nash equilibrium

1. [40 marks] There is ongoing vaccination during the Covid-19 pandemic. These vaccines require a first dose and second dose. To use derivatives, let's consider the total dosage as a continuous variable. So, let the total dosage or quantity of vaccine taken by person j be xj, j = 1,2. When person 1 takes the vaccine, he protects himself (less likely to fall sick from infection or get infected) and also protects person 2 (less likely for person 1 to infect person 2) and vice versa. Therefore, let each person's benefit from vaccination be B(x1 + X2) = aln (1 + x1 + X2), where a is a positive parameter. Vaccination could have costly (adverse) side effects. For a vaccine of quantity, x;, let the cost to each person be C(x;) = cx;, where c is a positive parameter j = 1,2. Assume that a > c. (i) Find the Nash equilibrium quantities {x1, x2} of this game. (ii) Now suppose that the vaccine only gives the benefit, aln(1 + x1 + X2), with probability, p. With probability, 1 - p, it gives no benefit (i.e., zero benefit), where 0 c and risk-neutrality. (iii) To get protection, suppose a person must get the vaccine every year and the efficacy of the vaccine in 2022 will be higher than its efficacy in 2021. How will the increase in efficacy affect the Nash equilibrium quantities of this game? Prove your answer. (iv) Carefully illustrate the equilibrium in (iii) in a diagram