Asked • 01/06/20

Positive externalities, public goods, game theory, Pigouvian Taxes and The NBA and extra credit question.

This is an extra credit 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)



 The total number of seats is equal to the sum of the number of seats that each one of them buys.


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


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

We have seen how setting a subsidy of 50% and taxing income or Taxing the consumption of other stuff at 100% and subsidizing income can generate the social optimum. This assumes that tax revenues and Transfers can be costlessly done. Perhaps there is a better way. We know we need to “tickle” Curry and Durand’s opportunity sets.


a)      Extra credit:  Suppose the government imposed a sales tax that was zero on the first 15 units and $1 for every unit of X greater than 15. What would Curry’s Budget Constraint look like? How much would be collected in Taxes? If we wanted to mimic this with a subsidy scheme How much would the lump sum taxes on Curry and Durand need to be?

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