When A Monopolist Increases Sales By One Unit

When a monopolist increases sales by one unit, the change in revenue is not simply the price of that unit; it is shaped by the downward‑sloping demand curve, the price elasticity of demand, and the marginal cost structure that together dictate the firm’s profit‑maximizing behavior.

Easier said than done, but still worth knowing.

Introduction

A monopoly faces a single market demand curve, unlike a perfectly competitive firm that can sell any quantity at the prevailing market price. Still, because the monopolist must lower the price to sell additional output, each extra unit sold affects the revenue earned on all previously sold units. Understanding what happens when a monopolist increases sales by one unit is essential for grasping concepts such as marginal revenue (MR), price discrimination, and the welfare implications of market power.

The Economic Logic Behind a One‑Unit Increase

1. The demand side

• Demand curve: In a monopoly, the demand curve ( P(Q) ) shows the highest price consumers are willing to pay for each quantity ( Q ).
• Price reduction: To sell the ((Q+1)^{\text{th}}) unit, the monopolist must lower the price from ( P(Q) ) to ( P(Q+1) ).
• Revenue impact: The price cut applies to all ( Q+1 ) units, not just the marginal unit.

2. Marginal Revenue (MR)

The formal definition of marginal revenue is

[ MR = \frac{d(TR)}{dQ} = \frac{d(P \cdot Q)}{dQ} ]

where total revenue ( TR = P(Q) \times Q ). Expanding the derivative gives

[ MR = P(Q) + Q \cdot \frac{dP}{dQ} ]

Because ( \frac{dP}{dQ} < 0 ) for a downward‑sloping demand, the second term is negative, making MR lower than price. When a monopolist increases sales by one unit, the incremental revenue equals this marginal revenue value.

3. The role of price elasticity

The price elasticity of demand ( \varepsilon = \frac{dQ}{dP}\frac{P}{Q} ) can be rearranged to express MR as

[ MR = P\left(1 + \frac{1}{\varepsilon}\right) ]

• If demand is elastic (( |\varepsilon| > 1 )), the term ( \frac{1}{\varepsilon} ) is negative but greater than (-1), so MR remains positive.
• If demand is unit‑elastic (( |\varepsilon| = 1 )), MR drops to zero.
• If demand is inelastic (( |\varepsilon| < 1 )), MR becomes negative, indicating that selling an additional unit actually reduces total revenue.

Thus, when a monopolist increases sales by one unit, the sign and magnitude of MR depend critically on the elasticity at the current output level And that's really what it comes down to. That's the whole idea..

Mathematical Derivation: A Step‑by‑Step Example

Consider a linear demand curve:

[ P = a - bQ \quad (a, b > 0) ]

Step 1: Compute Total Revenue

[ TR = P \times Q = (a - bQ)Q = aQ - bQ^{2} ]

Step 2: Derive Marginal Revenue

[ MR = \frac{dTR}{dQ} = a - 2bQ ]

Step 3: Introduce Marginal Cost (MC)

Assume constant marginal cost ( MC = c ) (with ( c < a ) to ensure a positive output) Easy to understand, harder to ignore..

Step 4: Profit‑maximizing condition

Set ( MR = MC ):

[ a - 2bQ^{} = c \quad \Longrightarrow \quad Q^{} = \frac{a - c}{2b} ]

Step 5: Determine the price for the optimal quantity

[ P^{} = a - bQ^{} = a - b\left(\frac{a - c}{2b}\right) = \frac{a + c}{2} ]

Interpretation of a one‑unit increase

If the monopolist raises output from ( Q^{} ) to ( Q^{}+1 ):

• New price: ( P_{new} = a - b(Q^{}+1) = P^{} - b )
• New total revenue: ( TR_{new} = (P^{} - b)(Q^{}+1) )
• Incremental revenue:

[ \Delta TR = TR_{new} - TR^{} = MR(Q^{}) - b ]

Because ( MR(Q^{}) = c ) by construction, the extra unit adds ( c - b ) to revenue. If ( b > c ), the extra unit actually decreases profit, confirming why the monopoly stops at ( Q^{} ).

Implications for Pricing Strategy

1. Why monopoly price exceeds marginal cost

Since MR lies below the demand curve, the equality ( MR = MC ) occurs at a quantity where price is higher than marginal cost:

[ P^{*} = MC + \frac{1}{2} (a - MC) > MC ]

The monopolist exploits its market power by restricting output, ensuring that each additional unit sold would lower revenue more than the cost saved.

2. The “markup” formula

From the elasticity expression of MR, rearranging gives the classic Lerner Index:

[ \frac{P - MC}{P} = -\frac{1}{\varepsilon} ]

When a monopolist evaluates the effect of increasing sales by one unit, this markup tells us how far above marginal cost the price must sit given the current elasticity Worth keeping that in mind..

3. Price discrimination and unit‑by‑unit sales

If the firm can segment markets and charge each consumer their reservation price, the effective MR for each additional unit can be kept close to the price of that unit, eliminating the revenue loss from a uniform price cut. In such cases, the impact of a one‑unit increase resembles that of a competitive firm—marginal revenue equals price for that segment.

Real‑World Examples

These cases illustrate that the specific shape of the demand curve determines whether increasing sales by one unit is beneficial, neutral, or detrimental to the monopolist’s profit Worth keeping that in mind. Surprisingly effective..

Frequently Asked Questions

Q1: Does a monopolist always lose profit when it sells one more unit?