Question

Consider the following Bertrand game involving two firms producing differentiated products. Firms have no costs of production. Firm 1's demand is \\[ q_{1}=1-p_{1}+b p_{2} \\] where \(b > 0 .\) A symmetric equation holds for firm 2 's demand. a. Solve for the Nash equilibrium of the simultaneous price-choice game. b. Compute the firms' outputs and profits. c. Represent the equilibrium on a best-response function diagram. Show how an increase in \(b\) would change the equilibrium. Draw a representative isoprofit curve for firm 1

Short answer

Answer: As the differentiation parameter \(b\) increases, the Nash equilibrium prices for both firms increase, while their outputs remain constant at a half. The profits for both firms also increase due to higher equilibrium prices. This increase in differentiation leads to steeper best-response functions, as each firm becomes less sensitive to the other firm's actions, allowing them to charge higher prices without losing too much market share.

Step-by-step solution

Find the profit functions of both firms.

To find the Nash equilibrium prices, we first need to find the profit functions for both firms. Since there are no costs, profit is simply given by revenue, which can be calculated as price times quantity. For firm 1, the profit function is: \\[ \pi_{1} = p_{1}q_{1} = p_{1}(1 - p_{1} + b p_{2}) \\] Similarly, for firm 2, the profit function is: \\[ \pi_{2} = p_{2}q_{2} = p_{2}(1 - p_{2} + b p_{1}) \\]

Find the Best-Response functions of both firms.

Now we need to find the best-response functions of both firms by maximizing their profits. To do this, we take the first derivative of the profit functions concerning their respective prices and set them equal to zero. For firm 1, the first-order condition is: \\[ \frac{\partial \pi_{1}}{\partial p_{1}} = 1 - 2 p_{1} + b p_{2} = 0 \\] For firm 2, the first-order condition is: \\[ \frac{\partial \pi_{2}}{\partial p_{2}} = 1 - 2 p_{2} + b p_{1} = 0 \\]

Solve the system of equations.

We now have a system of two linear equations with two unknowns (\(p_{1}\) and \(p_{2}\)). We can solve this system to find the Nash equilibrium prices. From the first-order condition for firm 1, we can solve for \(p_1\): \\[ p_{1} = \frac{1 + b p_{2}}{2} \\] Substitute this expression for \(p_1\) into the first-order condition for firm 2: \\[ 1 - 2 p_{2} + b \left(\frac{1 + b p_{2}}{2}\right) = 0 \\] Solve for \(p_2\): \\[ p_{2} = \frac{1}{2(2 - b)} \\] Substitute the expression for \(p_2\) back into the expression for \(p_1\): \\[ p_{1} = \frac{1 + b\left(\frac{1}{2(2 - b)}\right)}{2} = \frac{1}{2(2 - b)} \\] Thus, the Nash equilibrium prices are \(p_{1}^* = p_{2}^* = \frac{1}{2(2 - b)}\). #b. Computing outputs and profits#

Calculate companies' quantities.

Now that we have the Nash equilibrium prices, we can calculate the output of both firms by plugging these prices into their respective demand functions. For firm 1: \\[ q_{1}^* = 1 - p_{1}^* + b p_{2}^* = 1 - \frac{1}{2(2 - b)} + b\left(\frac{1}{2(2 - b)}\right) = \frac{1}{2} \\] For firm 2: \\[ q_{2}^* = 1 - p_{2}^* + b p_{1}^* = 1 - \frac{1}{2(2 - b)} + b\left(\frac{1}{2(2 - b)}\right) = \frac{1}{2} \\] Thus, the Nash equilibrium outputs are \(q_{1}^* = q_{2}^* = \frac{1}{2}\).

Calculate companies' profits.

Since the firms have no costs, their profits are equal to their revenues. We can use the Nash equilibrium prices and outputs to calculate these profits. For firm 1: \\[ \pi_{1}^* = p_{1}^* q_{1}^* = \left(\frac{1}{2(2 - b)}\right) \left(\frac{1}{2}\right) = \frac{1}{4(2 - b)} \\] For firm 2: \\[ \pi_{2}^* = p_{2}^* q_{2}^* = \left(\frac{1}{2(2 - b)}\right) \left(\frac{1}{2}\right) = \frac{1}{4(2 - b)} \\] Thus, the Nash equilibrium profits are \(\pi_{1}^* = \pi_{2}^* = \frac{1}{4(2 - b)}\). #c. Representing the equilibrium on a diagram#

Sketch the Best-Response functions and equilibrium.

To represent the equilibrium on a diagram, we first need to draw the best-response functions of both firms. The best-response function for firm 1 is obtained by solving its first-order condition for \(p_1\): \\[ p_{1} = \frac{1 + b p_{2}}{2} \\] Similarly, the best-response function for firm 2 is obtained by solving its first-order condition for \(p_2\): \\[ p_{2} = \frac{1 + b p_{1}}{2} \\] First, sketch the best-response functions for both firms on a diagram, with \(p_1\) on the vertical axis and \(p_2\) on the horizontal axis. Label these functions as \(BR_1(p_2)\) and \(BR_2(p_1)\), respectively. Next, draw a representative isoprofit curve for firm 1. This curve will be upward-sloping and convex, reflecting the trade-off between raising its price to increase its profit margin and losing market share to its competitor. Finally, plot the Nash equilibrium values of \(p_1\) and \(p_2\) that we obtained in part (a). This point will be the intersection of the best-response functions and will lie on the representative isoprofit curve of firm 1.

Analyze the effect of an increase in \(b\).

Now we examine the effect of an increase in the differentiation parameter \(b\). Recall that the Nash equilibrium prices are given by \(p_{1}^* = p_{2}^* = \frac{1}{2(2 - b)}\). As \(b\) increases, the denominator of the Nash equilibrium prices shrinks, causing the prices to increase. In other words, an increase in \(b\) results in higher Nash equilibrium prices for both firms. On the diagram, this implies that the best-response functions of both firms will become steeper as \(b\) increases. This is because an increase in differentiation makes each firm less sensitive to the other firm's actions, allowing them to charge higher prices without losing too much market share. In summary, an increase in \(b\) leads to higher Nash equilibrium prices, steeper best-response functions, and a shift in the representative isoprofit curve for firm 1.

Key concepts

Nash equilibrium

In the world of economics, a Nash equilibrium is a situation where two or more players in a game reach a point where no player can benefit by changing their strategy while the others keep theirs unchanged. It's like a stalemate in chess where any move changes the balance.

In the context of a Bertrand game with differentiated products, two firms decide on prices for their products at the same time. Here, the Nash equilibrium occurs when each firm's price is the optimal response to the other firm's price. At this point, neither firm can unilaterally lower its price to increase profit without reducing both firms' overall profits.

Nash equilibrium stands as a critical concept because it involves strategic decision-making where each firm anticipates the competitor's move, adjusting its pricing strategy accordingly. Once equilibrium is reached, both firms settle on prices that balance competitive and cooperative interactions within the market.

best-response functions

Best-response functions are the strategies that maximize a player's payoff, given the other players' strategies in the game. Think of it as a guide to what each firm should do when they know the other firm's potential moves.

For example, in our Bertrand game situation, each firm derives a best-response function by considering the price set by the other firm. Each firm calculates how its profits are affected by its competitor's price. It then adjusts its price to maximize its profit.

The process involves setting the derivative of the profit function to zero to find the optimal price. The best-response functions are critical in deriving the Nash equilibrium because they show how each firm can optimize its decision-making in response to the competitor's pricing.

These functions are visually represented through diagrams, aiding in understanding how changes affect each player's strategies and lead to equilibrium.

differentiated products

Differentiated products are variations of goods that are similar but have differences making them perfect substitutes. Consider how different brands of cereal may look the same but have unique flavors or health benefits.

In a Bertrand game, the differentiation parameter \(b\) influences how much one firm's product can substitute for another's. This parameter is crucial to the firms' pricing decisions, as a higher \(b\) means more differentiation, reducing the direct price competition between firms.

Increased differentiation allows firms to charge higher prices because they don't compete solely on price. Customers may prefer one product over another due to unique features, like brand loyalty or quality, allowing firms flexibility in their pricing strategies. Overall, understanding product differentiation helps firms in predicting competitor behavior and setting competitive prices.

isoprofit curves

Isoprofit curves are plots that represent combinations of prices yielding the same profit for a firm. Imagine contour lines on a map that show elevation; here, they show profit levels instead.

Within Bertrand competition, an isoprofit curve for a firm gives a visual representation of how different pricing strategies across its product line yield the same profit. The curve typically slopes upwards: as one firm's price increases, the other firm's price should also increase to keep profits constant.

By illustrating these curves on a diagram, firms can visually grasp how changes in one firm's pricing affect the other's profits. Isoprofit curves provide strategic insights into pricing decisions, helping firms understand the profitability landscape of the market and better anticipating competitive responses. They shed light on not just single points of equilibrium but also potential paths to reach or deviate from them.