Revenue

This section focusses on revenue including the formulae to calculate and understand the relationship between total revenue and average revenue and marginal revenue. It also explains the price elasticity of demand (PED) and its relationship to revenue concepts and the calculation of PED and its effect on revenue. 

Formulae to Calculate and Understand the Relationship Between

Total Revenue (TR)

• Definition: Total revenue is the total amount of money a firm receives from selling its goods or services.

Formula:

Total Revenue (TR)=Price (P)×Quantity (Q

$$\text{Total Revenue (TR)} = \text{Price (P)} \times \text{Quantity (Q)}$$ 

Explanation: Total revenue is calculated by multiplying the price per unit by the quantity sold. It reflects the total income generated from sales.

Example: If a company sells 100 units of a product at £5 each, its total revenue is:

$$TR = 5 \times 100 = £500$$ 

Average Revenue (AR)

• Definition: Average revenue is the revenue generated per unit of output sold. For most businesses, average revenue is simply the price at which the good or service is sold.

Formula:

Average Revenue (AR)=Total Revenue (TR)Quantity (Q)

$$\text{Average Revenue (AR)} = \frac{\text{Total Revenue (TR)}}{\text{Quantity (Q)}}$$ 

Explanation: Average revenue gives the revenue per unit sold, and for firms with a constant price, it is equal to the price (P). In perfectly competitive markets, AR and price are the same.

Example: If the firm’s total revenue is £500 and it sells 100 units, its average revenue is:

$$AR = \frac{500}{100} = £5 \text{ per unit}$$ 

In perfect competition, the AR curve is a horizontal line at the price level.

Marginal Revenue (MR):

• Definition: Marginal revenue is the additional revenue gained from selling one more unit of a product.

Formula:

$$\text{Marginal Revenue (MR)} = \frac{\Delta \text{Total Revenue (TR)}}{\Delta \text{Quantity (Q)}}$$ 

Explanation: Marginal revenue is the change in total revenue resulting from a one-unit change in quantity sold. It is the slope of the total revenue curve.

Example: If the total revenue from selling 100 units is £500, and the total revenue from selling 101 units is £505, then:

$$MR = \frac{505 - 500}{101 - 100} = £5$$ 

In a monopoly, marginal revenue tends to decrease as more units are sold, because the price is often lowered to sell additional units.

Relationship Between TR, AR, and MR

TR and AR: Average revenue is calculated by dividing total revenue by quantity. In many cases, particularly in a perfectly competitive market, average revenue is equal to the price of the good. Therefore, TR = AR × Q.

AR and MR: The relationship between AR and MR can differ depending on the market structure:

• In a perfectly competitive market, MR equals AR because firms are price takers, and they sell each unit at the same price.
• In a monopoly or monopolistic competition, MR will be lower than AR because the firm must reduce its price to sell additional units.

Example: In a monopoly, the firm may have to lower its price from £5 to £4 to sell an additional unit. In this case, the AR might still be £5, but the MR will be less than £5 due to the price cut on all units sold.

Price Elasticity of Demand (PED) and Its Relationship to Revenue Concepts

Price Elasticity of Demand (PED) measures how sensitive the quantity demanded is to a change in price. It plays a crucial role in understanding how changes in price affect total revenue.

Formula for PED:

$$\text{Price Elasticity of Demand (PED)} = \frac{\%\ \text{Change in Quantity Demanded (Q)}}{\%\ \text{Change in Price (P)}}$$ 

Explanation: PED quantifies the responsiveness of consumers to a change in price. If PED > 1, demand is elastic (sensitive to price changes). If PED < 1, demand is inelastic (less sensitive to price changes). If PED = 1, demand is unitary elastic.

Relationship Between PED and Total Revenue:

• Elastic Demand (PED > 1): If demand is elastic, a decrease in price will lead to a proportionally larger increase in quantity demanded, which increases total revenue. Conversely, increasing the price will lead to a decrease in total revenue because the percentage decrease in quantity demanded is greater than the percentage increase in price.
• Inelastic Demand (PED < 1): If demand is inelastic, a decrease in price will lead to a proportionally smaller increase in quantity demanded, which decreases total revenue. Increasing the price will lead to an increase in total revenue because the percentage decrease in quantity demanded is smaller than the percentage increase in price.
• Unitary Elastic Demand (PED = 1): If demand is unitary elastic, a change in price will not affect total revenue. The percentage change in price is exactly offset by the percentage change in quantity demanded.

Key Points:

• Elastic Demand (PED > 1): Lowering prices increases total revenue.
• Inelastic Demand (PED < 1): Raising prices increases total revenue.
• Unitary Elastic Demand (PED = 1): Total revenue remains unchanged with price changes.

Calculation of PED and its Effect on Revenue

Let’s work through an example of calculating Price Elasticity of Demand and examining its impact on Total Revenue.

Example 1: Elastic Demand (PED > 1)

A firm lowers the price of its product from £10 to £8, resulting in an increase in quantity demanded from 50 units to 70 units.

• Change in Quantity: 70 − 50 = 20 units
• Change in Price: 10 − 8 = £2

Percentage Change in Quantity:

$$\frac{20}{50} \times 100 = 40\% \text{ increase in quantity demanded}$$ 

Percentage Change in Price:

$$\frac{2}{10} \times 100 = 20\% \text{ decrease in price}$$

PED:

$$PED = \frac{40\%}{20\%} = 2 \quad \text{(Elastic demand, PED > 1)}$$ 

Effect on Total Revenue

Initial Total Revenue:

$$TR_{\text{initial}} = 10 \times 50 = £500$$ 

New Total Revenue:

$$TR_{\text{new}} = 8 \times 70 = £560$$ 

Conclusion: Since demand is elastic (PED > 1), the decrease in price leads to a more than proportionate increase in quantity demanded, which increases total revenue from £500 to £560.

Example 2: Inelastic Demand (PED < 1)

A firm increases the price of its product from £10 to £12, and quantity demanded decreases from 100 units to 90 units.

• Change in Quantity: 90 − 100 = −10 units
• Change in Price: 12 − 10 = £2

Percentage Change in Quantity:

$$\frac{-10}{100} \times 100 = -10\% \text{ decrease in quantity demanded}$$

Percentage Change in Price:

$$\frac{2}{10} \times 100 = 20\% \text{ increase in price}$$

PED:

$$PED = \frac{-10\%}{20\%} = -0.5 \quad \text{(Inelastic demand, PED < 1)}$$ 

Effect on Total Revenue

Initial Total Revenue:

$$TR_{\text{initial}} = 10 \times 100 = £1000$$ 

New Total Revenue:

$$TR_{\text{new}} = 12 \times 90 = £1080$$ 

Conclusion: Since demand is inelastic (PED < 1), the increase in price leads to a smaller decrease in quantity demanded, which increases total revenue from £1000 to £1080.

Summary

• Total Revenue (TR) is calculated as Price × Quantity.
• Average Revenue (AR) is calculated as Total Revenue ÷ Quantity, and for most firms, AR is equal to the price of the good.
• Marginal Revenue (MR) is the additional revenue from selling one more unit, and it can be calculated as the change in total revenue divided by the change in quantity.

Price Elasticity of Demand (PED) measures how quantity demanded responds to price changes and is crucial in determining how price changes affect total revenue:

• Elastic demand (PED > 1): Lower prices increase total revenue.
• Inelastic demand (PED < 1): Higher prices increase total revenue.
• Unitary elastic demand (PED = 1): Price changes have no effect on total revenue.