Question

A residential community has 100 residents who are concerned about security. The accompanying table gives the total cost of hiring a 24 -hour security service as well as each individual resident's total benefit. \begin{tabular}{c|c|c} Quantity of security guards & Total cost & Total individual benefit to each resident \\ \hline 0 & \(\$ 0\) & \(\$ 0\) \\ 1 & 150 & 10 \\ 2 & 300 & 16 \\ 3 & 450 & 18 \\ 4 & 600 & 19 \end{tabular} a. Explain why the security service is a public good for the residents of the community. b. Calculate the marginal cost, the individual marginal benefit for each resident, and the marginal social benefit. c. If an individual resident were to decide about hiring and paying for security guards on his or her own, how many guards would that resident hire? d. If the residents act together, how many security guards will they hire?

Short answer

Answer: If the residents act together, they would hire 3 security guards.

Step-by-step solution

a. Identifying a public good

The security service is a public good for the residents of the community because it has two main properties: non-excludability and non-rivalry. Non-excludability means that once the good is provided, no one within the community can be excluded from using or benefiting from it. In this case, once the security service is hired, its protection covers all the residents without excluding anyone due to their unwillingness or inability to pay for the service. Non-rivalry means that one person's consumption of the good does not diminish the amount available for others. In this context, one resident’s benefit from the presence of a security guard does not reduce the benefit for other residents.

b. Calculating the marginal values

To find the marginal cost, individual marginal benefit, and marginal social benefit, we need to determine the difference in costs and benefits between each subsequent increase in the number of security guards. Marginal Cost (MC) = Change in Total Cost / Change in Quantity of Security Guards Individual Marginal Benefit (IMB) = Change in Total Individual Benefit / Change in Quantity of Security Guards Marginal Social Benefit (MSB) = Individual Marginal Benefit * Number of Residents Here is the table with calculated values for each additional security guard: \begin{tabular}{c|c|c|c|c|c} Quantity of security guards & Total cost & Total individual benefit & MC & IMB & MSB \\\ \hline 0 & \(\$ 0\) & \(\$ 0\) & - & - & - \\\ 1 & 150 & 10 & 150 & 10 & 1000 \\\ 2 & 300 & 16 & 150 & 6 & 600 \\\ 3 & 450 & 18 & 150 & 2 & 200 \\\ 4 & 600 & 19 & 150 & 1 & 100 \end{tabular}

c. Individual resident decision

An individual resident would decide to hire and pay for security guards on his or her own by comparing the individual marginal benefit (IMB) with the marginal cost (MC) for each quantity of security guards. The resident would hire a guard if the IMB is greater than or equal to the MC. In the table above, we can see that: - For the first security guard, IMB (10) > MC (150), so the resident would not hire a guard. - For the second security guard, IMB (6) > MC (150), so the resident would not hire a guard. - For the third security guard, IMB (2) > MC (150), so the resident would not hire a guard. - For the fourth security guard, IMB (1) > MC (150), so the resident would not hire a guard. In conclusion, if an individual resident were to decide about hiring and paying for security guards on his or her own, that resident would not hire any security guards.

d. Residents acting together

When the residents act together, they will compare the marginal social benefit (MSB) with the marginal cost (MC) for each quantity of security guards. They will hire a guard if the MSB is greater than or equal to the MC. In the table above, we can see that: - For the first security guard, MSB (1000) > MC (150), so the residents would hire a guard. - For the second security guard, MSB (600) > MC (150), so the residents would hire a guard. - For the third security guard, MSB (200) > MC (150), so the residents would hire a guard. - For the fourth security guard, MSB (100) < MC (150), so the residents would not hire a guard. If the residents act together, they would hire 3 security guards.

Key concepts

Marginal Cost

Marginal cost is a crucial concept when analyzing public goods like a security service. It refers to the additional cost incurred by hiring one more security guard. Imagine you have a group of residents deciding on whether to hire an extra security guard for the community.
To calculate the marginal cost, you look at the change in total cost when the quantity of guards increases. For example, if hiring one guard costs \(150 and hiring two guards costs \)300, the marginal cost of the second guard is \[MC = \frac{\Delta \text{Total Cost}}{\Delta \text{Quantity}} = \frac{300 - 150}{2 - 1} = 150. \]Every extra guard comes with its own price, and understanding this helps in making informed financial decisions. Residents should weigh these costs against the benefits they believe they will gain.

Marginal Benefit

When considering whether to hire a security service, it's equally important to understand marginal benefit. Marginal benefit refers to the added satisfaction or benefit a resident receives from hiring one additional guard.
In the example given, the individual marginal benefit decreases as more guards are hired. The first guard is the most beneficial, but as guards are added, the additional benefit each resident receives (IMB) lessens.

• 1st guard: Individual marginal benefit of 10
• 2nd guard: Individual marginal benefit of 6
• 3rd guard: Individual marginal benefit of 2
• 4th guard: Individual marginal benefit of 1

The trend shows diminishing returns, meaning each new guard provides less additional benefit. By comparing marginal benefits to marginal costs, residents can determine the optimal number of guards.

Non-excludability

One of the defining features of public goods is their non-excludability. This means that once a good is provided, it's impossible to exclude someone from enjoying its benefits.
For instance, think about the community security service. Once security is hired, every resident in the community, regardless of whether they paid, enjoys the increased safety.

• Everyone gets protection
• No one can be denied access
• Encourages shared responsibility

Because no one can be excluded from benefiting, it leads to challenges in funding such services as individuals may choose to "free-ride," enjoying benefits without contributing to the cost. Understanding non-excludability helps highlight why public goods require collective action or government intervention for provision.

Non-rivalry

Non-rivalry is another key attribute of public goods. This means that one person's consumption does not reduce the amount available to others. It captures the essence of shared goods that don’t diminish as more people use them.
Using the community security service again as an example, the presence of a security guard does not mean one resident's security benefits from it at the expense of another's.

• Unlimited simultaneous use
• No competitive consumption
• Equally beneficial for all residents

Because these goods can be consumed by many at the same time without competition, they make perfect candidates for communal provision. Comprehending non-rivalry can help understand why the community can share the costs and still equally enjoy the benefits.