Return to Example 18.4, where each manager’s compensation is determined by Ii = ?i + ßixi ….

Return to Example 18.4, where each manager’s compensation is determined by Ii = ωi + βixi . Determine each manager’s wage, ωi , for three cases: α ∈ {0, .5, 1}. Explain your finding

((Example 18.4 Assume a setting patterned after Example 15.1. Each manager is now described by a risk aversion measure of ρ = .1 and a personal cost of cHi = 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. With the usual normalization, low input costs and the outside certainty equivalents are set to 0. For benchmark purposes, initially suppose there is no trade opportunity. We find, to no surprise, that β ∗ 1 = β ∗ 2 = .20, each manager incurs a risk premium of 20 and the firm’s cost is 160 = 2(20) + 2(60). Now introduce profitable trade by assuming R = 200, c = 100 and α = .5. The piece rates are unaltered, as the control problem surrounding input supply has not changed. The optimal transfer price is T ∗ = 150, which evenly splits the gains to trade between the two divisions. Each manager’s risk premium increases from 20 to 21.20, and the firm’s overall compensation cost increases to 162.40. The risk premia increase because, thanks to the possibility of profitable trade governed by α = .5, each manager’s compensation is now more risky. And with identical control problems in the two divisions, we equally allocate the profitable trade noise by having the divisions equally share the gains to trade.

In equilibrium, the managers are neither harmed by the additional risk nor explicit beneficiaries of profitable trade. The firm holds rational expectations, anticipates the probabilistic increment to their performance scores, and reflects this in the compensation function.14 The transfer price of 150, in turn, is designed to simultaneously motivate trade and to keep the managers’ respective evaluation measures, expressions (18.9) and (18.10), as informative as possible. Emphatically, the transfer price is designed to motivate communication and minimize noise. The managers learn each other’s private information from that communication, not from the price.15 ))

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