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Calculating The Monopoly Quantity, Price and Profit
Bertrand Duopoly with Differentiated Goods Assume that two firms produce differentiated versions of a good at identical and constant marginal cost,
MC1=MC2=50. The firms face demand curves as follows:
q1=1000−2p1+p2q2=1000−2p2+p1 a) Under what type of circumstances is Bertrand competition a reasonable assumption? b) Write down each firm's profit function. c) Derive and graph each firm's best response function. d) Find the Nash equilibrium quantities, prices and profits. e) Assume now that firm 2's factory burns down so that
q2=0, and firm 1 is left with a monopoly over its market. Calculate the monopoly quantity, price and profit for firm 1 , and consumer surplus. f) Contrast the outcomes under monopoly and Bertrand competition and explain your results. Calculating The Monopoly Quantity, Price and Profit
Step 1/2
a) Bertrand competition is a reasonable assumption when firms compete on the basis of price, and the goods are similar or differentiated, but not perfect substitutes. In this case, firms produce differentiated goods and compete on the basis of price, which makes Bertrand competition a reasonable assumption. b) The profit function of firm 1 can be written as: π 1 = (p 1 − 50) (1000 − 2p 1 p 2 )
Step 2/2
c) To derive the best response functions, we first need to find each firm's reaction function, which is the profit-maximizing price for each firm given the price of the other firm. To do so, we take the derivative of each firm's profit function with respect to its own price and set it equal to zero: For firm 1: ∂π 1 /∂p 1 =1000−4p 1 +p 2 −50=0 p 1 =1/4p 2 +238 For firm 2: ∂π 2 /∂p 2 =1000−4p 2 +p 1 −50=0 p 2 =1/4p 1 +238 We can then substitute each firm's reaction function into the other firm's profit function to get each firm's best response function: For firm 1: BR 1 : π 1 = (1/4p 2 +188−50)(1000−2(1/4p 2 +238)+p 2 ) π 1 = 125000−(125/2)p 2 +1/2p 2 2 −50000p 1 +100p 1 p 2 For firm 2: BR 2 : π 2 = (1/4p 1 +188−50)(1000−2(1/4p 1 +238)+p 1 ) π 2 = 125000−(125/2)p 1 +1/2p 1 2 −50000p 2 +100p 1 p 2 Calculating The Monopoly Quantity, Price and Profit
Bertrand Duopoly: Nash Equilibrium
Final answer
d) To find the Nash equilibrium, we need to find the intersection of each firm's best response function. Solving for the prices and quantities, we get: p 1 *=p 2 *=462.50 q 1 *=q 2 *=537.50 And the profits are: π 1 =π 2 = $31,406.25 e) With firm 2 out of the market, firm 1 faces a market demand of: q=1000−2p Firm 1's profit-maximizing quantity and price can be found by taking the derivative of the profit function and setting it equal to zero: ∂π 1 /∂q=(p−50)−2q=0 p 1 *=550 q 1 *=225 The consumer surplus is: CS=1/2(1000−550)(1000−2(225))= $31,875 And firm 1's profit is: π 1 = (550−50)(225)= $112,500 f) Under Bertrand competition, both firms capture some of the consumer surplus and the total surplus is lower than under monopoly Calculating The Monopoly Quantity, Price and Profit
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