# In this question, we’ll consider a version of the authority model where the principal works with two agents, each on a separate project. There is one principal who is in

In this question, we’ll consider a version of the authority model where the principal works with two

agents, each on a separate project. There is one principal who is in charge of two projects, i = 1, 2. The principal hires one agent for

each project; we identify the agent associated with project i as Agent i. Each Agent i separately chooses eEort £21′ to produce an idea for his own project 1′. Each Agent i’s

eﬂ’ort cost is is? . At the same time, the Principal P chooses effort levels E1 and E2 to devote to each project. Her total effort cost is :ﬂE1 + E2)2. The probability that Agent 1′ produces an idea for Project i equals his effort 8;. The probability that

P produces an idea for Project 1′ equals E,. For Project 1: if the Principal’s idea is implemented, then the Principal and Agent 1 both receive 1.

Ingent 1’s idea is implemented, then the principal receives 0 < a] < 1 and Agent 1 receives 1. For Project 2: as long as either the Principal’s or Agent 2’s idea is implemented, then both players

receive project values of 1 each. Notice that this means neither agent cares whose idea (his own versus the principal’s) is imple-

mented for their project. The principal does not care whose idea is implemented for project 2, but

prefers to implement his idea over Agent 1’s idea for project 1. P’s total payoff is the sum of his project values from both projects, minus her eﬂ’ort cost %(El + Elf.

Agent is payoff is his project value from his own project, minus his effort cost ail/2. The timing of the game is: 1. Each A, chooses effort level 2,. At the same time, P chooses E1 and E2.

2. Each player’s attempt (to generate an idea) either succeeds or fails.

3. For each project, P chooses which idea (if both are successful) to implement 4. Payoﬂs are realized. c) Work out the equilibrium levels of the agents’ effort choices £1 and £2, and the Principal’s effort choices, E1 and E2. (You’ll have to solve a system of four simultaneous equations, each of which

correspond to one of the ﬁrst-order conditions. nt: once you have the four equations, it ma},r be

convenient to ﬁnd a way to solve for E1 ﬁrst.)

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