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Problem 1: Covid-19 Vaccination Campaign [6 Points]

You were just hired by the CDC to lead the Covid-19 vaccination campaign. The private payoff from getting vaccinated is 7r = -c, where c is cost parameter that incorporates the risk of complications associated with vaccination （vaccines per se are free of charge）. The private payoff associated with not getting vaccinated is 7r = —bJ?（）（Vc— V）, where:

-I）is a cost parameter that incorporates the health risks associated with catching Covid-19;

-7?（）is the basic reproduction number （i.e. how transmissible the disease is）;

  • –      Vc is the critical level of immunity, or vaccination, the population needs to reach to eventually eradi­cate Covid-19 （i.e. the herd immunity threshold）;
  • –        V is the fraction of the population that is currently immune/vaccinated.

Together, /7/（）（Vr一 V）n represents the probability of catching Covid-19. For simplicity, assume that the immunity one develops to Covid-19 is permanent, and that catching the disease provides you with the same immunity that the vaccine does, so that V includes both vaccinated and infected individuals.

l.A. [0.5 point] Let S denote the fraction of the population that is susceptible. Find R, the effective repro­duction number and find Vc, the herd immunity threshold. How does Vc change as new variants that have higher a R（）become the new dominant strain? Show your work and explain in words.

l.B. Private incentives of vaccination.

  1. i)      [0.5 point] Find I the equilibrium level of vaccination that private individuals would choose on their own. Show your work and explain in words.
  1. ii)    [0.25 point] How does V* change if people falsely believe that complication costs c are much higher than what they actually are? Show your work and explain in words why a good estimate of the c is impor­tant.
  1. iii)     [0.25 point] For this subquestion only; suppose that a newly approved vaccine makes it such that c = 0. How does V* change and how does it compare to Vc? Show your work and explain in words what this means and what it implies if c > 0.

l.C. Social incentives of vaccination.

  1. i)      [1 point] Suppose that the total population is of size N. Find V**, the optimal level of vaccination in the population. How does it compare to the private vaccination level V* and what does this imply for the optimality of V*?

ii）   [0.5 point] For this subquestion only, suppose that R（）= 3, c = 5, and 6 = 4. How does V** compare to Vc? Is V** optimal? Explain in words whether or not you can conclude using the value of V** that Covid-19 should optimally be eradicated.

  1. iii)      [0.5 point] For this subquestion only, suppose that = 3, c = 5, and 6 = 8. How does V** compare to Vc? Explain in words whether V** optimal or not, and explain whether or not you can conclude using the value of V** that Covid-19 should optimally be eradicated.

l.D. Suppose that Rq = 3, c = 5, and 6 = 4. Suppose that the CDC wants to eradicate Covid-19 and that they expect eradication will occur T years from now.

i) [0.5 point] Use the Solver of choice (e.g. Excel, Google Sheets) and plot (1) the marginal private ben­efits of vaccination, (2) the marginal social benefits of vaccination, and (3) the marginal costs of vaccination as a function of the vaccination level V (i.e. benefits and costs are on the 〃-axis and vaccination level is on ⑦-axis). You can also plot this by hand, but make sure to include value for the z-axis and g-axis, and to draw your graph to scale. Plot all curves on the same graph and make sure to identify V*, V** and Vc on your graph. Hint: Use the fact that the marginal social benefits of vaccination are equal to the private benefits plus the marginal benefits that the unvaccinated individual incur.

ii) [1 point] What are the per-period costs of achieving eradication that the society incurs each year in the eradication phase (i.e. from £ = 0 to t = T)? What are the per-period benefits that the society incurs each year during the eradication phase? What are the per-period benefits that the society incurs in the post-eradication phase (i.e. in £ = T + 1 and after)?

iii) [1 point] Experts confirm to you that vaccinating at a rate of Vc for T = 50 years will eradicate Covid-19 in Period T. Assume the society’s discount rate is r = 0.1 or 10%. Should the CDC (a benevolent social planner) eradicate Covid-19? If so, how many years will it take for eradication to yield net benefits? If not, why should the CDC decide not to eradicate Covid-19?

Problem 2: The Neverending HIV/AIDS Epidemic [4 Points]

You were hired as a consultant by WHO to tackle the HIV/AIDS epidemic. Thankfully in many parts of the world, the epidemic has reached a steady state where the growth in susceptible individuals is equal to the growth in infected individuals. Let I denote the fraction of individuals that are infected (i.e. 0 < / < 1) and (1-1) denote the fraction of individuals that are susceptible. Let 0 denote the effective contact rate between the susceptible and infected individuals, and let 7 denote the rate of recovery (i.e., the rate at which infected individuals transition back into susceptible individuals).1 As such, the growth in I (that is, the change over time in the fraction of individuals that are infected, i; pronounced 〃I-dot〃)is given by:

I = (3(1- 1)1 -yl

2.A [0.5 point] Find an expression for the steady-state value of infected individuals, Iss・ Under which conditions does Iss = 0? Given your answer, how would WHO want its public health policies to affect 6 and 7? Show your work and explain in words.

2.B Suppose WHO is considering a nonpharmaceutical intervention in the form of a massive information campaign promoting the use of condoms, which would help reducing the effective contact rate /?.

  1. i)      [0.25 point] What will happen to the long-term steady-state level of infected individuals? Show your work in math and explain in words.
  1. ii)  [1 point] Now suppose that, before the information campaign, 0 = 2 and 7 = 1.25, and that experts predict the campaign would reduce contact rate 0 by 5% indefinitely. Assume the campaign costs $C today and would save lives that a have a constant value of a disability-adjusted life year (V-DALY) of $10,000. Assume the population contains 1,000 individuals and WHO’s discount rate is r = 0.1 or 10%. What is the value of Iss today? What would be the value of Iss if WHO decides to proceed with the information campaign? How high does C need to be for WHO to choose not to proceed with the campaign? Attach a print screen of your Excel sheet. Hint: Use a 300-year horizon to approximate the infection costs that are incurred indefinitely.

1 Infected individuals die from complications associated with HIV/AIDS and an equal number of individuals are bom into the susceptible population.

2.C Now suppose that after extensive research a vaccine for HIV/Al DS has finally been developed, ef­fectively creating a new class of individuals, the vaccinated individuals V. The vaccine is only effective against susceptible individuals (it provides them with indefinite immunity) and it has no effect on people that are already infected. Suppose that a proportion u of susceptible individuals are vaccinated every pe­riod. Because of this new class of individuals, I still represents the fraction of individuals that are infected, but (1 — I) now represents the fraction of individuals that are either susceptible S or vaccinated V. Hence, S + I + V = 1. Suppose that = 2, 7 = 1.25, that WHO’s discount rate is r = 0.1 or 10%, that the total population of interest is 1,000 people, and that the vaccine cost function is C(u) = lOOOw2.

i) [0.25 point] Write a new expression for I that incorporates the fact that some people will now be vaccinated, write an expression for the growth of the fraction of individuals that are susceptible S, and

.

write an expression for the growth of the fraction of individuals that are vaccinated V. Note: don’t input the parameter values yet—well do that later directly in the Solver.

ii) [1 point] Suppose WHO’s objective is to vaccinate systematically a fraction u = 0.1 of susceptible individuals each year. In what period T is the disease eradicated? What needs to be the value of a disability- adjusted life year (V-DALY) to make eradication a good investment? Attach a print screen of your Excel sheet. Hint: Input some arbitrary value of V-DALY and ask your Solver to find the one that makes it exactly worth it to eradicate HIV/AIDS.

  1. iii)     [1 point] As the only economist working for WHO怎 HIV/AIDS division, you are wondering if vaccinating systematically every year at a rate of u = 0.1 is the optimal thing to do. Using a solver, find the optimal vaccination level for each year, assuming that 0 <u(t) < 0.1. Does your answer differ from WHO’s proposed strategy in 2.C.(ii)? Explain why or why not your answer differs from WHO’s proposed strategy. Assume that vaccination cannot continue beyond T = 30 years and that V-DALY = $10,000. Attach a print screen of your Excel sheet showing the optimal vaccination level over 30 years.


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