Question

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. $$\begin{array}{ccc}\begin{array}{c}\text { Percent of } \\\\\text { population } \\\\\text { vaccinated } \end{array} & \begin{array}{c}\text { Deaths due to } \\ \text { smallpox }\end{array} & \begin{array}{c}\text { Deaths due to } \\\\\text { vaccination side } \\\\\text { effects }\end{array} \\\0 \% & 200 & 0 \\\10 & 180 & 4 \\\20 & 160 & 10 \\\30 & 140 & 18 \\\40 & 120 & 33 \\\50 & 100 & 50 \\\60 & 80 & 74\end{array}$$ 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

After calculating marginal benefits and costs for each increment, the optimal percentage of the population to be vaccinated is determined by the last increment where the net increase in human lives is positive.

Step-by-step solution

Calculating Marginal Benefit

To calculate the marginal benefit, subtract the number of deaths due to smallpox after vaccination from the number before. For example, for the first 10% increment, the marginal benefit is 200 (percent 0%) - 180 (percent 10%) = 20 lives saved.

Calculating Marginal Cost

To calculate the marginal cost, subtract the number of deaths due to vaccine side effects before vaccination from the number after. For the first 10% increment, the marginal cost is 4 (percent 10%) - 0 (percent 0%) = 4 lives lost.

Calculating Net Increase

The net increase in human lives is the marginal benefit minus the marginal cost. For the first 10% increment, it is 20 (marginal benefit) - 4 (marginal cost) = 16 lives saved.

Repeat Marginal Analysis for Each Increment

Repeat the steps for each subsequent 10% increment to calculate the marginal benefit, marginal cost, and net increase for each.

Determine Optimal Vaccination Percentage

Review the net increase for each increment. The optimal percentage for vaccination is the last increment that shows a positive net increase in lives saved. After this point, the marginal cost (additional deaths from side effects) exceeds the marginal benefit (lives saved from smallpox).

Key concepts

Marginal Cost

Marginal cost in microeconomics is a measure of the increase in total cost when one additional unit of output is produced. In the context of a vaccination program, the marginal cost is viewed in terms of the human lives lost due to the undesirable side effects of the vaccine. To calculate the marginal cost for each 10% increment of the population vaccinated against smallpox, we compare the number of deaths due to vaccine side effects at the current percentage with the previous one. For instance, if at 0% vaccination no lives are lost due to side effects and at 10% vaccination we have 4 deaths due to side effects, the marginal cost for the first 10% increment is 4 lives lost. It's essential to closely monitor the marginal cost to ensure that the intervention, while aiming to protect the population, does not result in a loss of more lives than it saves.

Understanding the marginal cost helps policymakers to weigh the consequences of their decisions and ensures resources are utilized where they offer the greatest benefit without imposing greater risks. This decision-making is a fundamental aspect of cost-benefit analysis and plays a crucial role in public health strategies.

Marginal Benefit

Marginal benefit in microeconomics refers to the additional satisfaction or utility gained from consuming or doing one more unit of a good or service. For a vaccination campaign, the marginal benefit can be computed by determining the reduction in deaths from the disease as vaccination rates increase. To illustrate, when the vaccination percentage increases from 0% to 10%, and the number of deaths due to smallpox decreases from 200 to 180, the marginal benefit is the 20 lives saved. This benefit highlights the positive impact of the vaccination program in protecting the population from the illness.

The concept of marginal benefit is crucial in finding the balance between the advantages of additional vaccination coverage and the risks associated with it. Policymakers must maximize this benefit up to the point where the marginal benefit equals the marginal cost, ensuring that each vaccination decision is justified by its positive outcome on public health.

Optimal Vaccination Percentage

The optimal vaccination percentage is the point at which the net benefit of vaccinating an additional portion of the population is maximized and the marginal benefit equals the marginal cost. At this juncture, extending vaccination coverage further would not result in a net increase in saved lives due to the rise in fatalities from vaccine side effects. The calculation of the optimal vaccination percentage involves conducting a marginal analysis for each successive increment of vaccinated individuals, considering both the reduction in disease-related deaths and the increase in deaths attributed to vaccination side effects.

Identifying the optimal level of vaccination is an example of applying microeconomic principles to public health policy. It helps in allocating health resources effectively and avoiding potential harm from over-vaccination. The ideal percentage strikes a balance between the benefits of disease prevention and the risks from vaccination, leading to the greatest net positive impact on community health.

Cost-Benefit Analysis

Cost-benefit analysis is a systematic approach for comparing the benefits and costs of a decision, project, or policy, which includes both qualitative and quantitative evaluations. In our case, the benefits are the lives saved from preventing smallpox, while the costs are the lives lost from vaccination side effects. It involves tabulating the benefits and costs over the range of possible vaccination rates and finding the point where net benefits (benefits minus costs) are highest.

This method provides a clear framework for making complex decisions where various factors must be weighed against each other. It allows for an objective comparison to determine if an investment (such as a vaccination program) yields a positive return, or in this case, a net increase in lives saved. The application of cost-benefit analysis in the healthcare sector is instrumental in making informed decisions about resource allocation and disease prevention strategies, ensuring that public health interventions do more good than harm.