# Positive Externalities, Public goods, Pigouvian Taxes and Subsidies Game theory and The NBA part 7.

This is the 7th part of 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!!).

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 they 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)

and

C=ln(Xc-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.

Formally,

S= Sc+Sd.

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.

We have already determined the MRS for both players = MRSCurry=(S-6)/(Xc-3) and MRSDurand=(S-6)/(Xd-3). Remember that Seats are “consumed” jointly while other stuff, (Xc and Xc) are consumed exclusively). The Reaction or “Best Response” Functions were determined to be

Sc= (39/2) – (1/2)Sd.     (1)

This is Curry’s Reaction or Best Response function to the number of seats Durand purchases.

Doing the same for Durand we see

Sd= 27/2 – ½(Sc)       (2)

We Know the Cournot Nash Solution has Sc=17, Sd=5  and Xc=Xd=19 We have also determined that Constrained bliss is When each player spends \$15 on other stuff (Xc=Xd=15) and 30 seats are rented with Curry covering 21 and Durand taking care of the remaining 9. Or Sc* =21 and Sd*=9. The Following relationships may be helpful.

Ln(16)=2ln(4)=4ln(2)

3ln(2)=ln(8)

Ln((24))=ln(2) ln(12)

Ln(12)=ln(4)+ln(3)

Ln(9)-ln(8)= ln(9/8)=-ln(8/9

.

We saw that Taxing Curry 10.5 dollars and taxing Durand 4.5 dollars and subsidizing the purchase of seats will Lead to the social optimum. Susidizing the good which there is too little of pushes us in the right direction

a)     Suppose instead the social planner wishes to tax the consumption of x with the tax proceeds going back to each player what tax rate would be Revenue neutral and generate the same outcome?