Positive Externailities, Pigouvian Taxes and Subsidies, Public Goods with a bit of Game Theory.. Part 1
This is a really neat exercise which covers elements of Imperfect markets (Duopsony), game theory, public economics, positive external benefits Pigouvian taxes and subsidies and elements of General Equilibrium and Cournot-Nash solutions. We will also have the opportunity to play with logarithms and Taylor expansions (yippee!!). There are multiple parts to this question so I will lay it out in pieces.
Suppose we have two players, Curry and Durand. They like to play one-on-one games for the young people of San Francisco to help promote the sport. To do this the must rent seats from a local arena They have preferences which could be characterized by the two utility functions:
D = ln(Xd-3)+ln(S-6)
The total number of seats is equal to the sum of the number of seats that each one of them buys.
Their incomes are given as Yc =36 and Yd = 24 the price of both seats, S and other stuff, X is equal to 1 so they can allocate their incomes across seats and other stuff according to their budget constraints which are given formally as, Yd= 24= Sd+Xd and Yc=36=Xc +Sc.
Note that looking at the utility functions if either one of these players increases their purchases of seats, the audience will be larger and they will both be happier. What we have here is joint consumption. We have a positive external benefit.
There are multiple parts to this question
a) Show Curry’s Budget and Consumption opportunity sets when Durand buys no seats, 5 seats and 9 seats.
b) Determine the marginal rates of substitution for both players and determine reaction functions for both players. Each player will decide how many seats they want to purchase themselves based upon how many seats the other player purchases. Use graphs and equations.