10.1 - Revenue - notes
10.1 - Revenue
Definitions of total, average, and marginal revenue
A firm's revenue represents the income generated from selling its goods or services. It is influenced by the quantity sold and the price achieved, which in turn depends on market demand.
Total revenue (TR)
Total revenue is the overall income received from sales over a specific period. It is also known as turnover.
Where:
- TR = Total revenue (£)
- Q = Quantity sold (units)
- P = Price per unit (£)
Average revenue (AR)
Average revenue is the income per unit sold, which equals the price.
Where:
- AR = Average revenue (£ per unit)
- TR = Total revenue (£)
- Q = Quantity sold (units)
Marginal revenue (MR)
Marginal revenue is the additional income from selling one more unit.
Where:
- MR = Marginal revenue (£)
- TRn = Total revenue at new sales level (£)
- TRn-1 = Total revenue at one unit less (£)
Revenue when price is constant
When price remains the same regardless of quantity sold, marginal revenue stays constant.
| Quantity sold | Price (£) | Total revenue (£) | Marginal revenue (£) |
|---|---|---|---|
| 0 | 8 | 0 | - |
| 1 | 8 | 8 | 8 |
| 2 | 8 | 16 | 8 |
| 3 | 8 | 24 | 8 |
| 4 | 8 | 32 | 8 |
Revenue when price changes to increase sales
When price must decrease to sell more, marginal revenue varies with quantity.
| Quantity sold | Price (£) | Total revenue (£) | Marginal revenue (£) |
|---|---|---|---|
| 0 | 300 | 0 | - |
| 1 | 250 | 250 | 250 |
| 2 | 220 | 440 | 190 |
| 3 | 200 | 600 | 160 |
| 4 | 180 | 720 | 120 |
| 5 | 160 | 800 | 80 |
| 6 | 140 | 840 | 40 |
How demand curves influence revenue
Demand curves illustrate the quantity a firm can sell at different prices, directly affecting revenue calculations:
- The demand curve shows the relationship between price and quantity demanded.
- Since price equals average revenue, the demand curve also represents the average revenue curve (labelled as AR).
- Total revenue at a given price is the area under the demand curve, calculated as quantity multiplied by price.
The diagram below shows this: at price P1 the firm sells quantity Q1, so total revenue is the shaded rectangle under the demand curve D (or AR), equal to Q1 multiplied by P1.
Revenue characteristics for price takers
Price takers cannot influence market price and must accept the prevailing rate. They operate in highly competitive markets.
Features of a price taker's demand curve
- The demand curve (D) is horizontal (perfectly elastic) at the market price P, meaning any price increase results in zero sales, while lowering price is unnecessary as all output sells at the market rate.
- Average revenue equals marginal revenue, so the same horizontal line is the D = AR = MR curve, as each additional unit sells at the same price.
- Total revenue (TR) rises linearly with quantity, forming an upward-sloping straight line from the origin.
The first diagram below shows the horizontal D = AR = MR curve at price P.
The second shows total revenue TR rising in a straight line from the origin.
Revenue characteristics for price makers
Price makers, such as those in monopolistic markets, can influence prices. They face less competition and can adjust prices to affect sales.
Features of a price maker's demand curve
- The demand curve slopes downwards, requiring price reductions to increase sales volume.
- If the demand curve is a straight line, price elasticity of demand (PED) varies along it:
- Elastic (PED < -1) in the upper section, where price cuts boost revenue significantly.
- Unit elastic (PED = -1) at the midpoint.
- Inelastic (PED > -1) in the lower section, where price cuts reduce revenue.
The diagram below shows this straight-line demand curve D. Demand is Elastic (PED < -1) in the upper section, Unit elastic (PED = -1) at the midpoint (price P, quantity Q), and Inelastic (PED > -1) in the lower section.
Maximising total revenue and the role of marginal revenue
Total revenue reaches its peak under specific conditions related to elasticity and marginal revenue, guiding firms on optimal output levels.
Conditions for maximising total revenue
- For a downward-sloping, straight-line demand curve, total revenue is maximised at the midpoint where PED = -1 (unit elasticity).
- At this point, further price reductions lead to disproportionate sales increases (elastic) or decreases (inelastic), but the maximum occurs at unit elasticity.
- The marginal revenue curve is twice as steep as the average revenue curve and crosses zero where total revenue peaks.
- When marginal revenue is zero, additional units do not increase total revenue; beyond this, marginal revenue becomes negative, reducing total revenue.
The diagrams below show this. The first plots total revenue (TR), which peaks at the output Q1 where PED = -1.
The second plots average revenue (AR) and marginal revenue (MR): MR is twice as steep as AR and crosses zero at the same output Q1, exactly where TR is at its maximum.
Worked example - Calculating total, average, and marginal revenue
A firm sells widgets. At 6 units, total revenue is £900. At 7 units, total revenue rises to £1015. Calculate the marginal revenue for the 7th unit, the average revenue at 7 units, and confirm the total revenue formula.
Step 1: Identify the values
- TR at 6 units = £900
- TR at 7 units = £1015
- Quantity at new level = 7 units
Step 2: Calculate marginal revenue
Step 3: Calculate average revenue