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.