Lenny D. answered • 01/07/20

Former Tufts Economics Professor and Wall Street Economist

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).**