10.3 The Monopoly Model
1. Introduction to Monopoly and Profit Maximization
Scenario: The Beerbrella Patent
Inheriting a business with a patent, like the Beerbrella, establishes a monopoly, meaning the firm is the sole producer in the market, giving it significant control over price and quantity.
Unlike firms in competitive markets that simply accept the market price, a monopolist faces a "double problem": determining both the quantity to produce and the price to charge.
Source: McMullin et al. Beerbrella US006637447B2, United States Patent and Trademark Office, 28 October 2003.
Applying the Marginal Decision Rule for Profit Maximization (Objective 3)
A monopolist maximizes profit by producing units until the marginal revenue (MRMR) of the last unit sold equals the marginal cost (MCMC) of producing it (MR=MCMR=MC).
While this profit maximization rule (MR=MCMR=MC) is consistent with principles from competitive markets, its application is more complex for a monopolist because the firm actively sets its own price, which directly influences its marginal revenue.
2. Monopoly, Market Demand, and Firm Demand
Monopolist's Firm Demand Curve
In a perfectly competitive market, individual firms are economically small and face a horizontal demand curve, meaning they can sell any quantity at the prevailing market price.
A monopolist is different; because it has its market entirely to itself, its firm's demand curve is the entire market demand curve.
This market demand curve, and thus the monopolist's firm demand curve, is downward-sloping.
Impact of Downward-Sloping Demand on Price and Quantity
The downward-sloping nature of the demand curve means that a monopolist can only sell a greater quantity of output by reducing its price.
Example (Figure 10.2 Implication):
If the Beerbrella monopoly sells quantity Q1Q1 units at a price P1P1 (point A).
To increase sales to Q2Q2 units, the firm must reduce its price to P2P2 (point B).
To sell an even larger quantity, Q3Q3, the price would need to be reduced further to P3P3 (point C).
Profit-maximizing monopoly firms choose their optimal price/output combination, but they are strictly constrained by the demand curve; they cannot charge a high price (P1P1) and simultaneously sell a high quantity (Q3Q3) if demand does not support that combination.
3. Relationship Between Price and Marginal Revenue When a Firm Faces a Downward-Sloping Demand Curve (Objective 1)
Marginal Revenue in Competitive vs. Monopoly Firms
For a perfectly competitive firm, marginal revenue is always equal to the market price because it can sell any quantity at that price without affecting it.
For a monopolist, however, marginal revenue changes each time the price is adjusted because the firm faces a downward-sloping demand curve, leading to MR<PMR<P (marginal revenue is less than price).
Explanation of the Relationship (Why MR<PMR<P)
To sell an additional unit, the monopolist must lower the price, not just for that specific unit, but for all units sold, including those that could have been sold at a higher price.
Example Breakdown (3 to 4 units):
Suppose a firm sells 3 units at P=7P=7, yielding Total Revenue (TRTR) = 2121. This is represented by a rectangle with height 77 and base 33.
To sell a 4th unit, the firm must reduce the price to P=6P=6. Total revenue becomes 4imes6=244imes6=24.
The Marginal Revenue (MRMR) for the 4th unit is TR4−TR3=24−21=3TR4−TR3=24−21=3.
Despite the 4th unit selling for P=6P=6, the MR is only 33. This is because the firm gained 66 from selling the 4th unit, but lost 33 (a 11 price reduction on each of the first 3 units) on the units it was already selling at a higher price.
Therefore, the marginal revenue (MRMR) obtained from selling an additional unit is always less than the price (PP) at which that unit (and all preceding units) is sold. This is a direct consequence of needing to reduce price to increase quantity demanded along a downward-sloping demand curve.
4. Relationship Between Total Revenue, Marginal Revenue, and Elasticity Along a Linear Demand Curve (Objective 2)
Elasticity's Variation Along a Linear Demand Curve
Demand is elastic in the upper half of a linear demand curve.
Demand is inelastic in the lower half of a linear demand curve.
Demand is unit elastic precisely at the midpoint of a linear demand curve.
Total Revenue (TR) Behavior
Total revenue starts at zero (at a very high price where quantity demanded is zero).
As price decreases and quantity increases through the elastic region, total revenue rises.
Total revenue reaches its peak at the midpoint of the demand curve, where demand is unit elastic.
As price continues to decrease and quantity increases through the inelastic region, total revenue declines, eventually returning to zero (at a zero price).
Monopolist's Operating Region for Profit Maximization
A profit-maximizing monopolist will never operate in the inelastic region (lower half) of the demand curve.
Reason: If a firm operates in the inelastic region, cutting output would both increase total revenue (due to inelastic demand) and decrease total costs, leading to unequivocally higher profits (π=TR−TCπ=TR−TC).
Therefore, a profit-maximizing monopolist will always operate in the elastic region (upper half) of the demand curve or, at most, at the point of unit elasticity where total revenue is maximized.
Marginal Revenue (MR) as the Slope of Total Revenue
Marginal revenue (MRMR) is defined as the change in total revenue (ΔTRΔTR) divided by the change in quantity (ΔQΔQ): MR=ΔTRΔQMR=ΔQΔTR. This means MRMR is the slope of the total revenue curve.
Relationship between MR, TR, and Elasticity:
When total revenue is increasing (elastic region of demand), MRMR is positive.
When total revenue is at its maximum (unit elastic point of demand), MRMR is zero.
When total revenue is decreasing (inelastic region of demand), MRMR is negative.
Rules for Drawing the Marginal Revenue Curve for a Linear Demand Curve
The marginal revenue curve starts at the same price intercept (vertical axis) as the demand curve.
The marginal revenue curve is twice as steep as the demand curve.
The marginal revenue curve cuts the horizontal (quantity) axis exactly halfway between the origin and where the demand curve cuts the horizontal axis.
1. Monopoly Profit Maximization: Applying the Marginal Decision Rule
Core Principle: Both perfectly competitive firms and monopolists apply the same principle to maximize profits: produce as long as marginal revenue (MRMR) exceeds marginal cost (MCMC), and stop when they are equal (MR=MCMR=MC).
Monopolist's Advantage: Inheriting a business with a patent (like the Beerbrella) establishes a monopoly, allowing the firm to be the sole producer and granting significant control over price and quantity.
Source Reference: McMullin et al. Beerbrella US006637447B2, United States Patent and Trademark Office, 28 October 2003.
2. Determining Profit-Maximizing Quantity and Price
How much to produce?
Apply the marginal decision rule: To maximize profits, produce the quantity where marginal revenue equals marginal cost (MR=MCMR=MC).
This point is identified as
point Ain a typical monopoly graph (like Figure 10.7 implied by the text).The profit-maximizing quantity is denoted as QM∗QM∗.
How much to charge?
Once QM∗QM∗ is determined, the monopolist charges the highest possible price that allows them to sell all units produced.
This price, PM^, is found by going vertically up from QM^ to the demand curve (identified as
point Bin Figure 10.7) and then horizontally to the price axis.
Key Insight: Only information regarding demand, marginal revenue, and marginal cost is necessary to determine the profit-maximizing price/quantity combination.
3. Computing Monopoly Profit
Information Needed: To calculate total profit, information about the average cost curve is required.
Average Cost: The average cost of producing QM∗QM∗ units is typically found on the average total cost (ATCATC) curve, denoted as ATCQMATCQM (as in Figure 10.8).
Profit Per Unit: This is the difference between the profit-maximizing price and the average total cost: (PM∗−ATCQM)(PM∗−ATCQM).
Total Profit: Total profit is calculated by multiplying the profit per unit by the profit-maximizing quantity: TotalProfit=(PM∗−ATCQM)imesQM∗TotalProfit=(PM∗−ATCQM)imesQM∗.
Graphical Representation: Total profit is measured as the area of a rectangle with a base equal to QM^ and a height equal to (PM−ATCQM)(PM−ATCQM). This area is often highlighted (e.g., green rectangle in Figure 10.8).
Profit Condition: A firm earns profits when the price (PM∗PM∗) is higher than the average cost of production (ATCQMATCQM).
4. Monopoly Losses and the Shutdown Decision
Profit Not Guaranteed: Even without competition, a monopoly is not guaranteed to earn profits. Success depends entirely on the strength of demand for the product and the cost of production.
Loss Scenario: If the demand for the product is weak or production costs are high, the profit-maximizing price (PM^) at quantity QM^ may be lower than the average cost of production (PM∗<ATCQMPM∗<ATCQM).
In such cases, the firm incurs losses. Losses occur when the average cost curve is above the demand curve at every level of possible output (as illustrated by the papier mache garden gnomes example in Figure 10.9).
Minimizing Losses: The firm should continue producing at QM^ and charging PM^ if the price is greater than the average variable cost (P>AVCP>AVC). This strategy aims to minimize losses, which are depicted as a red area in Figure 10.9.
Shutdown Rule: If the price is not greater than the average variable cost (PextextleAVCPextextleAVC), the firm should shut down operations and produce zero units to avoid incurring even larger losses.
5. Key Relationships: Price, Marginal Revenue, and Elasticity along a Linear Demand Curve
Marginal Revenue vs. Price: For a firm facing a downward-sloping demand curve (a monopolist), marginal revenue (MRMR) is always less than price (PP) (MR<PMR<P).
Explanation: To sell an additional unit, the monopolist must reduce the price not only for that extra unit but for all units sold, including those that could have been sold at a higher price.
Marginal Revenue and Elasticity: The relationship between marginal revenue and elasticity along a demand curve is:
MRMR is positive when demand is elastic (in the upper half of a linear demand curve).
MRMR is zero when demand is unit-price elastic (at the midpoint of a linear demand curve), where total revenue is maximized.
MRMR is negative when demand is inelastic (in the lower half of a linear demand curve).
Monopolist's Operating Region for Profit Maximization: A profit-maximizing monopolist will never operate in the inelastic region of the demand curve.
Reason: In the inelastic region, cutting output would both increase total revenue (due to inelastic demand) and decrease total costs, unequivocally leading to higher profits (π=TR−TCπ=TR−TC). Therefore, a monopolist operates in the elastic region or at most at the point of unit elasticity.
Rules for Drawing the Marginal Revenue Curve for a Linear Demand Curve:
The marginal revenue curve starts at the same price intercept (vertical axis) as the demand curve.
The marginal revenue curve is twice as steep as the demand curve.
The marginal revenue curve cuts the horizontal (quantity) axis exactly halfway between the origin and where the demand curve cuts the horizontal axis.