Question

Two firms produce the same product in a local market where the demand of product is P = 780−2Q - P is the price of the product and Q is the total amount of product produced by these two firms. Each firm needs to decide the quantity that they are going to produce given the local demand, and the cost of producing one unit is $1.

Write down the normal-form representation of the game.

Answer 1

It is given that there are 2 firms producing the same product. The demand function is given as P = 780 - 20. The MC of production is $1. Either the firms collude or compete. Case-1: If they collude, they work as monopolist. P= 780-29 TR = PQ =(780-29) = 7809-292 MR - AIR ΔΩ = 780-49 At optimum, MR = MC 780-4Q=1 779 = 49 7790 194.75 = Q Substitute this into demand function. P= 780-29 = 780–2(194.75) = 780-389.5 = $390.5 Thus, the optimal quantity produced by each firm is 97.375 unit and the price charged will be $390.5

The profit of each firm is calculated below: Profit = (P-MC) Quantity = ($390.5 -1)×97.375 = $37,927.56 Thus, the profit earned by each firm is $37,927.56 Case-2: If they compete. (Cournot equilibrium) The profit function of first firm is as follows: P= 780-29 where Q=91 +92 Thus, P=780-29, -292 TR = P(g) = (180 – 29.-292) 7 = TR -TC = (780 – 29. - 292)92-91 Derivate w.r.t q, and equate it to zero: ati = 780-49,-292-1 da, 0 = 779-49, -292 779 = 497 +292 779-292 = 49, Similarly, the best response function for firm-2 is as follows: 779 - 291 - 492 779 - 492 = 291 ...... (2) Substitute equation (2) into equation (1), one can get:

779–292 = 2(29) 779–292 = 2(779-492) 779 – 292 = 1,558 - 892 -29, +8q, = 1,558 – 779 692 = 779 779 926 91 = 72 = 129.83 Thus, the quantity produced by each firm in the Cournot equilibrium is 129.83 Substituting this into demand function, the optimal price is as follows: P=780-29 -292 = 780–2(129.83)-2(129.83) .= 780–259.67–259.67 = 260.67 Thus, the Cournot equilibrium price for this industry is $260.67 The profit earned by each firm is as follows: Profit =(P-MC) Quantity = ($260.67–1)x129.83 = $33,713.39 Thus, the profit earned by each firm is $33,713.39

The normal-form representation of game is as follows: Q2 Production 97.375 129.83 97.375 37927.56, 37927.56 4 2141.5, 31606 129.83 31606, 42141.5 33713.39, 33713.39 Q1 (NOTE: The values in the table are profit earned by respective firms.)