Question

Two firms set prices in a market with demand curve Q = 6 − p, where p is the lower of the two prices. If firm 1 is the lower priced firm, then it is firm 1 that meets all of the demand; conversely, the same applies to firm 2 if it is the lower priced firm. For example, if firms 1 and 2 post prices equal to 2 and 4 dollars, respectively, then firm 1–as the lower priced firm–meets all of the market demand and, hence, sells 4 units. If the two firms set the same price p, then they each get half of the market, that is, they each get (6−p )/2 . Suppose that prices can only be quoted in dollar units, such as 0, 1, 2, 3, 4, 5, or 6 dollars. Suppose, furthermore, that costs of production are zero for both firms. Finally, suppose that firms want to maximize their own profits.

Show that posting a price of 0 dollars and posting a price of 6 dollars are both dominated strategies. What about the strategy of posting a price of $4? $5?

Answer 1

The profit for the firm = Total revenue- Costs = PQ-0 = (6-p)*p - 0 (Since it costs 0)

Now if the price is set as 0, the total revenue will be 0 (As Q units sold at a price of 0 gives a revenue of 0) and if price is set at 6, the demand for the product Q = 6-p = 6-6=0 (i.e. Q demanded at a price of 6 is zero thus no units are sold at this price and thus the revenue and the profits is 0)
Since the profits in both the cases is zero, they are dominated strategies (In other cases either the profit will be positive, or 0 if the other firms gains the market, thus the strategy of choosing a price of 0 or 6 is weakly dominated as every other strategy leads to a payoff atleast equal or greater than 0)

For a price of 4 or 5 as the question says, if a firm prices it at 4 or 5, the other firm will simply price it at 3 and gain the entire market and get the profit 3*(6-3)=9 and the firm that sets the price 4 or 5 will get 0. Even at a price of 3, the other firm will set a price of 2 and get profit 2*4=8 instead of the profit (3*3/2= 4.5 if both firms shared the profit at a price of 3)

Now under Bertrand competition (i.e. the firm pricing lower captures the entire market), both will try to keep reducing the price and try to capture the entire market and increase the profits, in such a case eventually both the firm will reach at a price of p=1, and neither will reduce further as reducing further will bring the profits to 0. So they both set a price of 1 and each split the market as (6-1)/2 = 2.5
And get a profit of 2.5*1=2.5
Note: Units of quantity can be decimal for example 2.5 kg of rice.

Hope it helps. Do ask for any clarifications if required.