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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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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Opportunity Cost
Opportunity cost is a key concept in economics. It helps us understand what we are giving up when we make a choice. In this exercise, opportunity cost plays a role in deciding how much of the population to vaccinate against smallpox. If the opportunity cost is high, it means we are losing more than we gain.
When deciding on vaccination rates:
  • Opportunity cost refers to the lives lost due to vaccination side effects.
  • We compare these losses to the lives saved from preventing smallpox.
  • The goal is to choose a vaccination level where our sacrifice (or cost) doesn鈥檛 outweigh our gains.
Understanding opportunity cost helps us make better decisions, aiming for the greatest net positive effect.
Marginal Benefit
Marginal benefit is all about the additional advantage we get from one more unit of something. In terms of vaccination, it measures the extra lives saved as more people get vaccinated.
Here's how we look at it:
  • Calculate the difference in lives saved as vaccination rates increase by 10% increments.
  • For example, if vaccinating 10% of the population saves 20 lives compared to no vaccination, the marginal benefit is those 20 saved lives.
  • As the vaccination percentage increases, typically the marginal benefit starts to decrease.
Keeping an eye on marginal benefits helps understand how effective each additional increment of vaccination is in saving lives.
Marginal Cost
Marginal cost measures the additional cost incurred from producing one more unit. In the context of vaccination, it represents the extra lives lost due to side effects when more people get vaccinated. This aids in understanding where the trade-off between benefit and cost lies.
Key points include:
  • Calculate how many additional lives are lost due to vaccination side effects with each 10% increase in vaccination.
  • Identify when the marginal cost increases beyond the marginal benefit.
  • Recognize that even beneficial actions like vaccinations have costs.
Balancing marginal cost against marginal benefit is central to optimizing any decision-making process.
Optimal Decision-Making
When determining the optimal vaccination rate, it鈥檚 all about finding balance. This means looking for the point where the benefits of vaccinating more people just outweigh the costs associated with it. This optimal point ensures maximum positive impact with minimal negative repercussions.

The approach involves:
  • Assessing where the net increase in saved lives is highest as vaccination levels rise.
  • The optimal decision comes before the point where costs start increasing more than benefits.
  • In our original problem, the optimal vaccination rate was found to be at 50%, as it provided the highest net positive impact.
Implementing optimal decision-making in real-life scenarios helps in achieving effective results without unnecessary risks.

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Most popular questions from this chapter

You own and operate a bike store. Each year, you receive revenue of \(\$ 200,000\) from your bike sales, and it costs you \(\$ 100,000\) to obtain the bikes. In addition, you pay \(\$ 20,000\) for electricity, taxes, and other expenses per year. Instead of running the bike store, you could become an accountant and receive a yearly salary of \(\$ 40,000\). A large clothing retail chain wants to expand and offers to rent the store from you for \(\$ 50,000\) per year. How do you explain to your friends that despite making a profit, it is too costly for you to continue running your store?

Suppose you have just paid a nonrefundable fee of \(\$ 1,000\) for your meal plan for this academic term. This allows you to eat dinner in the cafeteria every evening. a. You are offered a part-time job in a restaurant where you can eat for free each evening. Your parents say that you should eat dinner in the cafeteria anyway, since you have already paid for those meals. Are your parents right? Explain why or why not. b. You are offered a part-time job in a different restaurant where, rather than being able to eat for free, you receive only a large discount on your meals. Each meal there will cost you \(\$ 2\); if you eat there each evening this semester, it will add up to \(\$ 200\). Your roommate says that you should eat in the restaurant since it costs less than the \(\$ 1,000\) that you paid for the meal plan. Is your roommate right? Explain why or why not.

Hiro owns and operates a small business that provides economic consulting services. During the year he spends \(\$ 57,000\) on travel to clients and other expenses. In addition, he owns a computer that he uses for business. If he didn't use the computer, he could sell it and earn yearly interest of \(\$ 100\) on the money created through this sale. Hiro's total revenue for the year is \(\$ 100,000\). Instead of working as a consultant for the year, he could teach economics at a small local college and make a salary of \(\$ 50,000\). a. What is Hiro's accounting profit? b. What is Hiro's economic profit? c. Should Hiro continue working as a consultant, or should he teach economics instead?

You have bought a \(\$ 10\) ticket in advance for the college soccer game, a ticket that cannot be resold. You know that going to the soccer game will give you a benefit equal to \(\$ 20\). After you have bought the ticket, you hear that there will be a professional baseball post-season game at the same time. Tickets to the baseball game cost \(\$ 20\), and you know that going to the baseball game will give you a benefit equal to \(\$ 35\). You tell your friends the following: "If I had known about the baseball game before buying the ticket to the soccer game, I would have gone to the baseball game instead. But now that I already have the ticket to the soccer game, it's better for me to just go to the soccer game." Are you making the correct decision? Justify your answer by calculating the benefits and costs of your decision.

In each of the following examples, explain whether the decision is rational or irrational. Describe the type of behavior exhibited. a. Kookie's best friend likes to give her gift cards that Kookie can use at her favorite stores. Kookie, however, often forgets to use the cards before their expiration date or loses them. Kookie, though, is careful with her own cash. b. In 2010 , the Panera Bread company opened a store in Clayton, Missouri, that allowed customers to pay any amount they like for their orders; instead of prices, the store listed suggested donations based on the cost of the goods. All profits went to a charitable foundation set up by Panera. In \(2011,\) the store was pleased with the success of the program. c. Rick has just gotten his teaching degree and has two job offers. One job, replacing a teacher who has gone on leave, will last only two years. It is at a prestigious high school, and he will be paid \(\$ 35,000\) per year. He thinks he will probably be able to find another good job in the area after the two years are up but isn't sure. The other job, also at a high school, pays \(\$ 25,000\) per year and is virtually guaranteed for five years; after those five years, he will be evaluated for a permanent teaching position at the school. About \(75 \%\) of the teachers who start at the school are hired for permanent positions. Rick takes the five-year position at \(\$ 25,000\) per year. d. Kimora has planned a trip to Florida during spring break in March. She has several school projects due after her return. Rather than do them in February, she figures she can take her books with her to Florida and complete her projects there. e. Sahir overpaid when buying a used car that has turned out to be a lemon. He could sell it for parts, but instead he lets it sit in his garage and deteriorate. f. Barry considers himself an excellent investor in stocks. He selects new stocks by finding ones with characteristics similar to those of his previous winning stocks. He chocks up losing trades to ups and downs in the macroeconomy.
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