A medical researcher is trying to cure a disease. For each unit of effort she puts into her work, she generates a utility benefit of 10 for each member of society. There are 1,000 people in society besides the medical researcher. The medical researcher doesn’t care about other people. She is in it for the glory. For each unit of effort she puts into her work, she gets a utility benefit of 1000 (which is inclusive of the 10 that she gets for being a member of society, plus a payoff of 990 in glory). If she exerts effort e, she also suffers cost e^2.
(a) Suppose a policymaker who was a committed utilitarian (including caring about the medical researcher’s glory, since the medical researcher cares about it) was to choose the level of effort the medical researcher exerts. That policymaker would add up the total utility in society (including the medical researcher’s utility) from any given level of effort and choose the level of effort that maximizes that aggregate utility.
i. What (social) utility function would the policymaker maximize?
ii. What level of effort, eFB, would the policymaker demand of the medical researcher?
iii. Suppose that the policymaker wanted to implement this first-best policy (eFB) by paying the medical researcher a subsidy σ for each unit of effort. What level of subsidy, σ FB, must the policymaker offer to get the medical research to choose the first-best level of effort?
(c) Suppose, now, that our policymaker can only pay for the subsidy through inefficient
i. What (social) utility function would the policymaker maximize?
ii. Will the subsidy she chooses, σSB, be higher or lower than the subsidy, σ FB, that implemented the first-best effort? Why?
iii. Calculate the second-best subsidy that the policymaker will implement.
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