Consider the following steps Celia chooses how much care, x ∈ [0,1], to take in programming her robot. This effort costs her x^2/2. Nature chooses whether the robot steps on Peter’s pet salamander, leading to emotional harm to Peter of H > 0 (with probability 1 - x). If the robot does step on the salamander then there is a chance of π that Celia will be identified as the culprit. If there is no accident (the salamander is not stepped on), then Celia’s payoff is V - x^2/2. Peter and Luke both get zero. If there is an accident, but Celia is identified as the culprit, then Celia gets V - x^2/2. Peter gets -H. Luke gets zero. If there is an accident, and Celia is identified as the culprit, then Like (the judge) decides a level of compensation D ∈ R+ for Celia to pay Peter. Celia gets V - x^2/2 - D. Peter gets D - H. Luke gets −(βH - D)^2. f) Now we turn to social otimality rather than equilibrium. Consider the value judgement that the optimal level of care is the one that maximises the expected sum of Celia's and Peter's payoffs. According to this approach, we can ignore Luke's payoffs (as well as any payoff to the salamander or the robot) when considering social optimality. Write down this expected sum. g) Take a first-order condition with respect to x. Find an expression for the socially optimal level of x, according to the value judgement we made in question (f).

f) The expected sum of Celia's and Peter's payoffs is: (1-x)(V-x^2/2) + x(1-π)(V-x^2/2) +…

Answered: Celia chooses how much care, x ∈ [0,1],…

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter7: Uncertainty

Section: Chapter Questions

Problem 7.3P

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Consider the following steps

Celia chooses how much care, x ∈ [0,1], to take in programming her robot. This effort costs her x^2/2.

Nature chooses whether the robot steps on Peter’s pet salamander, leading to emotional harm to Peter of H > 0 (with probability 1 - x). If the robot does step on the salamander then there is a chance of π that Celia will be identified as the culprit.

If there is no accident (the salamander is not stepped on), then Celia’s payoff is V - x^2/2. Peter and Luke both get zero.

If there is an accident, but Celia is identified as the culprit, then Celia gets V - x^2/2. Peter gets -H. Luke gets zero.

If there is an accident, and Celia is identified as the culprit, then Like (the judge) decides a level of compensation D ∈ R+ for Celia to pay Peter. Celia gets V - x^2/2 - D. Peter gets D - H. Luke gets −(βH - D)^2.

f) Now we turn to social otimality rather than equilibrium. Consider the value judgement that the optimal level of care is the one that maximises the expected sum of Celia's and Peter's payoffs. According to this approach, we can ignore Luke's payoffs (as well as any payoff to the salamander or the robot) when considering social optimality. Write down this expected sum.

g) Take a first-order condition with respect to x. Find an expression for the socially optimal level of x, according to the value judgement we made in question (f).

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