Question

Two firms are competing on price. If they have the same price, they share the market - otherwise the one with the lowest price captures all demand Market demand follows Q(P)=100−3P Cost is C(Q)=10Q Firms can only choose between the following prices: 10, 10.1, 10.2, 10.3. In the Nash equilibrium of this game, what price does each firm charge? Same suggestion as before: write down the payoff matrix

Answer 1

In the Nash equilibrium, Firm 1 will charge a price of 10 and Firm 2 will charge a price of 10.1.

Step-by-step explanation:

In a duopoly where firms are competing on price, and market demand follows the equation Q(P) = 100-3P, there are four possible prices for the firms to choose from: 10, 10.1, 10.2, and 10.3. To find the Nash equilibrium, we need to determine which price each firm will choose.

Since both firms have the same cost curve of C(Q) = 10Q, their profit will be equal to their total revenue minus their total cost. The total revenue can be calculated by multiplying the price by the quantity sold. To find the Nash equilibrium, we need to compare the profits for each combination of prices chosen by the two firms.

By calculating the profits for each combination of prices, we can find that in the Nash equilibrium, Firm 1 will charge a price of 10 and Firm 2 will charge a price of 10.1.

Answer 2

In a scenario where two firms compete on price in an oligopoly setting, the Nash equilibrium is found by considering a payoff matrix of profits at different price levels. Both firms will rationally settle at the highest price that allows them to capture the market without incentivizing the other to undercut.

Step-by-step explanation:

The student's question refers to a scenario where two firms are engaged in price competition within a market where the demand and the firm's costs are clearly defined. In this particular case, we are dealing with an oligopoly, since there are only two firms, which is not a perfect competition. A Nash equilibrium in this context represents a situation where no firm has an incentive to change its strategy given the strategy of the other firm.

To find the Nash equilibrium price that each firm charges, we should calculate the payoff matrix for each possible combination of prices the firms might charge ($10, $10.1, $10.2, and $10.3). The payoff for a firm is its profit, which is the total revenue (TR) minus the total cost (TC), where TR = Price × Quantity produced and TC = Average cost × Quantity produced. In this case, the average cost is given as AC = $10 per unit.

The firm with the lowest price will capture the whole market unless both firms set the same price, in which case they share the market. They will choose the highest possible price at which they can still attract the whole demand because setting a price below that would not increase their profit.

Here, both firms would end up charging the same price if they act rationally and understand the strategies of their competitor. The Nash equilibrium would be the highest price in the set of options that does not incentivize either firm to solely deviate to a lower price.