All Topics
economics-9708 | as-a-level
Your Flashcards are Ready!
15 Flashcards in this deck.
1.
The price system and the microeconomy
2.
The Macroeconomy
3.
International economic issues
4.
The macroeconomy
5.
The price system and the microeconomy
6.
Government macroeconomic intervention
7.
Basic economic ideas and resource allocation
8.
Government microeconomy intervention
9.
International economic issues
10.
Government microeconomic intervention
11.
Government macroeconomic intervention
Formulae and calculation of PED, YED, XED
Topic 2/3
or
3
Still Learning
I know
12
Previous
Next
Formulae and Calculation of PED, YED, XED
Introduction
Price elasticity of demand (PED), income elasticity of demand (YED), and cross elasticity of demand (XED) are fundamental concepts in microeconomics. Understanding these elasticity measures is crucial for analyzing how different factors influence consumer behavior and market dynamics. This article delves into the formulae and calculations of PED, YED, and XED, providing AS & A Level Economics students with comprehensive insights essential for mastering the subject.
Key Concepts
Price Elasticity of Demand (PED)
**Definition:** Price Elasticity of Demand (PED) measures the responsiveness of the quantity demanded of a good to a change in its price. It indicates how sensitive consumers are to price changes.
**Formula:**
$$
PED = \frac{\% \text{ Change in Quantity Demanded}}{\% \text{ Change in Price}}
$$
**Calculation:**
To calculate PED, use the following steps:
1. **Determine the Change in Quantity Demanded (\(\Delta Q\)):**
\[
\Delta Q = Q_2 - Q_1
\]
2. **Determine the Change in Price (\(\Delta P\)):**
\[
\Delta P = P_2 - P_1
\]
3. **Calculate the Percentage Changes:**
\[
\% \Delta Q = \left( \frac{\Delta Q}{Q_1} \right) \times 100
\]
\[
\% \Delta P = \left( \frac{\Delta P}{P_1} \right) \times 100
\]
4. **Apply the PED Formula:**
\[
PED = \frac{\% \Delta Q}{\% \Delta P}
\]
**Example:**
Suppose the price of coffee increases from $2 to $2.20 per cup, and the quantity demanded decreases from 1000 cups to 900 cups.
\[
\Delta P = 2.20 - 2.00 = 0.20 \quad (\text{\$})
\]
\[
\Delta Q = 900 - 1000 = -100 \quad (\text{cups})
\]
\[
\% \Delta P = \left( \frac{0.20}{2.00} \right) \times 100 = 10\%
\]
\[
\% \Delta Q = \left( \frac{-100}{1000} \right) \times 100 = -10\%
\]
\[
PED = \frac{-10\%}{10\%} = -1
\]
A PED of -1 indicates unitary elasticity, meaning the percentage change in quantity demanded is equal to the percentage change in price.
Income Elasticity of Demand (YED)
**Definition:** Income Elasticity of Demand (YED) measures the responsiveness of the quantity demanded of a good to a change in consumers' income. It indicates how demand for a good changes as income levels fluctuate.
**Formula:**
$$
YED = \frac{\% \text{ Change in Quantity Demanded}}{\% \text{ Change in Income}}
$$
**Calculation:**
To calculate YED, follow these steps:
1. **Determine the Change in Quantity Demanded (\(\Delta Q\)):**
\[
\Delta Q = Q_2 - Q_1
\]
2. **Determine the Change in Income (\(\Delta I\)):**
\[
\Delta I = I_2 - I_1
\]
3. **Calculate the Percentage Changes:**
\[
\% \Delta Q = \left( \frac{\Delta Q}{Q_1} \right) \times 100
\]
\[
\% \Delta I = \left( \frac{\Delta I}{I_1} \right) \times 100
\]
4. **Apply the YED Formula:**
\[
YED = \frac{\% \Delta Q}{\% \Delta I}
\]
**Example:**
If consumers' income increases from \$50,000 to \$55,000, and the quantity demanded for organic vegetables rises from 200 units to 220 units, then:
\[
\Delta I = 55,000 - 50,000 = 5,000 \quad (\text{\$})
\]
\[
\Delta Q = 220 - 200 = 20 \quad (\text{units})
\]
\[
\% \Delta I = \left( \frac{5,000}{50,000} \right) \times 100 = 10\%
\]
\[
\% \Delta Q = \left( \frac{20}{200} \right) \times 100 = 10\%
\]
\[
YED = \frac{10\%}{10\%} = 1
\]
A YED of 1 signifies unitary income elasticity, indicating that the quantity demanded changes by the same percentage as income.
Cross Elasticity of Demand (XED)
**Definition:** Cross Elasticity of Demand (XED) measures the responsiveness of the quantity demanded of one good to a change in the price of another good. It helps in understanding the relationship between two goods, whether they are substitutes or complements.
**Formula:**
$$
XED = \frac{\% \text{ Change in Quantity Demanded of Good A}}{\% \text{ Change in Price of Good B}}
$$
**Calculation:**
To calculate XED, use the following approach:
1. **Determine the Change in Quantity Demanded of Good A (\(\Delta Q_A\)):**
\[
\Delta Q_A = Q_{A2} - Q_{A1}
\]
2. **Determine the Change in Price of Good B (\(\Delta P_B\)):**
\[
\Delta P_B = P_{B2} - P_{B1}
\]
3. **Calculate the Percentage Changes:**
\[
\% \Delta Q_A = \left( \frac{\Delta Q_A}{Q_{A1}} \right) \times 100
\]
\[
\% \Delta P_B = \left( \frac{\Delta P_B}{P_{B1}} \right) \times 100
\]
4. **Apply the XED Formula:**
\[
XED = \frac{\% \Delta Q_A}{\% \Delta P_B}
\]
**Example:**
Consider two goods, butter and margarine. If the price of butter increases from \$3 to \$3.30 per loaf, and the quantity demanded for margarine increases from 500 units to 550 units:
\[
\Delta P_B = 3.30 - 3.00 = 0.30 \quad (\text{\$})
\]
\[
\Delta Q_A = 550 - 500 = 50 \quad (\text{units})
\]
\[
\% \Delta P_B = \left( \frac{0.30}{3.00} \right) \times 100 = 10\%
\]
\[
\% \Delta Q_A = \left( \frac{50}{500} \right) \times 100 = 10\%
\]
\[
XED = \frac{10\%}{10\%} = 1
\]
An XED of 1 indicates that the goods are substitutes; as the price of butter rises, consumers increase their demand for margarine.
Types of Elasticities
Understanding the different types of elasticities helps in categorizing goods based on their responsiveness to price and income changes.
1. **Elastic Demand:** PED > 1. Quantity demanded changes by a greater percentage than the price change.
2. **Inelastic Demand:** PED < 1. Quantity demanded changes by a smaller percentage than the price change.
3. **Unitary Elasticity:** PED = 1. Quantity demanded changes by the same percentage as the price change.
4. **Superior Goods:** YED > 1. Demand increases more than proportionally as income rises.
5. **Inferior Goods:** YED < 0. Demand decreases as income rises.
6. **Substitutes:** XED > 0. An increase in the price of one good leads to an increase in demand for another good.
7. **Complements:** XED < 0. An increase in the price of one good leads to a decrease in demand for another good.
Determinants of Elasticity
Several factors influence the elasticity of demand:
1. **Availability of Substitutes:** More substitutes lead to higher elasticity.
2. **Necessity vs. Luxury:** Necessities tend to have inelastic demand, while luxuries have elastic demand.
3. **Proportion of Income:** Goods that consume a larger portion of income tend to have more elastic demand.
4. **Time Period:** Demand elasticity typically increases over time as consumers can adjust their behavior.
5. **Definition of the Market:** Narrowly defined markets usually have more elastic demand compared to broadly defined markets.
Applications of Elasticity
Elasticity measures are vital for various economic decisions:
1. **Pricing Strategies:** Firms use PED to set optimal pricing to maximize revenue.
2. **Taxation Policies:** Governments assess elasticity to predict the impact of taxes on different goods.
3. **Subsidy Allocation:** Understanding YED helps in targeting subsidies to essential or inferior goods.
4. **Market Forecasting:** XED aids businesses in anticipating changes in demand due to price variations in related goods.
Advanced Concepts
Mathematical Derivation of PED
The PED can be derived using calculus to provide a more precise measurement, especially when analyzing continuous changes in price and quantity.
**Total Differentiation Approach:**
Starting from the demand function:
\[
Q = f(P)
\]
The PED is the ratio of the percentage change in quantity to the percentage change in price:
\[
PED = \frac{\frac{dQ}{Q}}{\frac{dP}{P}} = \frac{dQ/dP}{Q/P} = \frac{P}{Q} \cdot \frac{dQ}{dP}
\]
Where:
- \(\frac{dQ}{dP}\) is the derivative of the demand function with respect to price.
- \(P\) is the price level.
- \(Q\) is the quantity demanded.
**Example:**
Suppose the demand function is:
\[
Q = 100 - 2P
\]
Differentiating Q with respect to P:
\[
\frac{dQ}{dP} = -2
\]
At a price \(P = 20\):
\[
Q = 100 - 2(20) = 60
\]
Thus,
\[
PED = \frac{20}{60} \times (-2) = -\frac{40}{60} = -\frac{2}{3} \approx -0.67
\]
This indicates inelastic demand at \(P = 20\).
Income Elasticity and Normal vs. Inferior Goods
**Normal Goods:** These are goods for which demand increases as income increases (\(YED > 0\)).
- **Luxury Goods:** A subset of normal goods with \(YED > 1\).
- **Necessities:** Normal goods with \(0 < YED < 1\).
**Inferior Goods:** Goods for which demand decreases as income increases (\(YED < 0\)).
**Mathematical Representation:**
If \(Q = f(I)\), then:
\[
YED = \frac{I}{Q} \cdot \frac{dQ}{dI}
\]
**Implications:**
- **Luxury Goods:** Proportionately increase with income. Examples include high-end electronics, designer clothing.
- **Necessities:** Less sensitive to income changes. Examples include basic food items, utilities.
- **Inferior Goods:** Consumers reduce consumption as income rises. Examples include generic brands, used cars.
Cross Elasticity and Market Relationships
**Substitutes vs. Complements:**
- **Substitutes (\(XED > 0\)):** Goods that can replace each other. An increase in the price of one leads to an increase in demand for the other.
**Example:** Tea and coffee. If the price of tea rises, consumers may buy more coffee.
- **Complements (\(XED < 0\)):** Goods that are consumed together. An increase in the price of one leads to a decrease in demand for the other.
**Example:** Printers and ink cartridges. If the price of printers increases, the demand for ink cartridges may decrease.
**Unrelated Goods (\(XED = 0\)):**
No relationship exists between the two goods.
**Example:** Bread and bicycles. A change in the price of bread does not affect the demand for bicycles.
Elasticity and Total Revenue
**Understanding Total Revenue (TR):**
\[
TR = P \times Q
\]
The relationship between PED and total revenue is crucial for businesses:
1. **Elastic Demand (\(PED < -1\)):**
- Decrease in price leads to an increase in total revenue.
- Increase in price leads to a decrease in total revenue.
2. **Inelastic Demand (\(-1 < PED < 0\)):**
- Decrease in price leads to a decrease in total revenue.
- Increase in price leads to an increase in total revenue.
3. **Unitary Elasticity (\(PED = -1\)):**
- Changes in price do not affect total revenue.
**Graphical Representation:**
A demand curve with different elasticity segments can illustrate how total revenue changes with price movements.
**Example:**
Using the PED calculated earlier (\(PED = -0.67\)):
- Since \(|PED| < 1\), demand is inelastic.
- Increasing the price of the good will increase total revenue.
- Decreasing the price will decrease total revenue.
Interdisciplinary Connections
Elasticity concepts extend beyond economics into various fields:
1. **Marketing:** Understanding PED helps in pricing strategies and demand forecasting.
2. **Public Policy:** YED informs welfare policies and taxation on different income groups.
3. **Business Strategy:** XED aids in product positioning and competitive analysis.
4. **Environmental Economics:** Elasticity measures are used to assess the impact of environmental policies on resource consumption.
Complex Problem-Solving: Multi-Step Elasticity Calculation
**Problem:**
A smartphone manufacturer wants to analyze the impact of a price change on its sales and revenue. Currently, the price of the smartphone is \$500, and 10,000 units are sold per month. The company plans to decrease the price to \$450, anticipating that the quantity sold will increase to 12,000 units. Calculate the PED and determine the effect on total revenue.
**Solution:**
1. **Calculate \(\Delta P\) and \(\Delta Q\):**
\[
\Delta P = 450 - 500 = -50 \quad (\text{\$})
\]
\[
\Delta Q = 12,000 - 10,000 = 2,000 \quad (\text{units})
\]
2. **Calculate \(\% \Delta P\) and \(\% \Delta Q\):**
\[
\% \Delta P = \left( \frac{-50}{500} \right) \times 100 = -10\%
\]
\[
\% \Delta Q = \left( \frac{2,000}{10,000} \right) \times 100 = 20\%
\]
3. **Calculate PED:**
\[
PED = \frac{20\%}{-10\%} = -2
\]
4. **Interpretation:**
- PED = -2 indicates elastic demand.
- Since demand is elastic, decreasing the price will increase total revenue.
5. **Total Revenue Before and After:**
\[
TR_{\text{before}} = 500 \times 10,000 = 5,000,000 \quad (\text{\$})
\]
\[
TR_{\text{after}} = 450 \times 12,000 = 5,400,000 \quad (\text{\$})
\]
6. **Conclusion:**
- Total revenue increases from \$5,000,000 to \$5,400,000 due to the elastic nature of demand.
Comparison Table
| Elasticity Measure | Definition | Formula | Sign Indicator | Type of Goods |
|---|---|---|---|---|
| Price Elasticity of Demand (PED) | Responsiveness of quantity demanded to a change in price | \( \frac{\% \Delta Q}{\% \Delta P} \) | Negative (Law of Demand) | All goods |
| Income Elasticity of Demand (YED) | Responsiveness of quantity demanded to a change in income | \( \frac{\% \Delta Q}{\% \Delta I} \) | Positive for normal goods, Negative for inferior goods | Normal and inferior goods |
| Cross Elasticity of Demand (XED) | Responsiveness of quantity demanded of one good to a change in price of another good | \( \frac{\% \Delta Q_A}{\% \Delta P_B} \) | Positive for substitutes, Negative for complements | Substitutes and complements |
Summary and Key Takeaways
- PED, YED, and XED are essential measures of demand responsiveness.
- PED assesses how quantity demanded reacts to price changes, influencing pricing strategies.
- YED distinguishes between normal and inferior goods based on income fluctuations.
- XED identifies relationships between goods as substitutes or complements.
- Understanding elasticity aids in making informed economic and business decisions.
Coming Soon!
Examiner Tip
Tips
To remember the types of elasticities, use the mnemonic "PEY Crosses": Price (PED) usually Negative, Income (YED) Positive or Negative, Cross (XED) varies. Always double-check whether the good is a necessity or a luxury to determine YED. Practice with real-world examples to better understand XED relationships between products. For exams, clearly label graphs and include units in your calculations to avoid common pitfalls.
Did You Know
Did You Know
Did you know that the concept of elasticity was first introduced by the renowned economist Alfred Marshall in the late 19th century? Additionally, during the 1970s oil crisis, understanding price elasticity of demand (PED) became crucial for governments to implement effective pricing strategies. Another intriguing fact is that income elasticity of demand (YED) varies significantly across different cultures and economic systems, influencing global market trends.
Common Mistakes
Common Mistakes
Students often confuse the signs of PED, YED, and XED. For instance, they might assume all elasticity values are positive, neglecting that PED is typically negative due to the law of demand. Another common error is miscalculating percentage changes by forgetting to use the initial values as the base. Additionally, applying elasticity concepts to unrelated goods can lead to incorrect classifications of substitutes and complements.
FAQ
What does a PED of -1 signify?
A PED of -1 indicates unitary elasticity, meaning the percentage change in quantity demanded is equal to the percentage change in price.
How is YED used to classify goods?
YED distinguishes goods as normal (YED > 0) or inferior (YED < 0). Further, normal goods can be necessities (0 < YED < 1) or luxuries (YED > 1).
Can XED be zero?
Yes, an XED of zero means there is no relationship between the two goods; a price change in one does not affect the demand for the other.
Why is PED usually negative?
PED is typically negative due to the inverse relationship between price and quantity demanded, as stated by the law of demand.
How does time affect demand elasticity?
Demand elasticity generally increases over time because consumers have more time to adjust their behavior in response to price changes.
1.
The price system and the microeconomy
2.
The Macroeconomy
3.
International economic issues
4.
The macroeconomy
5.
The price system and the microeconomy
6.
Government macroeconomic intervention
7.
Basic economic ideas and resource allocation
8.
Government microeconomy intervention
9.
International economic issues
10.
Government microeconomic intervention
11.
Government macroeconomic intervention
Get PDF
How would you like to practise?
