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) The medical researcher’s payoff from exerting effort e is 1000e − e^2. What level of effort, e∗, will she exert?(b) 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. Would the policymaker like the medical researcher to exert more or less effort than your answer to part (a)? Why? (Use words, not math, to answer this.)iii. What level of effort, eFB, would the policymaker demand of the medical researcher?iv. 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 taxation. In particular, assume that if a total subsidy of eσ must be paid to the researcher, then a total amount of revenues 2eσ must be collected. The tax burden will be shared equally by everyone except the medical researcher (who pays no taxes). Assume taxes come linearly out of the payoffs of each member of society. Suppose the policymaker chooses the subsidy that maximizes the utilitarian payoff, taking account of these taxes and the level of medical research.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. 8.A medical researcher is trying to cure a disease. For each unit of effort sheputs into her work, she generates a utility benefit of 10 for each member ofsociety. 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 theglory. For each unit of effort she puts into her work, she gets a utility benefitof 1000 (which is inclusive of the 10 that she gets for being a member ofsociety, plus a payoff of 990 in glory). If she exerts effort e, she also sufferscost e².(a) The medical researcher's payoff from exerting effort e is 1000e - e².What level of effort, e*, will she exert?(b) Suppose a policymaker who was a committed utilitarian (includingcaring about the medical researcher's glory, since the medical researchercares about it) was to choose the level of effort the medical researcherexerts. That policymaker would add up the total utility in society(including the medical researcher's utility) from any given level of effortand choose the level of effort that maximizes that aggregate utility.i. What (social) utility function would the policymaker maximize?ii. Would the policymaker like the medical researcher to exert more orless effort than your answer to part (a)? Why? (Use words, not math,to answer this.) iii. What level of effort, eB, would the policymaker demand of themedical researcher?iv. Suppose that the policymaker wanted to implement this first-bestpolicy (eFB) by paying the medical researcher a subsidy o for eachunit of effort. What level of subsidy, oFB, must the policymaker offerto get the medical research to choose the first-best level of effort?(c) Suppose, now, that our policymaker can only pay for the subsidythrough inefficient taxation. In particular, assume that if a total subsidyof eo must be paid to the researcher, then a total amount of revenues 2eomust be collected. The tax burden will be shared equally by everyoneexcept the medical researcher (who pays no taxes). Assume taxes comelinearly out of the payoffs of each member of society. Suppose thepolicymaker chooses the subsidy that maximizes the utilitarian payoff,taking account of these taxes and the level of medical research.i. What (social) utility function would the policymaker maximize?ii. Will the subsidy she chooses, o SB, be higher or lower than thesubsidy, o FB, that implemented the first-best effort? Why?iii. Calculate the second-best subsidy that the policymaker willimplement.

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) The medical researcher’s payoff from exerting effort e is 1000e − e^2. What level of effort, e∗, will she exert? (b) 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. Would the policymaker like the medical researcher to exert more or less effort than your answer to part (a)? Why? (Use words, not math, to answer this.) iii. What level of effort, eFB, would the policymaker demand of the medical researcher? iv. 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 taxation. In particular, assume that if a total subsidy of eσ must be paid to the researcher, then a total amount of revenues 2eσ must be collected. The tax burden will be shared equally by everyone except the medical researcher (who pays no taxes). Assume taxes come linearly out of the payoffs of each member of society. Suppose the policymaker chooses the subsidy that maximizes the utilitarian payoff, taking account of these taxes and the level of medical research. 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.