Price Competition in a Duopoly

Price Competition in a Duopoly

• In a duopoly, two firms producing either differentiated or homogeneous products compete in prices.
• The unified model used to analyze price competition consists of firms 1 and 2 producing goods 1 and 2 at constant marginal costs C1 and C2, respectively.
• The substitution parameter D measures the substitutability between products (D = 0: totally differentiated, D = 1: exactly the same).

Maximization Program and Best Response Function

• Firm 1's profit maximization program involves choosing a price to maximize profit, given the price of firm 2.
• The first-order condition for profit maximization yields the best response function of firm 1, which is upward-sloping (strategic complements).
• If firm 2 increases its price, firm 1's best response is to also increase its price.

Strategic Complements and Reaction Functions

• Strategic complements occur when firms' strategies reinforce each other (e.g., price increases lead to further price increases).
• The reaction function of firm 1 is upward-sloping, meaning that an increase in firm 2's price leads to an increase in firm 1's price.
• The reaction function flattens as D decreases (products become more differentiated), and becomes horizontal when D = 0 (local monopolies).

Nash Equilibrium

• A Nash equilibrium occurs when both firms choose prices that are best responses to each other.
• The Nash equilibrium is found at the intersection of the reaction functions of both firms.
• The equilibrium prices increase with costs, and the price of one firm increases with the cost of the other firm due to strategic complements.

Symmetric Firms

• When firms are symmetric (same cost), they set the same price and quantity at equilibrium, earning the same profit.
• As D increases, the profit of symmetric firms decreases.

Homogeneous Products (Bertrand Model)

• In the Bertrand model, two firms produce the same product, and consumers only care about the price.

• The demand function assumes that consumers choose the firm with the lower price, and if prices are equal, the market splits between the two firms.

• The game is a simultaneous price-setting game, where firms cannot observe the other firm's price when making their decision.### Demand and Market Structure

• When firm I sets a price P1, the quantity demanded is 0 if Pi is higher than the price of firm J (PGA), and Alpha I (a share of the total demand) if Pi is lower than PGA.

• If both firms have the same price, firm I gets a share Alpha I of the total demand.

• There is a discontinuity at the level where prices are the same, where a small change in price can lead to a significant change in demand.

Nash Equilibrium in a Homogeneous Product Market

• The only Nash equilibrium is when both firms set a price equal to marginal cost (P1 = P2 = C).
• This is because any other pair of prices would lead to a profitable deviation by one firm, making it not a Nash equilibrium.
• The Nash equilibrium can be found by building the reaction functions of the two firms and finding the intersection point.

Reaction Function of Firms

• Firm I's reaction function is built by finding the best price for firm I given any price of firm J.
• If firm I is less expensive than firm J, it can get the whole market and prefers to have a high price (just below the price of firm J).
• However, firm I doesn't want to set a price below its costs (C) or above the monopoly price (PM).

Bertrand Paradox

• The Bertrand Paradox states that in a market with only two firms and homogeneous products, the prices go down to marginal cost, similar to a perfectly competitive market.
• This is because the assumptions of the model, including unlimited capacity, transparent market, and homogeneous products, lead to extreme competitive pressure.

Extension of the Model: Multiple Firms and Different Costs

• When there are more than two firms with different costs, the firm with the lowest marginal cost (firm 1) will set its price above its own costs but below the cost of the next efficient firm.
• All firms will set the same price at equilibrium, but only firm 1 will sell to consumers.
• The result is still a paradox because most firms don't get any profit and don't get any market.

Commodity Products

• A commodity is a product that is exactly the same from many firms.
• Firms should be careful not to let their product become a commodity, as it can lead to low prices and margins.
• A natural response to this is to try and make the products differentiated to relax price competition.

Price Competition in a Duopoly

• A duopoly consists of two firms producing either differentiated or homogeneous products that compete in prices.
• The unified model used to analyze price competition involves firms 1 and 2 producing goods 1 and 2 at constant marginal costs C1 and C2, respectively.
• The substitution parameter D measures the substitutability between products, ranging from 0 (totally differentiated) to 1 (exactly the same).

Maximization Program and Best Response Function

• Firm 1's profit maximization program involves choosing a price to maximize profit, given the price of firm 2.
• The best response function of firm 1 is upward-sloping, indicating that if firm 2 increases its price, firm 1's best response is to also increase its price.

Strategic Complements and Reaction Functions

• Strategic complements occur when firms' strategies reinforce each other, such as price increases leading to further price increases.
• The reaction function of firm 1 is upward-sloping, meaning that an increase in firm 2's price leads to an increase in firm 1's price.
• The reaction function flattens as D decreases (products become more differentiated), and becomes horizontal when D = 0 (local monopolies).

Nash Equilibrium

• A Nash equilibrium occurs when both firms choose prices that are best responses to each other.
• The Nash equilibrium is found at the intersection of the reaction functions of both firms.
• Equilibrium prices increase with costs, and the price of one firm increases with the cost of the other firm due to strategic complements.

Symmetric Firms

• When firms are symmetric (same cost), they set the same price and quantity at equilibrium, earning the same profit.
• As D increases, the profit of symmetric firms decreases.

Homogeneous Products (Bertrand Model)

• In the Bertrand model, two firms produce the same product, and consumers only care about the price.
• The demand function assumes that consumers choose the firm with the lower price, and if prices are equal, the market splits between the two firms.

Demand and Market Structure

• When firm I sets a price P1, the quantity demanded is 0 if Pi is higher than the price of firm J (PGA), and Alpha I (a share of the total demand) if Pi is lower than PGA.
• If both firms have the same price, firm I gets a share Alpha I of the total demand.

Nash Equilibrium in a Homogeneous Product Market

• The only Nash equilibrium is when both firms set a price equal to marginal cost (P1 = P2 = C).
• This is because any other pair of prices would lead to a profitable deviation by one firm, making it not a Nash equilibrium.

Reaction Function of Firms

• Firm I's reaction function is built by finding the best price for firm I given any price of firm J.
• If firm I is less expensive than firm J, it can get the whole market and prefers to have a high price (just below the price of firm J).

Bertrand Paradox

• The Bertrand Paradox states that in a market with only two firms and homogeneous products, the prices go down to marginal cost, similar to a perfectly competitive market.
• This is because the assumptions of the model, including unlimited capacity, transparent market, and homogeneous products, lead to extreme competitive pressure.

Extension of the Model: Multiple Firms and Different Costs

• When there are more than two firms with different costs, the firm with the lowest marginal cost (firm 1) will set its price above its own costs but below the cost of the next efficient firm.
• All firms will set the same price at equilibrium, but only firm 1 will sell to consumers.

Commodity Products

• A commodity is a product that is exactly the same from many firms.
• Firms should be careful not to let their product become a commodity, as it can lead to low prices and margins.