Suppose the inverse market demand equation is P = 80 ¡V 4(QA+QB), where QA is the output of firm A and QB is the output of firm B, and both firms have a constant marginal constant of $4. Firm B is the Stackelberg leader in this market.

(a)State the reaction function for firm A.
(b)Write down the profit equation for firm B.
(c)Calculate the profit maximizing output for firm B.
(d)State the output of firm A, and the Stackelberg equilibrium outputs.
(e)What is the industry output?
(f)Plot the Stackelberg equilibrium outputs on a graph (with the reaction functions).
(g)Compare the Cournot and Stackelberg equilibrium outputs. Explain why is it more profitable for the Stackelberg leader to produce more than the Cournot equilibrium output.

To answer these questions, we need to understand the concepts of reaction functions, profit equations, profit maximization, equilibrium outputs, and industry output in the context of the given scenario.

(a) The reaction function for firm A can be derived by setting the profit-maximizing output for firm A, QA, equal to zero in the inverse market demand equation. This means that firm A is acting as a follower and decides its output based on the output set by firm B.

Start with the inverse market demand equation: P = 80 - 4(QA + QB)
To find the reaction function for firm A, we assume that QB is given by firm B. Substituting this into the inverse demand equation:
P = 80 - 4(QA + QB)
P = 80 - 4QA - 4QB

Now, to solve for QA, set the derivative of firm A's profit equation with respect to QA equal to zero:
Marginal Revenue (MR) = Marginal Cost (MC)
MR = 80 - 8QA - 4QB
MC = $4 (given)

Setting MR equal to MC, we have:
80 - 8QA - 4QB = 4
-8QA - 4QB = -76
-2QA - QB = -19

This equation represents the reaction function for firm A.

(b) The profit equation for firm B can be derived by subtracting the cost equation from its revenue equation. Given that firm B has a constant marginal cost of $4, the profit equation becomes:
Profit (PB) = Revenue (RB) - Cost (CB)
RB = P * QB = (80 - 4(QA + QB)) * QB = 80QB - 4(QA + QB)QB
CB = $4 * QB = 4QB

Profit (PB) = 80QB - 4(QA + QB)QB - 4QB

(c) To calculate the profit-maximizing output for firm B, we need to find the value of QB that maximizes the profit function (PB). We can do this by taking the derivative of the profit equation with respect to QB, setting it equal to zero, and solving for QB.

(d) To find the output of firm A and the Stackelberg equilibrium outputs, we need to substitute the values of QB from the profit-maximizing output in (c) into the reaction function for firm A.

(e) The industry output is the sum of the outputs produced by firm A and firm B. Therefore, the industry output is QA + QB.

(f) To plot the Stackelberg equilibrium outputs on a graph, we need to create a graph with QA and QB on the axes. The reaction function for firm A and the profit-maximizing output found in (c) will help determine the specific points to plot on the graph.

(g) The Cournot and Stackelberg equilibrium outputs differ because of the leader-follower relationship between firms A and B in the Stackelberg model. In the Cournot model, each firm chooses its output simultaneously, assuming the output of the other firm remains constant. However, in the Stackelberg model, firm B, as the leader, sets its output first, taking into account the reaction of firm A.

The Stackelberg leader has the advantage of being able to set its output based on how firm A will react. As a result, the Stackelberg leader can strategically increase its output beyond the Cournot equilibrium level to maximize its profits. Firm B's increased output allows it to capture more market share and gain a pricing advantage over firm A. This enables the Stackelberg leader to achieve higher profits compared to the Cournot equilibrium output.