Question

Suppose that an industry is characterized as follows: $$\begin{array}{|ll|} \hline C=100+2 q^{2} & \text { each firm's total cost function } \\ \hline M C=4 q & \text { firm's marginal cost function } \\ \hline P=90-2 Q & \text { industry demand curve } \\ \hline M R=90-4 Q & \text { industry marginal revenve curve } \\ \hline \end{array}$$ a. If there is only one firm in the industry, find the monopoly price, quantity, and level of profit. b. Find the price, quantity, and level of profit if the industry is competitive. c. Graphically illustrate the demand curve, marginal revenue curve, marginal cost curve, and average cost curve. Identify the difference between the profit level of the monopoly and the profit level of the competitive industry in two different ways. Verify that the two are numerically equivalent.

Short answer

In the case of a monopoly firm, the optimal quantity is 15, the price is 60 and the profit is 350. While for a perfectly competitive market, the optimal quantity also turns out to be 15, with the price at 60 and the profit at 350, exactly the same as the monopoly scenario. When graphically displayed, the profit levels of both markets appear the same. Their numerical equivalence verifies this similarity.

Step-by-step solution

Calculate Monopoly Quantity (a)

Equating Marginal Cost(MC) to Marginal Revenue(MR) to maximize monopoly firm's profit, \(4q = 90-4Q\) -> \(Q = q\). So, \(4q = 90-4q\), which solves to \(q = 15\).

Calculate Monopoly Price (a)

Substitute \(q = 15\) into industry demand curve to get the price, \(P = 90 - 2(15)\), which evaluates to \(P = 60\).

Calculate Monopoly Profit (a)

First, calculate total revenue by multiplying price and quantity, \(TR = PQ = 60 * 15 = 900\). Then, calculate total cost using total cost function, substituting \(q=15, TC=100+2(15)^2 = 100 + 450 = 550\). Profit is total revenue minus total cost, thus Profit = 900 - 550 = 350.

Calculate Competitive Industry Quantity (b)

In a competitive market, price equals marginal cost. Solve \(P=MC\) for \(q\) using the given functions. \(90-2Q=4q\), as \(Q=q\) in a competitive market, this gives \(90-2q=4q\), which solves to \(q=15\). Thus, in a competitive market, quantity is 15.

Calculate Competitive Industry Price (b)

Substitute \(q=15\) into industry demand curve to get the price. \(P=90-2(15)\) evaluates to \(P=60\). The price in a competitive market is 60.

Calculate Competitive Industry Profit (b)

Calculate the total revenue by multiplying the price and quantity, \(TR=60*15=900\). Total cost will be the same as in Step 3. The profit is total revenue minus total cost. Profit = 900 - 550 = 350, the same as the monopoly.

Graphical Representation (c)

Create a graph with the given functions: MC, MR, average cost and the demand curve. The difference in profits can be seen as the rectangular area difference under the price and above the cost curves on the graph.

Numerical Verification (c)

If calculated correctly, the competitive and monopoly profit are found to be the same. Hence, there is no profit difference when verified numerically.

Key concepts

Demand Curve

In microeconomics, a demand curve represents the relationship between the quantity of a good that consumers want to purchase and the price of that good.
In a perfectly competitive market, each firm's price is dictated by the industry demand curve. This curve is usually downward sloping, indicating that as price decreases, demand increases and vice versa.
For our exercise, the industry demand curve is formulated as \(P = 90 - 2Q\). This equation tells us that when the quantity \(Q\) changes, the price \(P\) is adjusted accordingly. For instance, if no products are sold, the price is at a maximum of 90. As more units are purchased, the price per unit decreases.

• If the market is a monopoly, the demand curve represents the monopoly's entire market demand.
• If the market is competitive, the demand curve reflects the demand faced by the industry as a whole.

Understanding how to interpret and use the demand curve is crucial for firms when deciding their pricing and output strategies.

Marginal Cost

Marginal cost (MC) is the additional cost incurred by producing one more unit of a good or service. It is critical for firms to understand marginal cost as it directly influences both pricing and production decisions.
In this exercise, the marginal cost function is given by \(MC = 4q\), where \(q\) represents the quantity produced by each firm.
With this linear function, the cost of producing each additional unit increases by 4.

• For a monopoly, MC is crucial as it informs the decision on the optimal production level that maximizes profit when equated to marginal revenue.
• In a competitive market, firms produce where the price equals marginal cost to ensure no economic profit is being lost or otherwise gained.

Understanding marginal cost helps in identifying the optimal quantity of goods to produce to maximize profits or minimize losses.

Industry Demand

The industry demand is the total demand for a product across all firms within an industry. It encapsulates the cumulated demands of all buyers in the market and is a fundamental indicator in setting the price and production levels of goods.
In this setting, the industry demand is described by the equation \(P = 90 - 2Q\). This curve reflects the required adjustments in market price based on aggregate market demand levels.

• A monopoly controls the entire industry demand. Thus, the monopolist uses this curve to determine the price and output quantity that maximizes its profit.
• In a competitive market, each firm's output adjusts to changes in industry demand to ensure market equilibrium, where supply meets demand at the prevailing market price.

Understanding industry demand allows firms to respond effectively to market changes and to strategize in terms of pricing and outputs.

Marginal Revenue

Marginal Revenue (MR) is the additional income that a firm earns from selling one more unit of a good or service. It plays a pivotal role in determining the quantity of output a firm decides to produce.
Our exercise uses the marginal revenue function: \(MR = 90 - 4Q\). This function indicates that as the quantity sold increases, the additional revenue earned decreases.

• In a monopoly, marginal revenue is a significant driver of decisions because it helps identify the output level where MR equals MC, optimizing profit.
• However, in a perfect competition, the price is equal to marginal revenue because firms have to be price-takers.

Understanding and calculating marginal revenue is essential for profit maximization in different market structures, particularly for monopolistic enterprises.