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Chapter 9: Problem 8

The Centers for Disease Control and Prevention (CDC) recommended against vaccinating the whole population against the smallpox virus because the vaccination has undesirable, and sometimes fatal, side effects. Suppose the accompanying table gives the data that are available about the effects of a smallpox vaccination program. $$ \begin{array}{ccc} \begin{array}{c} \text { Percent of } \\\ \text { population } \\\ \text { vaccinated } \end{array} & \begin{array}{c} \text { Deaths due to } \\\ \text { smallpox } \end{array} & \begin{array}{c} \text { Deaths due to } \\\ \text { vaccination side } \\\ \text { effects } \end{array} \\ \hline 0 \% & 200 & 0 \\\ 10 & 180 & 4 \\\ 20 & 160 & 10 \\\ 30 & 140 & 18 \\\ 40 & 120 & 33 \\\ 50 & 100 & 50 \\\ 60 & 80 & 74 \\ \hline \end{array} $$ a. Calculate the marginal benefit (in terms of lives saved) and the marginal cost (in terms of lives lost) of each \(10 \%\) increment of smallpox vaccination. Calculate the net increase in human lives for each \(10 \%\) increment in population vaccinated. b. Using marginal analysis, determine the optimal percentage of the population that should be vaccinated.

Short Answer

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Step by step solution

01

a. Calculate Marginal Benefit, Marginal Cost, and Net Increase in Human lives

To calculate the marginal benefit, marginal cost, and net increase in human lives for each \(10\%\) increment in population vaccination, we will look at the differences in the death counts for every increment. Let's represent the number of deaths due to smallpox as \(D_s\) and the number of deaths due to vaccination side effects as \(D_v\). The net increase in human lives (\(NL\)) can be calculated with the following formula: $$NL = D_s - D_v$$ Marginal benefit (\(MB\)) and marginal cost (\(MC\)) can be calculated by finding the differences in each respective column as the percentage of the population vaccinated increases. We will now perform these calculations for each \(10\%\) increment: $$MB = (D_{s_{previous}} - D_{s_{current}})$$ $$MC = (D_{v_{current}} - D_{v_{previous}})$$
02

b. Determine the optimal percentage of the population that should be vaccinated

To find the optimal percentage of the population that should be vaccinated, we will look for the increment where the net increase in human lives is at its highest. In other words, the point at which the marginal benefit of vaccination is greater than the marginal cost but starts decreasing afterward. Calculate the marginal benefit, marginal cost, and net increase in human lives, then find the optimal vaccination percentage by comparing these values. After performing these calculations, we find that a \(50\%\) vaccination rate is optimal, as it produces the highest net increase in human lives saved and is the point where further vaccination would cause an increase in marginal cost that is higher than the marginal benefit.

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