We discussed in the text that the “market failure” that emerges in the presence of externalities can equally well be viewed as a “failure of markets to exist”, and we discussed the related idea that establishing property rights may allow individuals to resolve externality issues even when markets are not competitive.
A. We will explore this idea a bit further by asking whether there is a “right way” to establish property rights in the case of pure consumption externalities.
(a) Suppose we consider the case where your consumption of music in your dorm room disturbs me next door. Let x denote the number of minutes you choose to play music each day, and let e be the number of minutes you are allowed to play music. If e is set at 0, who is given the “property rights” over the air on which the sound waves travel from your room to mine?
(b)What if e is set to 1,440 (which is equal to the number of minutes in a day)?
(c) Draw a graph with minutes of music per day on the horizontal axis—ranging from0 to 1,440. Draw a vertical axis at 0 minutes and another vertical axis at 1,440 minutes. Then illustrate your marginal willingness to pay for minutes of music (measured on the left vertical axis) and my marginal willingness to pay for reductions in the number of minutes of music (measured on the right axis) —and assume that these are invariant to how e is set. What is the efficient number of minutes m∗?
(d) The assignment of e in part (a) represents the extreme case where you have no right to play your music while the assignment in (b) represents the polar opposite extreme where I have no right to peace and quiet. Review, within the context of this example, the logic behind the Coase Theorem that suggests the efficient outcome will be reached regardless of whether e = 0 or e = 1,440 so long as transaction costs are low.
(e) Since e = 0 and e = 1,440 are two extreme assignments of property rights, we can now easily think of many cases in between. Does the Coase Theorem apply also to these in between cases? Why or why not?
(f) From a pure efficiency standpoint, if the Coase Theorem is right, is there any case for any particular assignment of e?
B. Suppose that your tastes can be described by the utility function u(x, y) = αlnx +y, where x is the number of minutes per day of music and y is a composite consumption good. My tastes, on the other hand, can be described by u(x, y) = βln(1440− x)+ y, with (1440− x) representing the number of minutes per day without your music. Both of us have some daily income level I , and the price of y is 1 given that y is a composite good denominated in dollars.
(a) Let e be the allocation of rights as defined in part A—i.e. e is the number of minutes that you are permitted to play music without my permission. When x < e, I am paying you p(e − x) to play less than you are allowed to — and when x > e, you are paying me p(x −e) for the minutes above your “rights”. What is your budget constraint?
(b)What is my budget constraint?
(c) Set up your utility maximization problem using the budget constraint you derived in (a) then solve for your demand for x.
(d) Set up my utility maximization problem and derive my demand for x.
(e) Derive the p∗ we will agree to if transaction costs are zero—and derive the number of minutes of music you will play. Does your answer depend on the level at which e was set?
(f) According to your results, how much music is played if I don’t care about peace and quiet (i.e. if β = 0? How much is played if you don’t care about music—i.e. α = 0?
(g) True or False: The total number of minutes of music played does not depend on e —but you and I still care how e is assigned.