(a) Calculate the expected profits for Rachel on each case that reflects Emma's effort level. Which case does Rachel prefer? (b) Rachel's objective is to maximize the expected profit, subject to that Emma works for Rachel and Emma puts effort. However, effort level is not observable. Hence, Rachel needs to write a contract based on the observables. Let's say, Rachel pays Emma based on the outcome: when the profit is $0, when the profit is $2000, and when the profit is $3,000. Then Emma has three options: (i) Not to work for Rachel (ii) Work for Rachel without effort (iii) Work for Rachel with effort Find Emma's expected utility on each option. (c) Assuming Rachel wants Emma to put effort, her objective essentially becomes to find the lowest contingent payment scheme that is just enough for Emma to work for Rachel, and gives an incentive for Emma to put effort. Formally, we can write this as: min 0.1 +0.3xM +0.6 subject to and 0.1√ TL+0.3√/FM+0.6√TH-52 15, 0.1√√+0.3√√M+0.6√TH-5≥ 0.6√/TL+0.3√/M +0.1√/TH. (1) (2)
(a) Calculate the expected profits for Rachel on each case that reflects Emma's effort level. Which case does Rachel prefer? (b) Rachel's objective is to maximize the expected profit, subject to that Emma works for Rachel and Emma puts effort. However, effort level is not observable. Hence, Rachel needs to write a contract based on the observables. Let's say, Rachel pays Emma based on the outcome: when the profit is $0, when the profit is $2000, and when the profit is $3,000. Then Emma has three options: (i) Not to work for Rachel (ii) Work for Rachel without effort (iii) Work for Rachel with effort Find Emma's expected utility on each option. (c) Assuming Rachel wants Emma to put effort, her objective essentially becomes to find the lowest contingent payment scheme that is just enough for Emma to work for Rachel, and gives an incentive for Emma to put effort. Formally, we can write this as: min 0.1 +0.3xM +0.6 subject to and 0.1√ TL+0.3√/FM+0.6√TH-52 15, 0.1√√+0.3√√M+0.6√TH-5≥ 0.6√/TL+0.3√/M +0.1√/TH. (1) (2)
Chapter8: Game Theory
Section: Chapter Questions
Problem 8.12P
Related questions
Question
Please just answer b,c,d
Transcribed Image Text:Consider the following situation. Emma (the agent) works for Rachel (the principal). Emma's
work has three possible outcomes (that is, profit to Rachel):
where e = 0 means that Emma puts no effort on her job, and e = 1 means that Emma puts effort
on her job. Each cell shows the associated probabilities on each outcome.
Emma's utility function is given by
Profits
e=0
e=1
$0
p=0.6| p=0.1
$2,000 p=0.3 p=0.3
$3,000 p = 0.1| p=0.6
U =
where w is the payment from Rachel. If Emma does not work for Rachel, Emma will have an
outside option that gives her the utility level of 15.
√ if she puts no effort,
√-5 if she puts effort,
(a) Calculate the expected profits for Rachel on each case that reflects Emma's effort level. Which
case does Rachel prefer?
(b) Rachel's objective is to maximize the expected profit, subject to that Emma works for Rachel
and Emma puts effort. However, effort level is not observable. Hence, Rachel needs to write
a contract based on the observables. Let's say, Rachel pays Emma based on the outcome: IL
when the profit is $0, M when the profit is $2000, and when the profit is $3,000. Then
Emma has three options:
subject to
(i) Not to work for Rachel
(ii) Work for Rachel without effort
(iii) Work for Rachel with effort
Find Emma's expected utility on each option.
(c) Assuming Rachel wants Emma to put effort, her objective essentially becomes to find the
lowest contingent payment scheme that is just enough for Emma to work for Rachel, and
gives an incentive for Emma to put effort. Formally, we can write this as:
min 0.1 +0.3z +0.6FH
and
0.1√√TL+0.3√√FM+0.6√/TH - 5 ≥ 15,
0.1√TL+0.3√+0.6√/TH - 5 ≥ 0.6√TL +0.3√/FM +0.1√TH.
What is Constraint (1) called? What is Constraint (2) called?
(d) For your information, the solution for (c) is:
IL = $130.60
M = $400
TH=$459.16.
(1)
(2)
Transcribed Image Text:This contract will maximize Rachel's expected profits.
There is another way to achieve Rachel's objective. Instead of specifying the payment on
each outcome, Rachel can offer Emma a flat payment with a bonus. For example, Rachel
offers $225. This flat payment is enough for Emma to work for Rachel. However, if this is
the only payment, Emma does not have any incentive to put effort. For a better outcome,
Rachel needs to make sure Emma put her effort. To induce her effort, Rachel can promise
some bonus as some percentage of $B upon a better outcome. Let's say the expected profit
is $1200 higher with effort than without effort. Then Rachel sets B = 1200 and promise a
bonus of $xB, where 0 < x < 1, upon a better outcome. To induce Emma's effort, what is
the minimum z that Rachel promises to Emma?
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