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.