IB Economics - Linear Demand Functions

This study note for IB economics covers linear demand functions.

A linear demand function is a mathematical representation of the relationship between the quantity demanded (Qd) and the price (P) of a good, expressed as a straight line. The general form is Qd = a - bP.

Components:

• Qd: Quantity demanded.
• P: Price of the good.
• a: The intercept, representing the quantity demanded when the price is zero.
• b: The slope, representing the rate at which quantity demanded changes with respect to price.

Plotting a Demand Curve from a Linear Function

Example Function:

• Qd = 60 - 5P

Steps to Plot:

1. Identify the Intercept (a): In this example, a = 60. This is the quantity demanded when P = 0.
2. Identify the Slope (b): Here, b = 5. The slope indicates that for every unit increase in price, quantity demanded decreases by 5 units.
3. Create a Demand Schedule:
   • At P = 0, Qd = 60 - 5(0) = 60
   • At P = 5, Qd = 60 - 5(5) = 35
   • At P = 10, Qd = 60 - 5(10) = 10
   • At P = 12, Qd = 60 - 5(12) = 0

Plotting the Curve:

• On a graph, plot the points from the demand schedule and draw a line through them to illustrate the demand curve.

Identifying the Slope of the Demand Curve

• The slope of the demand curve is the coefficient of P in the linear demand function, which is -b.
• In Qd = 60 - 5P, the slope is -5. This indicates the steepness and direction of the curve.

Effects of Changes in "a" and "b" on the Demand Curve

Change in "a":

• Shift in the Demand Curve: If "a" changes, it shifts the entire demand curve.
  • Increase in "a": Shifts the demand curve to the right, indicating higher quantity demanded at each price level.
  • Decrease in "a": Shifts the demand curve to the left, indicating lower quantity demanded at each price level.

Change in "b":

• Change in Steepness: "b" affects the steepness of the demand curve.
  • Increase in "b": Makes the demand curve steeper, indicating a more significant change in quantity demanded with price changes.
  • Decrease in "b": Makes the demand curve flatter, indicating a less significant change in quantity demanded with price changes.

Real-World Examples

• Concert Tickets: If a concert's popularity increases (higher "a"), the demand for tickets at each price increases, shifting the demand curve to the right.
• Luxury Goods: For luxury goods, a higher price sensitivity (higher "b") results in a steeper demand curve as small price changes significantly impact quantity demanded.

Glossary of Key Terms

• Intercept (a): The quantity demanded when the price is zero.
• Linear Demand Function: A mathematical equation representing a straight-line relationship between price and quantity demanded.
• Quantity Demanded (Qd): The amount of a good consumers are willing and able to purchase at a given price.
• Slope (b): The rate at which quantity demanded changes with respect to price.
• Steepness: The angle of the demand curve, affected by the slope coefficient "b."

Related Topics for Further Exploration

1. Price Elasticity of Demand: How sensitive the quantity demanded is to price changes.
2. Income Elasticity of Demand: How changes in consumer income affect the quantity demanded.
3. Cross-Price Elasticity of Demand: How the price of one good affects the demand for another.
4. Consumer Surplus: The difference between what consumers are willing to pay and what they actually pay.
5. Market Equilibrium: How demand and supply interact to determine prices and quantities.

Worked Examples

Example 1:

• Qd = 100 - 10P
  • At P = 0, Qd = 100 - 10(0) = 100
  • At P = 5, Qd = 100 - 10(5) = 50
  • At P = 10, Qd = 100 - 10(10) = 0

Example 2:

• Qd = 80 - 2P
  • At P = 0, Qd = 80 - 2(0) = 80
  • At P = 20, Qd = 80 - 2(20) = 40
  • At P = 40, Qd = 80 - 2(40) = 0

Possible IB Economics Essay-Style Questions

1. Explain how to derive a demand curve from a linear demand function. Provide a detailed example.
2. Discuss the impact of changes in the "a" term of a linear demand function on the demand curve. Use diagrams to support your answer.
3. Evaluate the effect of varying the "b" coefficient in a demand function on the steepness of the demand curve. Provide examples.
4. Analyze how linear demand functions can be used to predict consumer behavior in different market scenarios. Use real-world examples.
5. Explain the significance of understanding linear demand functions for businesses setting pricing strategies. Discuss with reference to elasticity and revenue maximization.