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Organisational Economics Problem Set 3 Solutions

发布时间：2022-11-25

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Organisational Economics

Problem Set 3 Solutions

that ��[��] = ₃ ��. So, given an incentive s��ch=em��e[��]=−��1��, 2agent �� believes that his utility will be The first-order condition with respect to the agent’s choice �� is

the time. So, he expects that ��[��] = ��

��==3 ��[��.]��−’s1��u[t��ili]ty is

= (�� − ��2) 3 ��2.

The first-order condition with respect to(t1h−e p2r��i)n13c��ip2al=’s 0ch, oice �� is

and expects that ��[��] = ��. So, given an incentive sch1eme �� = ��, agent �� believes that his utility The first-order condition with respect to the��ag−en��t’=s c0h,oice �� is

time. So, he expects that ��[��] = ��

= �� =. ��’s[��u]ti−lit��y[i��s ]

= (�� − �� )�� .

The first-order condition with respect to t(h1e−p2ri��n)c��ip2 a=l’s0c,hoice �� is

t��hus chooses action ��, and expects that ��[��] = ₃ ��. So, given an incentive scheme �� = ��, agent

The first-order condition with respect to the agent’s choice �� is

time. So, he expects that ��[��] = �� = 0. �� also expects that the project will fail, and he will incur

the cost �� = 1, 100% of the time. ��’s uti��lit=y i��s [t��hu] s− ��[��] − 1

So, that incentive strength does not affect ’s utility; any choice of incentive strength is optimal for the principal.

d) If orders (to choose ), then his expected utility is1

�� = (1��1∗ ��−2��∗2) 3 ��2

If �� tries to persuade ��, then 1/2 of the time, persuasion1 succeeds and his expected utility is

utility from persuasion is 8 ��

− ₂ . He ch1ooses t1o per1suade the agent if and only if

1 8 ��21− 2 > 1 ��2, i.e.,

8 ��2 − 12 ��2 > 122, i.e.,

e) Persuasion, when it succeeds, convinces the agent that is the “better” action, i.e., the action that is more likely to succeed. On the other hand, giving an order results in the agent obeying (i.e.,

than if the principal orders the agent. For high , effort is more important to the principal; thus the principal prefers to motivate the agent using persuasion, rather than give the agent an order.

a) Consider each project �� ∈ {1, 2}. Because the principal has authority, the principal’s idea is imple-

Remember, also, the the agent gets payoff of 1 whenever either idea is implemented.

from project 2 is ��₂ + ��(��1 −+��(21)��−2.��H)i��s e+xp��ect+ed��(p1ay−o��ff i)s��th−us(��

+ �� )2/2.

Taking the first-order conditions for ��₁ a��n∗d+��2,∗w=e g1e−t �� ,

c) The four first-order conditions from (a) and (b) combine to form a system of four first-order con- ditions:

��11++=��(21²=−=1��1−−)/��2��1,2,,

Let’s solve this system step-by-step. Substituting (3) into (1) and (4) into (2), we get

��₁ +1(1 − ��)2��₂ = 1 −an��d..

(1)

(2)

(3)

(4)

(5)

(6)

We can solve (5) and (6) to get

= 1 − ��

1 1 + �� and ��₂ = 1 + �� .

��₂∗ =

1 +1− ��,

d) Note from Hongyi:

2 1 + ��

The result that is increasing in is counterintuitive, in the sense that the usual economic reasoning that we ask you to engage in would lead to the opposite conclusion. The reason for this counterintuitive result is technical and not relevant to this course: essentially, the equilibrium you have found in (c) is not the only equilibrium of the game, and in fact is “unstable” in a technical sense. Such unstable equilibria often generate counterintuitive economic results, which is exactly the case here. This also means that we don’t have a meaningful “economic” explanation for this result, and thus (in my view) there is no good answer for 2(d).

This is, of course, my fault for writing a problematic problem. All of you who submitted the problem set will automatically receive full credit for 2(d). My apologies for this mistake. We will, of course, be careful not to introduce such issues in the remaining problem set and the exam. If you have further questions, feel free to email me at [email protected].