Practice Problem for Multitasking

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Practice Problem for Multitasking
EC404: Information Economics

October 5, 2023
1 Problem

Consider the following multitasking Principal-Agent model. There are two types of actions that the agent can take, a1 and a2. If the agent takes actions a1 ≥ 0 and a2 ≥ 0, output is equal to f1a1 + f2a2, where f1 ≥ 0 (respectively, f2 ≥ 0) measures how effort a1 (respectively, effort a2)  translates into output. The principal does not observe output directly; instead, principal observes performance measure p = g1a1 +g2a2. The agent’s compensation is w = s+bp = s+b(g1a1 +g2a2). The agent’s cost of putting efforts a1 and a2 is

(i) Given this contract, what effort levels a1 and a2 will the agent take? How do these efforts depend on parameters g1 and g2?

Answer: the agent will choose a1 and a2 to maximize their expected utility

Taking first order conditions w.r.t. a1, we get
bg1 − a1 = 0 ⇐⇒ a1 = bg1.
Similarly, taking first order conditions w.r.t. a2, we get
bg2 − ka2 = 0 ⇐⇒ a2 = bg2.
(ii) Suppose that f1 = 0 and g2 = 0. Based on your answer to part (i), what bonus rate b ≥ 0 should the principal choose? Explain. Answer: If f1 = 0 and g2 = 0, the principal should set b = 0. Intuitively, if b > 0, the agent will only put effort on task a1 when g2 = 0. But when f1 = 0 the principal only cares about task a2. (since output is f2a2. )

2023-10-11