Question

A single firm monopolizes the entire market for widgets and can produce at constant average and marginal costs of \\[A C=M C=10.\\] Originally, the firm faces a market demand curve given by \\[Q=60-P.\\] a. Calculate the profit-maximizing price-quantity combination for the firm. What are the firm's profits? b. Now assume that the market demand curve shifts outward (becoming steeper) and is given by \\[Q=45-0.5 P.\\] What is the firm's profit-maximizing price-quantity combination now? What are the firm's profits? c. Instead of the assumptions of part (b), assume that the market demand curve shifts outward (becoming flatter) and is given by \\[Q=100-2 P.\\] What is the firm's profit-maximizing price-quantity combination now? What are the firm's profits? d. Graph the three different situations of parts (a), (b), and (c). Using your results, explain why there is no real supply curve for a monopoly.

Short answer

2. Calculate the firm's profits for each of the three demand curves. 3. Explain why there is no real supply curve for a monopoly.

Step-by-step solution

1a: Finding the profit-maximizing price-quantity combination for the first demand curve

The original demand curve is given by: \\[Q = 60 - P\\] First, we need to find the total revenue (TR) function. To do this, we can rewrite the demand curve as: \\[P = 60-Q\\] Then, multiply it by Q to find the total revenue: \\[TR = P \times Q = (60 - Q)Q\\] To find the marginal revenue (MR), we can take the derivative of TR with respect to Q: \\[MR = \dfrac{d(TR)}{dQ} = 60 - 2Q\\] Now, we can set the MR equal to the MC to find the profit-maximizing quantity: \\[60 - 2Q = 10\\] \\[Q = 25\\] Now that we have the profit-maximizing quantity, we can plug it into the demand curve to find the price: \\[P = 60 - 25 = 35\\] So, the profit-maximizing price-quantity combination for the first demand curve is P = 35 and Q = 25.

1b: Calculating the firm's profits for the first demand curve

To find the firm's profits, we need to calculate the difference between the total revenue and total cost. The total revenue (TR) can be found as: \\[TR = P \times Q = 35 \times 25 = 875\\] The total cost (TC) can be found using the average cost (AC) and the quantity produced: \\[TC = AC \times Q = 10 \times 25 = 250\\] Thus, the firm's profits are: \\[\text{Profit} = TR - TC = 875 - 250 = 625\\]

2a: Finding the profit-maximizing price-quantity combination for the second_demand curve

The second_demand curve is given by: \\[Q = 45 - 0.5P\\] First, we need to find the total revenue (TR) function. To do this, we can rewrite the demand curve as: \\[P = 90 - 2Q\\] Then, multiply it by Q to find the total revenue: \\[TR = P \times Q = (90 - 2Q)Q\\] To find the marginal revenue (MR), take the derivative of TR with respect to Q: \\[MR = \dfrac{d(TR)}{dQ} = 90 - 4Q\\] Now, set the MR equal to the MC to find the profit-maximizing quantity: \\[90 - 4Q = 10\\] \\[Q = 20\\] Now that we have the profit-maximizing quantity, we can plug it into the demand curve to find the price: \\[P = 90 - 40 = 50\\] So, the profit-maximizing price-quantity combination for the second_demand curve is P = 50 and Q = 20.

2b: Calculating the firm's profits for the second_demand curve

To find the firm's profits, we need to calculate the difference between the total revenue and total cost. The total revenue (TR) can be found as: \\[TR = P \times Q = 50 \times 20 = 1000\\] The total cost (TC) can be found using the average cost (AC) and the quantity produced: \\[TC = AC \times Q = 10 \times 20 = 200\\] Thus, the firm's profits are: \\[\text{Profit} = TR - TC = 1000 - 200 = 800\\]

3a: Finding the profit-maximizing price-quantity combination for the third_demand curve

The third_demand curve is given by: \\[Q = 100 - 2P\\] First, we need to find the total revenue (TR) function. To do this, we can rewrite the demand curve as: \\[P = 50 - 0.5Q\\] Then, multiply it by Q to find the total revenue: \\[TR = P \times Q = (50 - 0.5Q)Q\\] To find the marginal revenue (MR), take the derivative of TR with respect to Q: \\[MR = \dfrac{d(TR)}{dQ} = 50 - Q\\] Now, set the MR equal to the MC to find the profit-maximizing quantity: \\[50 - Q = 10\\] \\[Q = 40\\] Now that we have the profit-maximizing quantity, we can plug it into the demand curve to find the price: \\[P = 50 - 20 = 30\\] So, the profit-maximizing price-quantity combination for the third_demand curve is P = 30 and Q = 40.

3b: Calculating the firm's profits for the third_demand curve

To find the firm's profits, we need to calculate the difference between the total revenue and total cost. The total revenue (TR) can be found as: \\[TR = P \times Q = 30 \times 40 = 1200\\] The total cost (TC) can be found using the average cost (AC) and the quantity produced: \\[TC = AC \times Q = 10 * 40 = 400\\] Thus, the firm's profits are: \\[\text{Profit} = TR - TC = 1200 - 400 = 800\\]

4: Graph the three different situations and explain why there is no real supply curve for a monopoly

A supply curve shows the relationship between the quantity supplied and the price in the market. However, in a monopoly, the firm chooses both the quantity and the price that maximize its profit. As a result, there is no unique relationship between quantity and price, as each demand curve results in a different price-quantity combination. When the monopolist faces a different demand curve, the profit-maximizing price and output may change. In other words, there is no single, upward-sloping supply curve for a monopolist because the monopolist's supply decision is tied directly to the market demand curve, which can have many different shapes. Thus, a monopoly doesn't have a real supply curve that can be represented as a simple functional relationship between quantity and price.

Key concepts

Demand Curve Analysis

In a monopoly, the demand curve represents the relationship between the price consumers are willing to pay and the quantity of goods the monopolist can sell. For the monopolist, understanding this curve is crucial. The slope and position of the demand curve dictate the firm's ability to set prices.
Initially, with a curve like \(Q = 60 - P\), the demand is relatively elastic, meaning consumers are sensitive to price changes.
As the curve becomes more steep or flat, indicated by the equations \(Q = 45 - 0.5P\) and \(Q = 100 - 2P\) respectively, the firm faces different price sensitivities.
The steeper the curve, the less sensitive consumers are to price changes, allowing the firm potentially higher profit margins.
The flatter the curve, the more choice consumers have to go elsewhere if prices rise, challenging the monopolist to consider competitive pricing strategies to maintain sales.

Marginal Revenue

Marginal Revenue (MR) is the additional revenue the firm earns from selling one more unit of a product. For a monopoly, the MR curve is often below the demand curve because the price must drop to sell additional units.
This is shown mathematically by first transforming the demand equation, as seen with \(Q = 60 - P\), into a total revenue function \(TR = (60 - Q)Q\), and then deriving the MR by differentiating TR with respect to Q, giving \(MR = 60 - 2Q\).
Understanding MR is critical as the firm will set production where marginal revenue equals marginal cost (MC, the cost of producing one more unit), maximizing profit.
This difference between demand and MR highlights that a monopolist always decreases price on previous units sold to sell an additional unit, affecting both pricing strategy and output decision.

Monopolist Pricing Strategy

A monopolist's pricing strategy focuses on maximizing profits, which involves finding the equilibrium where marginal revenue equals marginal cost. In our scenarios, with a constant MC of 10, calculations for each demand curve reveal the optimal price and quantity combinations.
For example, solving \(60 - 2Q = 10\) yields a Q of 25 and a corresponding price of 35 from the initial demand \(Q = 60 - P\).
In contrast, a steeper curve like \(Q = 45 - 0.5P\) results in a lower quantity (Q = 20) but a higher price (P = 50), demonstrating how demand characteristics impact strategic decisions.
The monopolist must gauge market reactions and adjust pricing accordingly, balancing between setting a high price and encouraging higher sales volumes.

Total Revenue and Total Cost

Total Revenue (TR) is calculated by multiplying the selling price by the quantity sold. In a monopoly, it varies according to the price-quantity point the firm chooses along its demand curve.
For example, TR for Q = 25 from \(Q = 60 - P\) at price 35 is accomplished by multiplying 35 by 25, resulting in 875.
Total Cost (TC), on the other hand, is given by multiplying the average cost with the quantity produced, shown here as \(10 \, \text{(AC)} \times 25 = 250\).
Consequently, profits are calculated as TR minus TC, offering insights into the profitability of different strategic decisions, which in turn guides future decisions in production and pricing.

Supply Curve for Monopolies

Unlike competitive markets, monopolies do not have a traditional supply curve that denotes a direct relationship between price and quantity supplied. A monopoly's choice is any price-quantity combination on its demand curve that maximizes profits.
Why? Because the monopolist is both price setter and quantity provider, driven by the idea of balancing MR and MC.
Adjustments in demand, such as shifts or changes in elasticity, alter the firm's optimal output and pricing, making each scenario unique and non-linear.
Thus, while competitive firms supply more as prices rise, a monopolist adjusts output based on changes in profit-maximizing conditions influenced by demand variations, lacking a predictable supply response.