Problem 4 Inoculation (16 points) Consider a population of individuals with a normalized size of 1 that is faced with a viral infection. Each individual is exposed to the virus with probability MR), where R 6 [0,1] is the fraction of the population that is immune. It is assumed that MR) 2 1 R. Individual \"i E [0,1] can achieve immunity through a vaccination. The vaccination cost differs across individuals by C(i) : i. If the individual vaccinates, exposure to the virus does not result in illness and utility is regardless of exposure given by U, = 1 C(16) If the individual does not vaccinate, the outcome is now uncertain. With probability MR), the individual is exposed and become sick with a resulting utility of U, = 0. With probability 1 MR) the individual escapes exposure and remains healthy with resulting utility U, = 1. It is assumed that the expected utility over this uncertainty can be written as EU, 2 AU?) - 0 + (1 MR - 1 = 1 MR). To summarize, individual 21's expected utility as a function of the vaccination choice, 'U E {0,1}, is, 1 3' if \"U = 1 EUZ('I.'|R) : 1 (1 1-!)A(R) _ 118(1) : {R if \"U : 0 where 'U : 1 represents the choice to vaccinate and i: : 0 is the choice not to. 1. Taking everybody else's choice of whether to vaccinate or not as given and fully reected in a fraction R of the population being immune, determine individual -i's best response !,(R) for any 3' E [0, 1]. 2. Determine the Nash equilibrium. What fraction of the population will in equilibrium choose to vaccinate, R\"? Problem 4 Inoculation (16 points) Consider a population of individuals with a normalized size of 1 that is faced with a viral infection. Each individual is exposed to the virus with probability MR), where R 6 [0,1] is the fraction of the population that is immune. It is assumed that MR) 2 1 R. Individual \"i E [0,1] can achieve immunity through a vaccination. The vaccination cost differs across individuals by C(i) : i. If the individual vaccinates, exposure to the virus does not result in illness and utility is regardless of exposure given by U, = 1 C(16) If the individual does not vaccinate, the outcome is now uncertain. With probability MR), the individual is exposed and become sick with a resulting utility of U, = 0. With probability 1 MR) the individual escapes exposure and remains healthy with resulting utility U, = 1. It is assumed that the expected utility over this uncertainty can be written as EU, 2 AU?) - 0 + (1 MR - 1 = 1 MR). To summarize, individual 21's expected utility as a function of the vaccination choice, 'U E {0,1}, is, 1 3' if \"U = 1 EUZ('I.'|R) : 1 (1 1-!)A(R) _ 118(1) : {R if \"U : 0 where 'U : 1 represents the choice to vaccinate and i: : 0 is the choice not to. 1. Taking everybody else's choice of whether to vaccinate or not as given and fully reected in a fraction R of the population being immune, determine individual -i's best response !,(R) for any 3' E [0, 1]. 2. Determine the Nash equilibrium. What fraction of the population will in equilibrium choose to vaccinate, R\"? WW 8' = fol EU,[v,|R)di, where u,- is individual 's vaccination choice and R = I; vidi is the fraction of the population that is choosing to vaccinate. To simplify matters a hit, make the observation that for purposes of maximizing total surplus, if it is optimal to vaccinate individual i, then any individual with index less than i must also vaccinate as part of maximizing S. Therefore, restate total surplus as a function of R E [0, 1] where it is directly imposed that U,- = 1 for all i E [0, R] and u,- = 0 for all i E (R, 1]. With that total surplus can be written as, R 1 R 1 R 3(3):] (1i)di+/ Rdi=f ans] (iif as D R D R I] 1 R = [115' +R[z']}, ill-El = (R 0) +R(1 R) le 0) o = R + R R? $32 3 2 2R ER . In case the integrations are intimidating, the surplus can also be illustrated as follows, EU, Total surplus is the combined area of the gray and orange boxes. The gray box is the combined utility of the individuals who are vaccinating. And the orange area is the combined utility of the agents that are not vaccinating. In the above example, 8(R), is drawn for an example where R 3 1/2. The gray area is a combination of a rectangle with area R[1 R) and a triangle with area %R [1 (1 3.)] = %R2. The orange area is a rectangle with area R(1 R). Thus, we have that S(R) = 2R(1 R) + %R2 2 2R gRg, which conrms the calculation using the integrals. (a) What is the total surplus associated with the Nash equilibrium you found in question 2? (13) Suppose a social planner can dictate the fraction of the population that must be vac cinated in service of maximizing total surplus. What fraction would the social planner choose? (c) Discuss why the social plarnler is choosing a larger fraction. WW 8' = fol EU,[v,|R)di, where u,- is individual 's vaccination choice and R = I; vidi is the fraction of the population that is choosing to vaccinate. To simplify matters a hit, make the observation that for purposes of maximizing total surplus, if it is optimal to vaccinate individual i, then any individual with index less than i must also vaccinate as part of maximizing S. Therefore, restate total surplus as a function of R E [0, 1] where it is directly imposed that U,- = 1 for all i E [0, R] and u,- = 0 for all i E (R, 1]. With that total surplus can be written as, R 1 R 1 R 3(3):] (1i)di+/ Rdi=f ans] (iif as D R D R I] 1 R = [115' +R[z']}, ill-El = (R 0) +R(1 R) le 0) o = R + R R? $32 3 2 2R ER . In case the integrations are intimidating, the surplus can also be illustrated as follows, EU, Total surplus is the combined area of the gray and orange boxes. The gray box is the combined utility of the individuals who are vaccinating. And the orange area is the combined utility of the agents that are not vaccinating. In the above example, 8(R), is drawn for an example where R 3 1/2. The gray area is a combination of a rectangle with area R[1 R) and a triangle with area %R [1 (1 3.)] = %R2. The orange area is a rectangle with area R(1 R). Thus, we have that S(R) = 2R(1 R) + %R2 2 2R gRg, which conrms the calculation using the integrals. (a) What is the total surplus associated with the Nash equilibrium you found in question 2? (13) Suppose a social planner can dictate the fraction of the population that must be vac cinated in service of maximizing total surplus. What fraction would the social planner choose? (c) Discuss why the social plarnler is choosing a larger fraction