Yale lectures 5 and 6

• Nash Equilibrium – no individual can do strictly better by deviating – self enforcing in agreements

• Investment game – all invest or no invest. Guess and check. Converges.

Mark best alternative given the other player has made a decision.

Terms

• What is a fixed point?• What does he term a silly Nash

Equilibrium?• In investment game (get profit of $5

if 90% invest, lose $10 if not), what were the two equilibrium?

• Why is it not a prisoners dilemma?• What is the “guess and check”

method of finding a NE?

5,5 -10,00,-10 0,0

Cournot competition• There is more than one firm and all firms produce a

homogeneous product, i.e. there is no product differentiation;

• Firms do not cooperate, i.e. there is no collusion;• Firms have market power, i.e. each firm's output

decision affects the good's price;• The number of firms is fixed;• Firms compete in quantities, and choose quantities

simultaneously;• The firms are economically rational and act

strategically, usually seeking to maximize profit given their competitors' decisions.

Take aways

• We won’t worry too much about the economic aspects of the model, but want to look at what Nash Equilibrium means.

• Does it mean profit is maximized?• Can we agree to do something different and

have that agreement adhered to?

• p1 = firm 1 price, p2 = firm 2 price

• q1 = firm 1 quantity, q2 = firm 2 quantity • c = marginal cost, identical for both firms • profit is given by Π1 = q1(P(q1 + q2) − c)• Equilibrium prices will be: • p1 = p2 = P(q1 + q2) = a – b(q1 + q2)

• u1 = [p]q1 – c*q1

• Can find best response• u1 = (a – b(q1 + q2))* q1 – c*q1 = a q1 – bq1

2 -bq1q2– c*q1

• taking derivative with respect to q1

• 0=a – 2bq1 -bq2– c

• q1 = (a-c)/2b - q2/2

dotted lines show how a series of best responses gets us back to the NE instead of the agreed production level

All along this line, the monopolyquantity is being produced,which is optimal for producers

So what is NE• Consider Q the total amount to produce to

maximize profit• (a – b(Q))* Q – c*Q = aQ-bQ2-cQ• taking derivative a -2bQ –c = 0• Q = (a-c)/2b • So if each does the same, q1 = (a-c)/4b • Would get higher utility if could agree to both

producing less.