In exercise 21.3, we began to investigate different ways of assigning property rights in the presence of externalities.

A. Consider again the case of you playing music that disturbs me.

(a) Begin with the assumptions in exercise 21.3 that led to the graph you drew in part (d) of that exercise. Then suppose that the transaction cost of getting together is k. In your graph, indicate for what range of e such a transaction cost will prohibit the efficient outcome from being reached?

(b) If e is assigned outside that range, what will be the outcome?

(c) Next, suppose income€”or endowment€”effects are important; i.e. tastes are not quasilinear. Did we allow for that in exercise 21.3?

(d) Suppose in particular that such endowment effects matter for you but not for me €” with music a normal good for you. Illustrate in a graph what happens to the amount of music as e increases. What happens to p?

(e) If endowment effects matter similarly for you and me, might it be the case that the agreed upon price is unaffected by e? What about the agreed upon amount of music?

(f) Is the Coase Theorem wrong in cases where endowment effects impact the amount of music that is played as property rights are assigned differently?

(g) True or False: As long as transactions costs are zero, we will reach an efficient outcome€”but that outcome (i.e. the amount of music played) might differ depending on whether income effects are important.

B. Suppose first that our tastes are again those given in part B of exercise 21.3.

(a) If you have not done exercise 21.3, do so now and check whether the level of music played will depend on the assignment of property rights e in the absence of transactions costs.

(b) Next, suppose that €” instead of the tastes in exercise 21.3 €” your tastes can be described by the utility function u(x, y) = x^αy^(1ˆ’α) (where α lies between 0 and 1). My tastes remain unchanged. How much music will be played? Does your answer depend on e €”and does the equilibrium price p^ˆ— depend on e? . (21.26)

(c) Next, suppose that my utility function is also Cobb-Douglas, taking the form u(x, y) = (1440ˆ’ x)^βy^(1ˆ’β). Derive again the amount of music that will be played (assuming zero transactions costs). Does your answer depend on e? Does the equilibrium price depend on e?

(d) Explain your results intuitively.

(e) In section 21B.3.4, we went through a numerical exercise to illustrate how the establishment of property rights in the presence of externalities will resolve the €œmarket failure€ in a simple exchange economy. Review the example in the text prior to proceeding. Note that in the text we assigned the property rights in the new market to person 2€”the victim of the externality. But we could have assigned property rights in many other ways (as suggested in our music example). Define x₃ once again as the impression of person 1€™s consumption of x1 on person 2; i.e. x₃ = x²₁₁. We can establish a market for the good x₃ by endowing individual 1 with e₃ units of x₃. This means that individual 1 can produce up to e₃ units of x₃ €” which is the same as saying that individual 1 can consume up to e₃ units of x₁ €” without having to pay the market price p₃. But if he wants to produce any more x₃, he must pay individual 2 p₃ for each additional unit above e₃. Similarly, under the endowment of e₃ for individual 1, individual 2 must pay p₃ per unit to individual 1 for any amount of x₃ that falls below e₃ €” and receives p₃ for any amount of x₃ above e₃. In the numerical example of the text, what did we implicitly set e₃ to?

(f) Write down individual 1€™s budget constraint when he is assigned e₃ in property rights. (Hint: If x₁ < e₁, individual 1 will earn p₃(e₃ ˆ’ x₁) but if x₁ > e₁, he will have to pay p₃(x₁ ˆ’e₃) which is equivalent to saying he will earn p₃(e₃ ˆ’x₁).)

(g) Next, write down individual 2€™s budget constraint.

(h) If you substitute your answer to (e) into the budget constraints in (f ) and (g), you should end up with the budget constraints we used in the numerical example of the text. Do you?

(i) Now suppose that

Suppose further that p₁ = 0,

p₂ = 1 and p₃ = p, and that

Can you now interpret the general equilibrium model as modeling our case of you (person 1) bothering me (person 2) with music?

(j) Solve for p and x₃ (which is equal to x₁¹). Do you get the same answer as you got when you assumed Cobb-Douglas tastes for both of us in part (c)?

Transcribed Image Text:

(1-5) (1-a) and u = (1440– x3)P x,