Consider the following steps: 1. Celia chooses how much care, x ∈ [0, 1], to take in programming her robot. This effort costs her x^2/2. 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 not 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 Luke (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. h) What would β have to equal, in order for Celia to choose the socially optimal level of x in a Subgame Perfect Equilibrium ? We are looking for a mathematical expression, rather than a number i) What is the intuition for the result you got in (h)?
Consider the following steps: 1. Celia chooses how much care, x ∈ [0, 1], to take in programming her robot. This effort costs her x^2/2. 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 not 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 Luke (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.
h) What would β have to equal, in order for Celia to choose the socially optimal level of x in a Subgame Perfect Equilibrium ? We are looking for a mathematical expression, rather than a number
i) What is the intuition for the result you got in (h)?
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