Lenny D. answered • 01/06/20

Former Tufts Economics Professor and Wall Street Economist

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**