Positive Externalities, Public Goods, Game Theory Cournot-Nash Equilibria Pigouvian Taxes and Subisidies and the NBA part 5

Question

This is  the fifth 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)

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)

.

a)     Would it be possible for current in Durand to achieve this outcome without government intervention? That is, is there a collusive solution?

b)     Suppose that Curry has secretly signed a deal with the Laker’s and this will be his last exhibition in San Francisco. How much should he cheat?

c)      Suppose the social planner (instead of telling these two players what they needed to consume by fiat) chooses to subsidize the purchase of seats. She would fund the subsidy with the tax on their incomes leaving their effective purchasing powers unchanged (at the social optimum). What would the subsidy rate be and how much would each pay in taxes?  

Answer 1

a)     Would it be possible for current in Durand to achieve this outcome without government intervention? That is, is there a collusive solution? The answer is Yes. They could easily come to an agreement where they both agreed to either limit their consumption of X to 15 or that Curry would buy 21 seats and Durand would buy 9. If the signed an enforceable contract to the effect with some form of Punitive non-performance clause. We could achieve the same outcome.

b)     Suppose that Curry has secretly signed a deal with the Laker’s and this will be his last exhibition in San Francisco. How much should he cheat?

Simply go back to Curry’s Reaction function and see that When Sd=9, Sc = 39/2-(1/2)(9/2)= 30/2=15.

To figure out the gain from cheating.

So instead of 21 he will only buy 15 seats. Total seats will drop from 30 to 24. His consumption of other stuff rises from 15 to 21. His new Utility Would be Ln(24-6) +ln(21-3) = 2Ln(18) =2ln(2)+2ln(9). His Old Utility = ln(24)+ln(12) =  = ln(2) +2(ln(12) = ln(2)+2ln(4) +2ln(3) =5(ln2)+ln(9). The change in his Utiilty is therefore 2ln(2) -5 ln() +2ln(9)-ln(9) = -3ln(2)+ln(9)

=-ln(8)+ln(9) = Ln(9/8) >0

c)      Suppose the social planner (instead of telling these two players what they needed to consume by fiat) chooses to subsidize the purchase of seats. She would fund the subsidy with the tax on their incomes leaving their effective purchasing powers unchanged (at the social optimum). What would the subsidy rate be and how much would each pay in taxes?  

There are many ways to skin this kitty Cat. We Know we need MRS =Px/(Ps(1-s)) = 1/(1-s)  

So,

 (S-6)/(xd-3)=(S-6)/(Xc-3) =(1/(1-s)  =(30-6)/(15-3) = 24/12 =2  so 1-s =1/2 or s= 50 cents.

For Curry. He would buy 21 Sc with half coming out of his pocket (21)/(2) the other (21)/2 Is paid for by the social planner. So, if we taxed Curry 21/2 in a lump sum fashion he would have an effective budget constraint of 25.5 =Xc +(1/2)Sc. He will buy 21 seats and spend the remaining $15 on other stuff (Xc). Durand will be buying 9 seats. So he will be getting (9/2) at the cash register in subsidies. If we taxed him 9/2 dollars his budget constraint becomes Yd=19.5 = Xd+(1/2)sd. He will buy 9 seats and spend the remaining 15 dollars on other stuff.