Positive Externalities, Public Goods, Pigouvian Taxes and Subsidies, Game Theory and the NBA part 4

Question

This is the fourth 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

.

a)      What would happen to the utility of each player if they both agreed to purchase one more seat and one less unit of x? 

b)     Determine the social marginal rate of substitution.

c)      Define the social planner’s problem and calculate the solution for the optimal quantity of S. How much s would each player purchase question how much would each player consume a good X?

Answer 1

a)      What would happen to the utility of each player if they both agreed to purchase one more seat and one less unit of x? 

MU x = 1/16 and MUs= 1/16 if They both buy one less X and One more seat. There will be two more seats in total. So for Curry ∆U= Mux(∆Xc) +MUs (∆S). If the both agree to buy one more seat, ∆S=-2(∆Xc) =2 so ∆U= (1/16)*2 +(1/16)*(-1) = +1/16 The same is true for Curry. The astute student will see that we have actually done is used a first order Taylor approximation about S-22 and X=19.

b)     Determine the social marginal rate of substitution. 

The Impact of an increase in seats is the sum of the individual impacts. That it is, “society" benefits twice as much as each individual.  So SMU= 2PMU= 2/(S-6) so social MRS= (S-6)/(2(Xc-3). Collectively, the value of other stuff is half what an extra seat is worth

c)      Define the social planner’s problem and calculate the solution for the optimal quantity of S. How much s would each player purchase question how much would each player consume a good X?

Here we need to choose S ,Xc and Xd which Maximizes C+D= ln(Xc-3) +ln(Xd-3) +2ln(S-6) subject to 36=Xc + Sc , 24=Xd+Sd and S= Sc+Sd.

Our first order conditions have (S-6)/(2(Xd-3))=(S-6)/(2(Xc-3))=1

Or 2(Xd-3)=2(Xc-3))=S-6. So 2Xd-6=2Xc-6= S-6 or,

Xd =Xc= S/2 = Sd/2 +Sc/2 We can substitute this into the two budget constraints and get

Curry: 36 = Sc/2 +Sd/2 + sc/2) = 3/2Sc +1/2Sd or Sc =(2/3)36 –(2/3)(1/2)Sd =24-(1/3)Sd

Durand 24= Sd/2 +Sc/2 +Sd or Sd=16-(1/3)Sc

So with Sd= 16 – (1/3)Sc and Sc= 24-1/3Sd we can solve for Sd then Sc

Sd= 16 =(1/3)((24-(1/3)Sd) = 16 -8 +(1/9)Sd or

(8/9)Sd= 8 or Sd=9. When Sd = 9, Sc = 24-(1/3)9=21 and S= 21+9. =30 and Xd=Xc=S/2 =15