second period performance measure. You should verify that the optimal piece rates here are β ∗ 1 = β ∗ 2 = .20 and that the manager’s risk premium totals 40. There is no inherent short-run versus long-run conflict here. Dealing with the input supply incentives leads, naturally, to a balanced view of the first period’s short-run and long-run tasks. This, recall, is the meaning of θ = 1; it implies the performance scores will capture short-run and long-run activities in unbiased fashion. Literally, input incentives lead to β ∗ 1 = β ∗ 2 = .20, and the balance requirement in (18.7) is redundant. Balance is not a control issue here.))

((Example 18.2 Continuing with the same setting, now let parameter θ = .80. This means the evaluation measures provide a downward biased estimate of the first period’s long-run activities. We know from Example 18.1 that setting β1 = β2 = .20 is the least costly way to motivate high input in each period. Unfortunately, these piece rates, coupled with θ ∗ 2 = .25. We are forced to more heavily weight second period performance in order to compensate for this bias. Coupled with β ∗ 1 = .20, we thus restore the necessary balance of β ∗ 1 = θβ∗ 2 = .80(.25) = .20. The solution is not without cost, as increasing the second period’s piece rate increases the manager’s risk premium from 40 to 51.25. The control hot spots, which correspond to the positive shadow prices in the design program, are now input supply in the first period, constraint (18.8), and the short-run versus long-run balance constraint (18.7).)

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