microeconomics 4

Exercise 1 (Training and payment system, By Kim Swales) Two players: The employee (Raquel) and the employer (Vera). Raquel has to choose whether to pursue training that costs $1; 000 to herself or not. Vera has to decide whether to pay a xed wage of $10; 000 to Raquel or share the revenues of the enterprise 50:50 with Raquel. The output is positively aected by both training and revenue sharing. Indeed, with no training and a xed wage total output is $20; 000, while if either training or prot sharing is implemented the output rises to $22; 000. If both training and revenue sharing are implemented the output is $25; 000. 1. Construct the pay-o matrix 2. Is there any equilibrium in dominant strategies? 3. Can you nd the solution of the game with Iterated Elimination of Dominated Strategies? 4. Is there any Nash equilibrium?

Exercise 2 (Cournot duopoly with asymmetric rms) In a market characterized by the following (inverse) demand function P = 40 Q two rms compete a la Cournot. Firm A has production cost described by the cost function cA (qA) = 20qA, while rm B's cost function is cB (qB) = q2 B. 1. Which rms has increasing marginal cost? Which one has constant marginal cost? 2. Dene the reaction functions of the rms. 3. Compute the Cournot equilibrium quantities and price.

Exercise 5 (A prisoner's dilemma game, by Kim Swales) Firms Alpha and Beta serve the same market. They have constant average costs of $2 per unit. The rms can choose either a high price ($10) or a low price ($5) for their output. When both rms set a high price, total demand = 10,000 units which is split evenly between the two rms. When both set a low price, total demand is 18,000, which is again split evenly. If one rm sets a low price and the second a high price, the low priced rm sells 15,000 units, the high priced rm only 2,000 units. Analyse the pricing decisions of the two rms as a non-co-operative game. 1. In the normal from representation, construct the pay-o matrix, where the elements of each cell of the matrix are the two rms' prots. 2. Derive the equilibrium set of strategies. 3. Explain why this is an example of the prisoners' dilemma game.

3. Cournot equilibrium is identied by the quantities that are mutually best responses for both rms; so, they are obtained by the solution of the following two-equation system: ( 100 7

Exercise 2 (Simultaneous-move games) Construct the reaction functions and nd the Nash equilibrium in the following normal form games. Will and John 1 John Will Left Right Up 9; 20 90; 0 Middle 12; 14 40; 13 Down 14; 0 17;2 Will and John 2 John Will Left Centre Right Up 2; 8 0; 9 4; 3 Down 3; 7 2; 10 2; 15 Will and John 3