Question

Aaron plays the saxophone but his neighbor Bruno hates music. Aaron and Bruno have the following utility functions: UA = 100 + 10z - 0.1z2 and UB = 100 - 10z; where z is the number of hours spent by Aaron playing the saxophone (and UA includes the cost of the saxophone). Determine:

a) [3 marks] The number of hours of saxophone played by Aaron when the external effect on Bruno is ignored.

b) [3 marks] The socially optimal level of hours of saxophone that should be played by Aaron.

c) [5 marks] The optimal Pigouvian tax needed to reach the social optimum.

d) [7 marks] The outcome with Coasian bargaining when the property right is assigned to the saxophone player. [Hint: Calculate the utility gain of Aaron and the utility loss of Bruno. What is the minimum amount of money Aaron needs as compensation? How much Bruno is willing to pay to reduce the number of hours of saxophone?].

e) [7 marks] The outcome with Coasian bargaining when the property right is assigned to the neighbor.

Answer 1

• Answers for : (a.) 50  (b.) 0 (c.) 10
• answers for rest and explanation is given below :

UA= 100 + 102 0.1 22 UR = 100 - 102 the utility maximization Aaron Sawes paoblem : V 12) = 100 + 10 2 - 0.12? max {2>,o} The first -arder condition is VA 1o – 0.2 z dz which gives the optimal no. of hours z* = 40 - 50 - ® 0.2 6 The socially optimal level of hours وهلو& may TVAlz) + UB (2) J = (100 + 10Z -0.1 22) {23,0} (100-102) 200 -0.922 - © 2

This function is decreasing in z , Assume Aaron has to pay a tax © for each hour. The utility maximization problem is now max UA (Zit) = 100+ 10Z -0.32?_tz 230 The first order condition becomes duA - 1o - 0.22-E to dz thus the social Zso =0 is optimum of reached in

@ When the property right is assigned to saxophone player , the chooses to play . hours. The neighbour (Bruno)'s utility lo se is Us (50) - UB (O) = -500 . is : The Aaron's utility gain VA (50)- UA (O) = 250 > Aason will require payment of atleast 250 & Bruno will be willing to pay upto 50o .. Depending upon the bargaining power, the outcome of coarian bargaining is that : ce the Bruno pays to the Aarth anything fw 250 and 500 for not playing & the Radon will accept. Thus, socially efficient outcome wall be achieved. &

assigned to ban When the property sight is to neighbour, the chooses Saxophone. The Aaron's utility loss is! VA (0) - VA (50) = - 250. The is ! neighbour is utility gain UR (O) - UB (50) = 500. To remove the music ban, the heighbor Laill require payment of atleast 500, but Aaron will be willing to pay only apto 250. - Thus, the in place out come music ban will remain & the socially efficient will be achieved.