Positive Externalities, Public Goods, Pigouvian Taxes and Subsidies, Duopsony, Game Theory, Cournot-Nash Equilbria and NBA basketball part 3

Question

This is the third 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 = MRS^Curry=(S-6)/(Xc-3) and MRS^Durand=(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)

Sd= 27/2 – ½(Sc)       (2)

a)      Determine how many seats each player purchases and how many units of X each player consumes.

b)     Show that this is a Nash equilibrium. That is, at this point, neither player would want to adjust their current consumption mix.

Answer 1

a)      Determine how many seats each player purchases and how many units of X each player consumes.

If we plug the right hand side of 2) into 1) we get

Sc= (39/2) –(1/2) (27/2 – ½(Sc))= (68/4)-27/4) +(1/4)Sc

Collecting terms.

3/4Sc= 51/4 or Sc= 17. Plug 17 in for Sc in Durand’s reaction function 1), and get

Sd=27/2/ - ½ (17) = 10/2 or sd = 5.

If Durand buys 5 seats and has 24 to spend, he will buy 19 X or Xd=19.

With Curry, He has 36 to spend and buys 17 seats which means he also buys 19 units of other stuff.

b)      Show that this is a Nash equilibrium. That is, at this point, neither player would want to adjust their current consumption mix.

 When S= 22 and Xd=Xc = 19 , the MRS for Curry and the MRS for Durand are = (22-6)/(19-3) =1. Further note that Curry’s utility is Ln(22-6) +ln(19-3) = 2ln(16). Durand’s utility is also 2ln(16). If Curry were to expand his consumption of other stuff (Xc) by one and reduce his spending on seats by one Total seats, S would drop from 22 to 21. Durand would be unambiguously worse off. Through no action of his own, there will be fewer fans there to watch him play.  His utility would drop by ln(22-16)-ln(21-6). Durand would most like retaliate and reduce his spending on seats as well  (remember his reaction function). Curry will find that his utility also falls. The Utility he get from consuming X rises from Ln(19-3) to Ln(20-3) the change in Utility from fewer seats Is ln(21-6) -ln(22-6). His new utility is  ln(17) + ln(15) < 2ln(16) (we know this from  the convexity of U).