Question 10.8

1. Suppose there are 10 Homebodies and 6 Revelers. Determine the marginal social benefit schedule of money spent on the party. What is the efficient level of spending?

   With 10 Homebodies and 6 Revelers, the marginal social benefit schedule of money spent on the party is as shown in the accompanying table.

   Money spent on party	Marginal social benefit
   $0
   	(10 × $0.05) + (6 × $0.13) = $1.28
   1
   	(10 × $0.04) + (6 × $0.11) = $1.06
   2
   	(10 × $0.03) + (6 × $0.09) = $0.84
   3
   	(10 × $0.02) + (6 × $0.07) = $0.62
   4

   The efficient spending level is $2, the highest level for which the marginal social benefit is greater than the marginal cost ($1).

2. Suppose there are 6 Homebodies and 10 Revelers. How do your answers to part a change? Explain.

   With 6 Homebodies and 10 Revelers, the marginal social benefit schedule of money spent on the party is as shown in the accompanying table.

   Money spent on party	Marginal social benefit
   $0
   	(6 × $0.05) + (10 × $0.13) = $1.60
   1
   	(6 × $0.04) + (10 × $0.11) = $1.34
   2
   	(6 × $0.03) + (10 × $0.09) = $1.08
   3
   	(6 × $0.02) + (10 × $0.07) = $0.82
   4

   The efficient spending level is now $3, the highest level for which the marginal social benefit is greater than the marginal cost ($1). The efficient level of spending has increased from that in part a because with relatively more Revelers than Homebodies, an additional dollar spent on the party generates a higher level of social benefit compared to when there are relatively more Homebodies than Revelers.

3. Suppose that the individual marginal benefit schedules are known but no one knows the true proportion of Homebodies versus Revelers. Individuals are asked their preferences. What is the likely outcome if each person assumes that others will pay for any additional amount of the public good? Why is it likely to result in an inefficiently high level of spending? Explain.

   When the numbers of Homebodies and Revelers are unknown but residents are asked their preferences, Homebodies will pretend to be Revelers to induce a higher level of spending on the public party. That’s because a Homebody still receives a positive individual marginal benefit from an additional $1 spent, despite the fact that his or her individual marginal benefit is lower than that of a Reveler for every additional $1. In this case the “reported” marginal social benefit schedule of money spent on the party will be as shown in the accompanying table.

   Money spent on party	Marginal social benefit
   $0
   	16 × $0.13 = $2.08
   1
   	16 × $0.11 = $1.76
   2
   	16 × $0.09 = $1.44
   3
   	16 × $0.07 = $1.12
   4

   As a result, $4 will be spent on the party, the highest level for which the “reported” marginal social benefit is greater than the marginal cost ($1). Regardless of whether there are 10 Homebodies and 6 Revelers (part a) or 6 Homebodies and 10 Revelers (part b), spending $4 in total on the party is clearly inefficient because marginal cost exceeds marginal social benefit at this spending level.

   As a further exercise, consider how much Homebodies gain by this misrepresentation. In part a, the efficient level of spending is $2. So by misrepresenting their preferences, the 10 Homebodies gain, in total, 10 × ($0.03 + $0.02) = $0.50—that is, they gain the marginal individual benefit in going from a spending level of $2 to $4. The 6 Revelers also gain from the misrepresentations of the Homebodies; they gain 6 × ($0.09 + $0.07) = $0.96 in total. This outcome is clearly inefficient—when $4 in total is spent, the marginal cost is $1 but the marginal social benefit is only $0.62, indicating that too much money is being spent on the party.

   In part b, the efficient level of spending is actually $3. The misrepresentation by the 6 Homebodies gains them, in total, 6 × $0.02 = $0.12, but the 10 Revelers gain 10 × $0.07 = $0.70 in total. This outcome is also clearly inefficient—when $4 is spent, marginal social benefit is only $0.12 + $0.70 = $0.82 but marginal cost is $1.