 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, . The firms face demand curves as follows:  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 , 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-by-step

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
• Explanation for step 2
Bertrand Duopoly: Nash Equilibrium