Two firms sell identical products and engage in Bertrand price competition. Each
ID: 2428665 • Letter: T
Question
Two firms sell identical products and engage in Bertrand price competition. Each firm has constant average and marginal cost c = 10. The market demand curve is q(p) = 100 ? p.
1. Calculate the Nash Equilibrium prices for each firm. What is the total quantity produced? What is the deadweight loss relative to perfect competition?
2. Suppose a third firm with constant average and marginal cost c = 15 enters the market. Repeat part (A).
3. Suppose a fourth firm invents a new production technology, with c = 5, and enters the market. Repeat (A)
Explanation / Answer
In bertrand Price competition model, The firms decide a price and let the market decide the quantity sold. Therefore, each firm forcasts the price set by the other firms and wants to set its own price to maximise profts. Ultimately, Bertrand equilibrium turns out to be competetive equilibruim because each firm wants a bigger share of the market and it does so by undercutting the other firm's price by a marginal amount, until each firm competes with each other and compettive equilibrium is achieved.
Therefore in Bertrand equilibrium, competitve equlibrium is achieved . That means, at equilibrium, price = marginal cost.
1. Here the marginal cost (MC) = 10. Therefore putiing P = MC, we get price = 10.
Here the demand function is q(P) = 100 - P
Putting P = 10 we will get:
q = 100 - 10 = 90.
Therefore total quantity in the market is 90. Since both firms are identical, they will produce equal amounts.
Therefore each firm will produce 90/2 = 45 units of good. This is the Nash equlibrium with each firm producing 45 units,
Deadweight loss comparitive to perfect competition is zero. As in perfect competition too price = marginal cost and Bertrand competition follows the equlibrium of a perfect competitive market.
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2. If the third firm enters the market, the firms already existing will be producing at Price = MC = 10. This price is very low for this new firm as its marginal cost is higher. Therefore, this new firm will produce nothing to avoid losses.
Hence the total quanitity sold in the market will remain at 90 and the first two firms will produce 45 units each.
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Now a new firm has entered the market. However this time its marginal cost is the lowest among all the firms at MC = 5. Therefore in order to get the higher market share, this firm will sell at price equal to its marginal cost, that is it will set P = 5.
The other firms cannot produce at this price because their cost is too high and they wil incur losses. Therefore they will produce nothing.
According to the demand the total quantity now is:
q = 100 - 5 =95.
Therefore the quantity increases. And it will all be produced by this new firm.
Hence Quantity = 95, price = 5.
Quantity profuced by frims with Mc = 10 will be zero and quantity produced by firm with MC = 5 will produce 95 units of good.
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