I am having problems answering the question below can you please help?
With probability p, you will catch a disease that reduces your income from y, its level when you are healthy, to y-k, where k > 0. A vaccine is available, at cost c, that reduces the probability of your catching the disease from p to q<p.
a) Suppose that you know the values of p, q, y, k, and c, so that the only thing about which you are uncertain is whether you will catch the disease. Write the condition that determines whether or not you should buy the vaccine.
b) Now suppose that you know y, k, and c, but neither p nor q . Which is more relevant to your decision, the percentage amount by which the vaccine reduces the probability of catching the disease (what is usually reported in the press), or the absolute amount? Explain.
c) How do your answers to (a) and (b) change if you are a risk-averse expected-utility maximizer?
a) To determine whether or not you should buy the vaccine, you need to compare the expected cost of getting the disease without the vaccine to the expected cost of buying the vaccine.
Let's break this down:
- Without the vaccine, the expected cost of getting the disease is the probability of catching the disease (p) multiplied by the reduction in income (y - k). So the expected cost without the vaccine is p * (y - k).
- With the vaccine, the probability of catching the disease is reduced to q. So the expected cost with the vaccine is q * (y - k) plus the cost of buying the vaccine (c).
If the expected cost without the vaccine is higher than the expected cost with the vaccine, then it is financially favorable to buy the vaccine.
So, the condition that determines whether or not you should buy the vaccine is:
p * (y - k) > q * (y - k) + c
If this condition is true, you should buy the vaccine. If not, you should not buy the vaccine.
b) When you only know y, k, and c, and not p or q, the percentage amount by which the vaccine reduces the probability of catching the disease is more relevant to your decision.
The percentage reduction tells you how much risk is being mitigated by the vaccine. It allows you to evaluate the potential impact of the vaccine in relative terms, compared to the baseline risk (which you don't know).
The absolute amount, on the other hand, tells you the actual reduction in probability, but without knowing the baseline risk (p), it is difficult to assess its significance. The absolute reduction alone does not provide enough context to make an informed decision.
Therefore, when you don't know p or q, the percentage amount by which the vaccine reduces the probability of catching the disease is more useful for decision-making.
c) If you are a risk-averse expected-utility maximizer, your answers to (a) and (b) may change because of your risk-aversion.
In (a), your risk-aversion would make you more inclined to buy the vaccine even if the cost difference between the two options is relatively small. This is because the potential loss of income from the disease (y - k) would have a more significant impact on your well-being due to your risk-aversion.
In (b), your risk-aversion would likely make the percentage reduction in probability even more relevant to your decision. As a risk-averse individual, you would be more concerned about minimizing the probability of catching the disease, even if the absolute amount of reduction is low. This is because your aversion to uncertainty and potential losses would influence your preference for mitigating risk.
Therefore, as a risk-averse expected-utility maximizer, you would be more likely to buy the vaccine in both cases (a) and (b) compared to if you were not risk-averse.