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Organisational Economics Problem Set 2 Solutions

发布时间：2022-11-25

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Organisational Economics

Problem Set 2 Solutions

Question 1

2 2 . To do so, we have�� to=w��ri[t��e ]ou−t��₁ in terms of2��₂. Remember that

We may calculate the various components of agent 1’s payoff:

Var(��₁) = Var[�� + ��(1��₁ + ��) + ��(2��₂ + ��)]

= (�� + ��) .

�� + ��₁

+ ��₂

Thus the payoff of agent 1 is ��1 = �� + ��1 + ��2 − (�� + ��)2/2 − ��21/2.

2 = ��(��(��++��¹��+1 +2��−2 −(��(��++��)��2)/²2/2−−��2��/²12/)2/)2−−��22//22 given �� = 1/2 Agent 2 chooses ��₂ to maximize this ob��je��ctive; the first-order condition tells us

*by Hongyi Li and Adam Solomon

We already know from (a) that agent 1’s payoff can be written as

c) 2The principal chooses an incentive scheme are Agent 1 which ensures that ��₁ = ��; but not for agent

Let’s further rewrite this expression in terms of �� and ��:

= ��[��1]1+ ��[��2] − ��[��₁] + ��[��₁]₂ − Var[��₁]/2 − ��21/2

= ��

+1�� −

2 Var[��₁₂]/2 − ��₁/2

1 2 Var[��₁]/2 − ��₁/2

The principal chooses �� and �� to ma��xi��m=ize1 t−hi(s��o+bj��e)ct−iv��e;=th0e, first-order conditions are

Solving these simultaneous equations, we get �� = 1/2 and �� = 0.

d) Suppose the principal were to offer agent 1 an incentive scheme with relative performance eval-

reduces agent 1’s payoff, he will withhold effort. To avoid this effect, the principal does not engage

e) Rehashing our calculation��s f=ro��m��([a��) ]an−d��2(/b2),=ag��en[��t 2]’s−p��a2y/o2ff is (in terms of ��₂)

so ��∗2 = ��.

��2 = �� − ��₂ = 0,

f ) The calculations are identical to those of (b); so ��₁ = ��.

= ��[��₁]1+ ��[��₂] − ��[��₁] + ��[��₁] − [��₁]/2 − ��21/2

(1)

= ��[��₁] + ��[��₂] − [��₁]/2 − ��21/2Var

(2)

The principal chooses ��

= ��₁ + ��₂ − [��₁]/V2a−r ��21/2

��

(3)

(4)

(5)

and

to maximize this objective; the first-order conditions are

�� = 1∗ − (�� + ��) =∗ 0. ∗

Solving these simultaneous equations, we get �� = 0 and �� = 1. Notice that as a result, ��₁ = �� = 0.

about Agent 1, and thus exerts effort to increase Agent 1’s payoff∗). When A∗gent 2 is highly altruistic

for higher effort when using �� to induce ��₂. Consequently, �� is a cheaper instrument than ��, and

Question 2

a) The agent’s payoff is

= �� + ��₁��₁ + ��₂��₂ − ��(V��a1r[+��1��2)+2��2/��2]/−2 −(��2(��+12 +��2��)/22)/.2

b) Applying the∗ usual trick,∗we rewrite the principal’s objective as choosing ��₁ and ��₂ to maximize

= ��₁ − ��(��21 + ��₂)12��2/22 − (��21 + ��₂)1/2. 2

Solving the simultaneous equatio2ns, we get

+¹ ��₂)��²2 − ��₂

=¹ 0;

and

Similarly, the agent’s cor��r∗es=po��n∗d=ing 1eff+or��t choices are∗ ∗

��2

1 1 1 + 2��2 and ��2 = ��2 = − 1 + 2��2 .

c) The efficient choices ��e1ff, ��e2ff, ��e1ff, ��e2ff maximize the total payoff

Notice that

∗ 1 + ��2 2 2

��₁ = 1 + 2��2 < 1 = ��e1ff

∗

because 1 + ��

��a2nd

< 1 + 2�� ,

��₂ = − 1 + 2��2 < 0 = ��e2ff.

d) As with (b), but keeping in mind that now �� = 1, we may rewrite the principal’s objective as choos-

= ��1 + ��22 − ��(��11 + ��22))22��22//22 − ((��1221 ++��22))//22

��1 = 1 − ��(��¹ + ��²)��2 − ��¹ = 0;

and

Solving the simultaneous equa��t��io2ns, we get 1 2 1 2

e) The efficient choices ��e1ff, ��e2ff, ��e1ff, ��e2ff

��∗1 = ��∗2 = 1 + 2��2 .

taking first-order conditions with1 resp2ect to 1��₁ an2d ��₂, we get 1 2

f ) Indeed,

that is, effort in task 1 is lower when �� = 1 than when �� = 0.

When the importance of task 2 increases ( increases), the principal wants to provide incentives to the agent for task 2. However, if the principal offers strong incentives in both tasks, the agent will be

perfectly correlated, and thus reinforce each other rather than cancelling each other out. To reduce the loss from such noisiness, the principal optimally weakens incentives on task (so that the agent is exposed to less noise), especially when the principal plans to offer relatively strong incentives on

g) Remember that inefficiently low effort levels occur because the principal deliberately weakens in- centives to reduce the noise that the agent is exposed to. In order for the equilibrium effort levels to equal the efficient effort levels, it must be that the incentive scheme that induces the agent to exert the efficient effort levels is completely “noiseless”, i.e., has zero variance. This only occurs if the incentive strength on task is exactly the negative of the incentive strength on task (that is,

1 2 ), in which case the noise from the two outputs cancels out perfectly. In turn, this occurs

if and only if the principal seeks to induce opposite efforts in the two tasks; that is, when �� = −1.