Derek is a fan of loud music that he plays on his stereo system. His utility function is defined by equation:
U(q,y)=10q-(.5q^2)+y
where q is the number of hours of music he plays, and y is his consumption of all other goods. Each hour of music he plays costs him $2 in electricity, but his income is large enough that he could always afford to play music 10 hours a day if he so wished.
a. Find Luis' demand for hours of music as a function of the price of electricity.
b. Luis has a roommate, Maria, who has the same utility function as Luis. She doesn't have a stereo system, nor does she pay the electricity bill, but she listens to the music. How many hours of music maximizes social surplus, and what (Pigouvian) tax/subsidy enables this allocation to be attained?
c. Maria moves out, and Gordon moves in with Luis. Gordon's utility function is defined
by the equation:
u(q,y)=-2q-(.5q^2)+y
Now which allocation maximizes total surplus, and what (Pigouvian) tax/subsidy enables
this allocation to be attained?
d. Assume that the Pigouvian tax you found in part c. is not in place. Suppose that Luis and Gordon can costlessly negotiate to change the hours of music Luis plays to any amount that they can decide upon. What is this number of hours, how much is Gordon willing to pay for this change, and how little is Luis willing to accept?