## **Learning Outcomes**

After reading this article, you will be able to explain the relationship between price and demand, calculate and interpret price elasticity, and use demand equations to determine the optimum selling price and output for a product. You will develop the ability to apply demand-based pricing models—critical for profit maximisation—in quantitative ACCA Performance Management exam scenarios.

## **ACCA Performance Management (PM) Syllabus**

For ACCA Performance Management (PM), you are required to understand demand-driven approaches to pricing and how price–volume relationships inform optimum pricing. Specifically, you must be able to:

- Describe the factors influencing product or service pricing
- Calculate and interpret price elasticity of demand
- Derive and use linear demand equations for pricing decisions
- Calculate the optimum selling price and output by equating marginal cost and marginal revenue
- Use algebraic and tabular methods to determine profit-maximising price–volume combinations
- Evaluate pricing strategies in various market conditions

## **Test Your Knowledge**

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision._

1. What is the formula for the price elasticity of demand, and what does a value above 1 mean for a product?
2. If demand at \$40 per unit is 800 units and at \$36 per unit is 1,000 units, derive the straight-line demand equation relating price (P) and quantity (Q).
3. Given a demand equation of $P = 100 - 0.1Q$ and a marginal cost of \$30, how do you find the profit-maximising quantity and price?
4. Explain the significance of the marginal revenue curve in determining optimal pricing for a monopoly product.

## **Introduction**

Pricing is a critical decision area for every business. The price set for a product not only affects the volume sold but directly impacts profitability. In the ACCA Performance Management exam, you must be able to analyse how demand responds to changes in price, quantify this relationship, and apply mathematical models to find the price and output that maximises profit.

Economists have established that there is usually an inverse relationship between the selling price of a product and the quantity demanded. This relationship is linear in many exam questions and can be captured mathematically, allowing you to apply systematic calculations for optimal pricing.

> **Key Term: demand curve**
> A mathematical relationship showing how the quantity demanded of a product varies with its selling price, typically represented as a straight line.

## **THE PRICE–VOLUME DEMAND EQUATION**

Firms facing imperfect competition (most real-world businesses) cannot simply set any price and sell any volume they wish. Instead, reducing price generally increases demand and raising price reduces it.

The relationship can be modelled as a straight-line equation:

$$
P = a - bQ
$$

Where:

- $P$ is the selling price per unit,
- $Q$ is the quantity demanded,
- $a$ is the price at which demand would fall to zero (the intercept),
- $b$ is the slope of the curve (the change in price per unit increase in quantity).

> **Key Term: marginal revenue (MR)**
> The extra revenue from selling one additional unit. For a linear demand curve:
>
> $$
> P = a - bQ
> $$
>
> the marginal revenue is:
>
> $$
> MR = a - 2bQ
> $$

## **PRICE ELASTICITY OF DEMAND**

Price elasticity measures the sensitivity of quantity demanded to a change in price. This is a key consideration for managers, as it affects both revenue and pricing decisions.

> **Key Term: price elasticity of demand**
> The percentage change in quantity demanded divided by the percentage change in price; a measure of how responsive demand is to price changes.

If elasticity is greater than 1 (elastic), a small price cut leads to a large increase in demand—total revenue rises when price is reduced. If elasticity is less than 1 (inelastic), a price cut increases demand only slightly—total revenue falls.

### Calculating Price Elasticity

$$
\text{Price elasticity of demand} = \frac{\%\ \text{change in quantity demanded}}{\%\ \text{change in price}}
$$

A negative sign is normal (price and demand move in opposite directions), but the sign is often ignored in interpretation.

## **OPTIMUM PRICING: MR, MC AND PROFIT MAXIMISATION**

Profit maximisation occurs where marginal revenue equals marginal cost (MR = MC). At this point, producing and selling one extra unit neither adds nor subtracts from profit; to produce more would decrease profit, less would also decrease profit.

The process:

1. Find the demand equation ($P = a - bQ$) from two price–quantity pairs.
2. Calculate the marginal revenue equation ($MR = a - 2bQ$).
3. Set $MR = MC$ (where $MC$ is marginal cost per unit) and solve for the optimal $Q$.
4. Substitute $Q$ into the demand equation to get the profit-maximising price.
5. Calculate the resulting contribution/profit as needed.

### **Worked Example 1.1**

A company finds that it can sell 1,200 units per month if the price is \$60, but only 800 units per month if the price is \$70. Marginal cost per unit is \$30. What is the profit-maximising price and quantity?

> **Answer:**
>
> 1. Calculate the slope:
>    $$
>    b = (70 - 60)/(800 - 1200) = 10 / (-400) = -0.025
>    $$
> 2. The demand equation is:
>    $$
>    P = a - bQ
>    $$
>    Use one data point to solve for a:
>    \$60 = a - (-0.025 \times 1,200)
>    \$60 = a + 30
>    $$
>    a = 60 - 30 = 30
>    $$
>    Correction: This cannot be right—re-express: \$60 = a - (-0.025) \times 1,200 = a + 30
>    $$
>    a = 60 - 30 = 30
>    $$
>    (but that gives zero demand at P = 30, which can't be right).
>    Let's check carefully:
>    Step 1:
>    $$
>    b = (70 - 60)/(800 - 1,200) = 10 / (-400) = -0.025
>    $$
>    (correct).
>    Step 2:
>    $$
>    P = a - bQ
>    $$
>    using (Q, P) = (1,200, \$60):
>    \$60 = a - (-0.025 \times 1,200)
>    \$60 = a + 30
>    $$
>    a = 60 - 30 = 30
>    $$
>    But clearly, that is problematic; the intercept a is usually higher than the observed prices. Let's use the (800, \$70) point:
>    \$70 = a - (-0.025 × 800) = a + 20
>    $$
>    a = 70 - 20 = 50
>    $$
>    The two results are different—this suggests:
>    $$
>    b = (60-70)/(1,200-800) = (-10)/400 = -0.025
>    $$
>    but:
>    $$
>    P = a - bQ
>    $$
>    Let’s sum up:
>    $$
>    b = (70-60)/(800-1,200) = 10/(-400) = -0.025
>    $$
>    Using (1,200, \$60):
>    \$60 = a - (-0.025 \times 1,200) \Rightarrow a = 60 - 30 = 30
>    Using (800, \$70):
>    \$70 = a - (-0.025 \times 800) \Rightarrow a = 70 - 20 = 50
>    There's a mistake:
>    $$
>    P = a - bQ
>    $$
>    If:
>    $$
>    b = (70 - 60)/(800 - 1,200) = 10/(-400) = -0.025
>    $$
>    Let's clarify. Use the standard method:
>    $$
>    b = (P_2 - P_1)/(Q_2 - Q_1)
>    $$
>    $$
>    P_1 = 70, Q_1 = 800
>    $$
>    $$
>    P_2 = 60, Q_2 = 1,200
>    $$
>    $$
>    b = (60 - 70) / (1,200 - 800) = -10/400 = -0.025
>    $$
>    Use:
>    $$
>    P = a - bQ
>    $$
>    \$60 = a - (-0.025 \times 1,200)
>    \$60 = a + 30
>    $$
>    a = 60 - 30 = 30
>    $$
>    So, equation:
>    $$
>    P = 30 - (-0.025)Q = 30 + 0.025Q
>    $$
>    But this implies price rises as demand rises, which is not logical.
>    Therefore, the correct approach is:
>    $$
>    b = (P_2 - P_1)/(Q_2 - Q_1) = (-10)/(400) = -0.025
>    $$
>    So:
>    $$
>    P = a - 0.025Q
>    $$
>    Use (Q, P) = (800, 70):
>    \$70 = a - 0.025 \times 800 = a - 20
>    $$
>    a = 70 + 20 = 90
>    $$
>    Now:
>    $$
>    P = 90 - 0.025Q
>    $$
>    $$
>    MR = a - 2bQ = 90 - 0.05Q
>    $$
>    Set:
>    $$
>    MR = MC
>    $$
>    \$90 - 0.05Q = 30 \rightarrow 0.05Q = 60 \rightarrow Q = 1,200
>    Plug back into demand equation for price:
>    $$
>    P = 90 - 0.025 \times 1,200 = 90 - 30 = 60
>    $$
>    So, maximum profit is at quantity 1,200 and price \$60.
>    Always double-check your calculations and substitute points into your derived equation.
>
> **Exam Warning**
> Always check your demand equation by substituting both sets of data points. Many students lose marks through calculation errors in b or a.

## **THE TABULAR APPROACH TO PRICING**

Sometimes, demand and cost data are provided in table form, not as equations. In these situations, you should calculate profit at each output level and select the price–quantity combination yielding the highest profit.

### **Worked Example 1.2**

A firm has the following estimated sales and cost data for its new product:

| Selling Price | Units Sold | Total Cost |
| ------------- | ---------- | ---------- |
| \$50          | 5,000      | \$180,000  |
| \$45          | 7,000      | \$210,000  |
| \$40          | 9,000      | \$250,000  |

Which price maximises profit?

> **Answer:**  
>
> For each option, calculate total profit:
>
> - \$50: Revenue \$250,000 (5,000×\$50), Profit \$70,000 (\$250,000–\$180,000)
> - \$45: Revenue \$315,000, Profit \$105,000 (\$315,000–\$210,000)
> - \$40: Revenue \$360,000, Profit \$110,000 (\$360,000–\$250,000)
>   \$40 maximises profit, yielding \$110,000.

## **SUMMARY: DEMAND-BASED PRICING AND EXAM FOCUS**

Understanding how price and quantity interact via the demand curve and price elasticity is essential for ACCA students. The ACCA PM exam expects you to fluently derive the demand equation, compute elasticity, assess the effect of price changes on revenue, and confidently find the optimal profit-maximising price and output using both algebraic and tabular methods.

### **Revision Tip**

Carefully lay out all working when deriving demand and MR functions—show all steps to minimise errors and gain follow-through marks, even if your initial answer is incorrect.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Explain the inverse relationship between price and demand using linear demand equations
- Define and calculate price elasticity of demand and explain its significance
- Apply the marginal revenue and marginal cost method for profit-maximising price and output
- Use both algebraic and tabular methods to select optimal pricing and output
- Interpret demand and revenue models in ACCA PM assessment scenarios

## **Key Terms and Concepts**

- demand curve
- marginal revenue (MR)
- price elasticity of demand


Pricing decisions - Demand, elasticity, and price–volume relationships