problem: Consider two firms producing an identical product in a market where the demand is described by p = 1; 200 - 2Y . The corresponding cost functions are c_{1}(y_{1}) = y^{2}_{1} and c_{2}(y_{2}) = 3y^{2}_{2}.
(a) State the profit maximization problem of firm 1 and use the first order condition to derive firm 1's reaction function.
(b) State the profit maximization problem of firm 2 and use the first order condition to derive firm 2's reaction function.
(c) Solve the system of reaction functions to identify how much each firm is producing, what is the market quantity, the market price, and the corresponding individual and collective profits.
(d) Assuming that the firms can coordinate their actions, what are the individual quantities produced, the market quantity, the market price, and the resulting joint profits?
(e) How will the firms distribute the joint profits (Hint: Find the minimum amount that each firm is willing to accept and the maximum amount available for each firm under the cartel agreement ). Is this form of cooperation sustainable? describe.
problem: Two firms, producing an identical good, engage in price competition. The cost functions are c_{1} (y_{1}) = 1:17y_{1} and c_{2} (y_{2}) = 1:19y_{2}, correspondingly. The demand function is D(p) = 800 - 50p. The firm that charges the lowest price gets the entire demand, while, if prices are equal, each firm gets exactly one half of the total demand. Firms can only charge prices that correspond to denominations of Canadian dollars (i.e., prices change by one cent).
(a) Suggest an equilibrium pricing scenario (i.e., one price per firm). Given the prices you propose, find the quantities that each firm is selling, the total market quantity, and the corresponding profits. Justify why the prices you propose are equilibrium prices (by showing that NO firm wants to deviate from the prices you propose).
(b) Repeat part (a) assuming that the Canadian economy run out of 1 cent and 5 cent coins (i.e., prices can change only in multiples of twenty-five cents).
problem: Consider a two-player game where player A chooses "Up," or "Down" and player B chooses "Left," "Center," or "Right". Their payo§s are as follows: When player A chooses "Up" and player B chooses "Left" player A gets $5 while player B gets $2. When player A chooses 1"Up" and player B chooses "Center" they get $6 and $1 correspondingly, while when player A chooses "Up" and player B chooses "Right" player A gets $7 while player B gets $3. Moreover, when player A chooses "Down" and player B chooses "Left" they get $6 and $2, while when player A chooses "Down" and player B chooses "Center" they both get $1. Finally, when player A chooses "Down" and player B chooses "Right" player A loses $1 and player B gets $1. Assume that the players decide simultaneously (or, in general, when one makes his decision doesnít know what the other player has chosen).
(a) Draw the strategic form game.
(b) Is there any dominant strategy for any of the players? Justify your answer.
(c) Is there any Nash equilibrium in pure strategies? Justify your answer fully and discuss your result.
When an action is never chosen by a player it is because this action is DOMINATED by another action (or by a combination of other actions). Dominated strategies are assigned a probability of 0 in any Nash Equilibrium in mixed strategies. Given this observation answer the following parts of this problem:
(d) Find the best response functions and the mixed strategies Nash Equilibrium if each player randomizes over his actions.
(e) Show graphically the best responses and the Nash Equilibria (in pure and in mixed strategies).