In the introduction, we mentioned that, while we often treat public and private goods as distinct concepts, many goods actually lie in between the extremes because of “crowding”.
A: We can think of the level of crowding as determining the optimal group size for consumption of the good — with optimal group size in turn locating the good on the continuum between purely private and purely public goods.
(a) One way to model different types of goods is in terms of the marginal cost and marginal benefit of admitting additional group members to enjoy the good. Begin by considering a bite of your lunch sandwich. What is the marginal benefit of admitting a second person to the consumption of this bite? What is therefore the optimal “group size”—and how does this relate to our conception of the sandwich bite as a private good?
(b) Next, consider a chess club. Draw a graph with group size N on the horizontal axis and dollars on the vertical. With additional members, you’ll have to get more chess-boards — with the marginal cost of additional members plausibly being flat. The marginal benefit of additional members might initially be increasing, but if the club gets too large, it becomes impersonal and not much fun. Draw the marginal benefit and marginal cost curves and indicate the optimal group size. In what way is the chess club not a pure public good?
(c) Consider the same exercise with respect to a movie theater that has N seats (but you could add additional people by having them sit or stand in the isles). Each customer adds to the mess and thus the cleanup cost. What might the marginal cost and benefit curves now look like?
(d) Repeat the exercise for fireworks.
(e) Which of these do you think the market and/or civil society can provide relatively efficiently —and which might require some government assistance?
(f) Why do you think fireworks on national holidays are usually provided by local governments —but Disney World is able to put on fireworks every night without government help?
B: Consider in this part of the exercise only crowding on the cost side — with the cost of providing some discrete public good given by the function c(N) = FC +αNβ with α > 0 and β ≥ 0. Assume throughout that there is no crowding in consumption of the public good.
(a) Derive the marginal cost of admitting additional customers. In order for there to be crowding in production, how large must β be?
(b) Find the group membership at the lowest point of the average cost function. How does this relate to optimal group size when group size is sufficiently small for multiple providers to be in the market?
(c) What is the relationship between α, β and FC for purely private goods?
(d) Suppose that the good is a purely public good. What value of α could make this so? If α > 0, what value of β might make this so?
(e) How does α affect optimal group size? What about FC and β? Interpret your answer.