1. An economy comprises two consumers, 1 and 2, with two consumption goods bi-cycles (b) and wheat . Both consumers have the same utility function μ Bi-cyclesandwheat areproducedbytwofirmswhichuseonlylabouraccordingtotheproductionfunctions b =l and l Both firms are owned by consumer 1, and consumer 2 owns 200 units of labour. (a) Find the production possibility frontier for this economy. (b) Find the competitive equilibrium. (c) Find competitive equilibrium if every consumer owns 100 units of labour and owns one firm. (d) Find the Pareto efficient allocations for this economy. 2. Assume that there are four firms supplying a homogenous product. They have identical cost functions given by C (Q) = 40 Q. If the demand curve for the industry is given by µ = 100 – Q, find the equilibrium industry output if the producers are Cournot competitors. What would be the resultant market price? What are the profits of each firm? Section B Answer all the questions from this section. 5×12=60 (a) Distinguish between pure strategy Nash equilibrium and mixed strategy equilibrium. When would you use mixed strategy equilibrium? (b) Find all the Nash equilibrium of the following game: Player 2 Left Right Player1 Up (5,4) Down (1,3) (4,1) (2,2)
1. (a) A production–possibility frontier (PPF) is a graph representing production tradeoffs of an economy given fixed resources. The graph shows the various combinations of amounts of two commodities that an economy can produce (e.g., number of guns vs kilos of butter) using a fixed amount of each of the factors of production. Graphically bounding the production set for fixed input quantities, the PPF curve shows the maximum possible production level of one commodity for any given production level of the other, given the existing state of technology. (b) In this case, the competitive equilibrium can't be found, as there is not enough data. (c) If every consumer owns 100 units of labour and owns one firm, the competitive equilibrium will change. (d) We can't find the Pareto efficient allocations for this economy, because there is not enough data.
2. An essential assumption of this model is the "not conjecture" that each firm aims to maximize profits, based on the expectation that its own output decision will not have an effect on the decisions of its rivals. The market price is set at a level such that demand equals the total quantity produced by all firms. Each firm takes the quantity set by its competitors as a given, evaluates its residual demand, and then behaves as a monopoly.
If the firms have identical cost functions given by C (Q) = 40 Q, so every firm produce the quantity, for which MR = MC. MC = C' = 40 MR = TR' = (P*Q)' = ((100 - Q)*Q)' = 100 - 2Q So, 100 - 2Q = 40, Q = 30 So, the equilibrium industry output if the producers are Cournot competitors is 30*4 = 120 units. The market price is P = 100 - 30 = $70. Total profits of each firm are: TP = TR - TC = P*Q - TC = 70*30 - 40*30 = $900.
Section B (a) A game can have either a pure-strategy or a mixed Nash Equilibrium. (In the latter a pure strategy is chosen stochastically with a fixed probability). Nash proved that if we allow mixed strategies, then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium. (b) The Nash equilibrium of the following game is in the point (2,2).