Consider a Cournot duopoly. The inverse demand function of the market is given by p = 10-Q, where p is the market price, and Q = 9₁ +92 is the aggregate output. The marginal costs of the two firms are c₁= 1 and c₂ = 4. (a) Solve for the Nash equilibrium of the game including firm out- puts, market price, aggregate output, and firm profits. (b) Now suppose these two firms play a 2-stage game. In stage 1, they produce capacities 9₁ and 92, which are equal to the Nash equilibrium quantities of the Cournot game characterised by part (a). In stage 2, they simultaneously decide on their prices p₁ and P2. The marginal cost for each firm to sell up to capacity is 0. It is impossible to sell more than capacity. The residual demand for 10 - Piāj if Pi > Pj 10-Pi firm ij, is Di (Pi, Pj) 2 if Pi = pj. (Note, Pj if Pi < Pj 10 - Pi here we assume that the efficient/parallel rationing applies). Prove that it is a Nash equilibrium of the second stage subgame that each firm charges the market clearing price pi = 10-91-92, for i=1,2 (i.e., this is a sufficient condition). = (c) This part continues from part (b). Prove that each firm charges the market clearing price pi 10-91-92, for i = 1, 2, is the unique pure strategy Nash equilibrium (note: you need to prove this condition is necessary)

Consider the following game matrix with two players- Player 1 and Player 2 with their respective strategies. Which of the following statements is correct such as x>y>b>a Players Player2 Player 1 A B C D (Y, Y) (b, x) (x, a) (AE) is the one of many Nash equilibria (C) is the unique Nash equilibria (C. D) is the unique Nash equilibria (A. E) and (B. F) are two Nash equilibria E (a, b) (x, y) (y, a) F (a,x) (a, b) (b, b)

Suppose that Flashfry and Warmbreeze are the only two firms in a hypothetical market that produce and sell air fryers. The following payoff matrix gives profit scenarios for each company (in millions of dollars), depending on whether it chooses to set a high or low price for fryers.   Warmbreeze Pricing High Low Flashfry Pricing High 11, 11 2, 15 Low 15, 2 8, 8   For example, the lower-left cell shows that if Flashfry prices low and Warmbreeze prices high, Flashfry will earn a profit of $15 million, and Warmbreeze will earn a profit of $2 million. Assume this is a simultaneous game and that Flashfry and Warmbreeze are both profit-maximizing firms. If Flashfry prices high, Warmbreeze will make more profit if it chooses a    price, and if Flashfry prices low, Warmbreeze will make more profit if it chooses a    price.   If Warmbreeze prices high, Flashfry will make more profit if it chooses a    price, and if Warmbreeze prices low, Flashfry will make more profit if…

Suppose that there are only two firms in a market in which demand is given by p = 64 - Q, where Q is the total production of the two firms. Each firm can choose either a low level of output, qL = 15, or a high level of output, qH = 20. The unit cost of production for both firms is $4. Write down the normal-form representation of the game in which the strategic variable for each firm is the quantity of output and the firms make their choices simultaneously. Find the pure strategy Nash equilibrium of this game (quantities produced and market price).