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Tutor: Cheney Wan
Date: 21/10/2023
UNSW ECON3123 Organisational Economics
Problem set 3
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Problem set 3
Question 1
In this question, we will consider a model of incentives and disagreement where the principal can
either choose to give the agent an order, or persuade the worker. Unlike the model covered in class,
orders are binding.
There is a (P)rincipal and an (A)gent. A works on a project: he chooses effort e at cost and
a binary decision d(either −1 or 1). There is a binary state of the world θ(either −1 or 1) that
is not known until after the agent makes his choices. The project is successful if and only if the
action matches the state (d=θ), in which case the principal receives revenue v=Be , where B>0 .
Otherwise, if the project fails, then the principal receives v=0, and the principal also incurs an
additional cost of c=1.
P can offer A an incentive scheme of the form .
So, the principal’s and agent’s payoffs are
P and A disagree on how likely each state of the world is. P believes that θ=1 occurs with probability 1;
A believes that occurs θ=1 with probability 1/3.
At the start of the game, P chooses one of two options.
• P can give A an order i(either −1 or 1). If A receives an order, he cannot disobey, and must choose d=i;
however, he can choose any effort level he likes.
• Or, P can persuade A. If P chooses to persuade, then she is successful 50% of the time: A changes his mind,
and believes (like P) that “θ=1 will occur with probability 1”. The other 50% of the time, persuasion fails: A
continues to believe that “θ=1 will occur with probability 1/3”.
The game proceeds as follows:
Step 1. The principal chooses whether to give the agent an order i, or to persuade the agent. If the
principal chooses to persuade the agent, then she learns whether persuasion was successful.
Step 2. The principal offers the agent an incentive contract .
1Step 3. The agent chooses a decision d, and an effort level e≥0. If the principal gave an order, the agent must
obey.
Step 4. The state of the world is revealed, and the project succeeds or fails (based on whether the
agent chose the correct action). The principal pays the agent his wage τ.
We’ll work this out step-by-step:
a) Suppose the principal chooses to give A an order. What order does P choose? What effort level does the
agent exert in step 3, given the principal’s choice of incentive strength b in step 2? What incentive strength
b∗ does the principal optimally choose in step 2?
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P会选择什么 order, 此时努力程度和激励机制是多少？
order可以是 i=-1或 1，agent必须遵守，对应的 d为-1或 1；i为-1时，principal的角度来看，项目会失
败，因此 principal会倾向 order i=1，此时 d为 1，算出此时的 u，求导得出努力程度，从而得出激励 b
b) Suppose the principal chooses to persuade A, and that persuasion succeeds. What effort level does
the agent exert in step 3, given the principal’s choice of incentive strength b in step 2? What incentive
strength b∗ does the principal optimally choose in step 2?
劝说成功，agent相信成功概率为 1，此时努力程度和激励机制是多少
C) Suppose the principal chooses to persuade A, and that persuasion fails. What effort level does the agent
exert in step 3, given the principal’s choice of incentive b strength in step 2? Show that the principal will
always receive the same payoff, regardless of the choice of incentive strength.
劝说失败，agent相信成功自己成功概率，倾向选择 d=-1, 算出此时 agent的努力程度，此时 principal
的 payoff为-1
d) For what values of B does the principal choose to persuade the agent, rather than order the agent?
e) Explain, in words, why the principal prefers to persuade the agent (rather than order the agent) when B is
large.
B越大，则 agent的努力程度越大，此时命令会让 agent的努力程度相对劝说变小，所得相对劝说下要
小
Question 2
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 effort to produce an idea for his own project i. Each Agent ’s effort cost is
. At the same time, the Principal P chooses effort levels and to devote to each project. Her
total effort cost is . (Note that this means that there is a strong crowding-out effect for the
principal: if he exerts more effort in one task, it becomes more costly for him to exert effort in the other task.)
The probability that Agent i produces an idea for Project i equals his effort . The probability that P
produces an idea for Project i equals .
For Project 1: if the Principal’s idea is implemented, then the Principal and Agent 1 both receive 1.
If Agent 1’s idea is implemented, then the Principal receives 1 and Agent 1 receives 1/2.
For Project 2: if the Principal’s idea is implemented, then the Principal and Agent 2 both receive 1.
If Agent 2’s idea is implemented, then the Principal receives 0 < a<1 and Agent 2 receives 1.
P ’s total payoff is the sum of his expected project values from both projects, minus her effort cost
. Each Agent ’s payoff is his expected project value from his own project, minus his
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effort cost .
The timing of the game is:
1. Each chooses effort level . At the same time, P chooses and .
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. Payoffs are realized.
We’ll go through the problem set-by-step.
此题建议大家分别把 P-authority和 A-authority的情况先分别列清楚，再去做下面的小问
a) Write down ’s expected utility as a function of and . Calculate ’s choice of as a
function of .
分别看 P-authority和 A-authority下 u1和 u2的式子
b) Write down P’s expected utility as a function of his and the agents’ effort choices. Calculate P’s
choice of and as functions of and .
同理，分别看 P-authority和 A-authority下 U的式子
c) Work out the equilibrium levels of the agents’ effort choices and , and the Principal’s effort
choices, and . (You’ll have to solve a system of four simultaneous equations, each of which
correspond to one of the first-order conditions. (Hint: Once you have the four equations, it may
be convenient to find a way to solve for first.)
此时有 4个式子，两种情况都是，计算出均衡条件下这四个的值
d) You should find that agent 1’s equilibrium effort ∗ is increasing in a, which measures how well
aligned the Principal is with Agent 2 over the outcome of Project 2. Explain, in words, why the principal’s
alignment in one project increases the other agent’s effort choice in the other project.
会发现 e1会随着 a的增加而增加，说明 principal此时的结果和 Agent2的目标一致，解释为什么会这样
（挤出效应去解释）