Question

Due Date: June 21, 2019 1. Consider the market for ayran. There are two firms producing ayran: Merve (firm 1) and Sutas (firm 2). Inverse demand function for ayran is given as: P = 110 qi-92- Both firms have a cost function of C(x) = x2, where x is the amount produced. (a) What is the Nash equilibrium if they choose the level of production at the same time? What are the profits of each firm? (b) Suppose now Merve has an opportunity. Before market opens, it can invest in new technology that produces more efficiently. If it invests in this technology, the cost function for Merve becomes 2/2. Suppose that price of this technology is F = $350. Should Merve invest in this C(x) == technology? (c) Consider another scenario. Suppose now that both firms have this opportunity. Sutas also can invest to decrease the cost function. The choice of technology is made at the same time. 1. What is the equilibrium profits if both firms choose to invest? 2. What is the equilibrium profits if both firms choose not to invest? 3. What is theequilibrium profits if one firm chooses to invest and one firm chooses not to invest? (d) Draw the matrix of the game, where Merve is the row player and Sutas is the column player and choices are invest and not invest. What is the equilibrium of this game?

Answer 1

Ginen P: 110- -a2 So Protit e Fisum 1 TR- Tc 2 =110--a/2)a, a oa-2aaa can bo NUsiten as, rotit, assuming the Both titems uill marimise thei aLLantities ol it's 9ival to be onstant Hence Fisest order conditiony asre dT, and ニ0 dar anuauy Simuta ean and Solve 4a,-au 110 ェA ェ400 ニ0 H0-440-au 16a = 0

15a = 330 a 22 gof, either W 22 Substituti It0-49, -,O 110-8 a22 (10-a,-22) a Firm 1: T 2 66X22 484 11 968 Firm 2 Protit 464 66X22 Equilibnum, Nash So in au,- 22 amd a 22 R, 968 amd 968 (b) I mesrue invests in meus technolec than, cost becomas, Ci 350+a2 its total 2. TRI Tc' Se prsotit o merue Fimi) (10-a,-a,Ja 3 5o-a2 2 2

-3at Lohesas lor fim aa-350 2 protit nuiu be as t s, T10-a-a) ie 10a- 2a for Nash Eatuiibrium, daur SO, da Q = lio-3a, O And J dar 1i0-49-, o im ultanao us ty Solue and sin 3a6 -a O +/ 0-330 - a t 12au2 0 aj 20 in eauaticon Put a 20 au, 10- 4a Z10-2O - 80

Prair of meiue (Firm o - 2 = (1o-,-a,) ,-350 60x30 350-900 2 = 1800-3 50-450 = 1e00-8o0 =1000 Ard Proit d Fisem 2 (110-0,-a,)au 400 60X20- 1200-apo we Obserue thauf tor Meaua This meauns, by installing meur teuhnology Cam inurease its frotit from$96s to $1000. merue Should install tha neustechnolag merue Hene inuest tnan Ce It both tirms choose 4o 350+ a 2. C, = 3 50 t a 2 C - Se Protits uiu be, (i0-a,-a2a,-3 So -a 2aa,a3 so 110a 2au,a,-3So And

In Nash eauilibrium, and d da/ aay So dn I10-34,-CU o Amd da 0-34/2-a,o d tarsously, Solve and Simul X 3 11 0 110-330 a+9a, = o 27.5 2) ue Substitul a= 27.5 in 10-3au2 =110 - 3メ27.5 = 110-82 5 27.5 Hane (110-a,-a,.a,-35o- ev =110-55)x 27 S-350- 398 -125 T84 375 2-

N(10-,-a1) aj 350 2 378-125 SSA27 5- 350- -1 484.375 Henu eauiibrium bsolits uohen both tisns chose to inuest) 84.315 784.375 TR2 tirm cheose o Eaui tibyum proit Sot to inues, LRefer when both to part (a)) = 968 T 968 Eauilibrium brotit choese to inuast, CRetur pat b) Ohen cne firm of the (Firm 1 1000 1000 (Firm 2 invest)

d) SULTAS Invest Not Invest (1000, 800) Invest 784.375,784-375) (reo, 1000) (968, 968) Not invest bayory tar () marked cells in the mahrix asto is the Nash Eauuibri'um, au mone o the im has incentiue to deuiate from their initial stsategt The tenu, (Invest, Not Inuust inuest (Not inuest, is the Nash Eauilibrium MERVE