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Yale lectures 5 and 6

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?

vernon-beasley
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Yale lectures 5 and 6

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  1. 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.

  2. 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?

  3. 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.

  4. 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?

  5. 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 – bq12 -bq1q2– c*q1 • taking derivative with respect to q1 • 0=a – 2bq1 -bq2– c • q1 = (a-c)/2b - q2/2

  6. All along this line, the monopoly quantity is being produced, which is optimal for producers dotted lines show how a series of best responses gets us back to the NE instead of the agreed production level

  7. 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.

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