Introductory quantitative methods (HL only)

IB Economics HL· 45 min read

1. Linear equations for demand and supply★★☆☆☆HL only⏱ 15 min

📘 Definition

General form of a linear economic equation

$Q = a + b P$

Where $Q$ is quantity (dependent variable, plotted on the horizontal axis), $P$ is price (independent variable, plotted on the vertical axis), $a$ is the quantity intercept, and $b$ is the slope of the line.

Example:

For demand, $b$ is negative (inverse relationship between price and quantity), while for supply $b$ is positive (direct relationship).

In economics, we follow a convention opposite to standard mathematics: price is plotted on the vertical axis and quantity on the horizontal axis, even though price is typically the independent variable. This reversal changes how we calculate slope, so it is important to remember this convention at all times.

📐 Worked Example

A linear demand curve is given as $Q_{d} = 100 - 2 P$ . Write the inverse demand function (with $P$ as the subject) and identify the intercept on the price axis.

1
Start with the given direct demand function:
2
$Q_d = 100 - 2P$
3
Rearrange to isolate $P$ on the left-hand side:
4
$2P = 100 - Q_d$
5
Divide both sides by 2 to get the inverse form:
6
$P = 50 - 0.5Q_d$
7
The price intercept occurs when quantity demanded $Q_{d} = 0$ . Substitute $Q_{d} = 0$ :
8
$P = 50 - 0.5(0) = 50$

2. Calculating and interpreting slope★★☆☆☆HL only⏱ 15 min

Slope measures the rate of change between two variables. For any straight line, slope is constant along the entire curve. Following the economics axis convention, slope is calculated as the change in price (vertical variable) divided by the change in quantity (horizontal variable).

📐 Worked Example

Given two points on a linear supply curve: when $P = 10$ , $Q_{s} = 20$ ; when $P = 15$ , $Q_{s} = 40$ . Calculate the slope of the supply curve following economics convention.

1
Write the slope formula for economics axis convention:
2
$\text{slope} = \frac{\Delta P}{\Delta Q} = \frac{P_2 - P_1}{Q_2 - Q_1}$
3
Substitute the values from the two points:
4
$\Delta P = 15 - 10 = 5 \quad ; \quad \Delta Q = 40 - 20 = 20$
5
Calculate the final slope value:
6
$\text{slope} = \frac{5}{20} = 0.25$
7
Interpret the result: For every 1 unit increase in quantity supplied, price increases by 0.25, which matches the expected positive slope for a supply curve.

3. Algebraic solution for market equilibrium★★★☆☆HL only⏱ 20 min

Market equilibrium is defined as the point where quantity demanded equals quantity supplied. With linear demand and supply equations, we can solve for equilibrium price ( $P^{*}$ ) and equilibrium quantity ( $Q^{*}$ ) directly, without relying on graphing.

📐 Worked Example

Given $Q_{d} = 120 - 3 P$ and $Q_{s} = 30 + 2 P$ , calculate the equilibrium price and equilibrium quantity.

1
Set quantity demanded equal to quantity supplied, the equilibrium condition:
2
$120 - 3P = 30 + 2P$
3
Rearrange to collect like terms (constants on left, $P$ terms on right):
4
$120 - 30 = 3P + 2P \quad \rightarrow \quad 90 = 5P$
5
Solve for equilibrium price $P^{*}$ :
6
$P^* = \frac{90}{5} = 18$
7
Substitute $P^{*}$ back into either the demand or supply equation to find $Q^{*}$ :
8
$Q^* = 120 - 3(18) = 120 - 54 = 66$
9
Check your result by substituting into the other equation to confirm:
10
$Q^* = 30 + 2(18) = 30 + 36 = 66$
11
Final result: Equilibrium price = $18, equilibrium quantity = 66 units.

✓ Quick check

Test your understanding with this quick question:

If $Q_{d} = 50 - P$ and $Q_{s} = 2 P - 10$ , what is the equilibrium quantity?
- A) 20
- B) 30
- C) 40
- D) 10
Reveal answer
B) 30 —
Correct. First solve for equilibrium price: $50 - P = 2 P - 10 \to 60 = 3 P \to P = 20$ . Substitute back to get $Q = 50 - 20 = 30$ .

4. Common Pitfalls

Wrong move:

Calculating slope as $Δ Q /Δ P$ instead of $Δ P /Δ Q$

Why:

Economics reverses the standard math axis convention, so slope calculation is flipped from what you learned in general math

Correct move:

Always calculate slope as change in the vertical variable (price) divided by change in the horizontal variable (quantity): $Δ P /Δ Q$

Wrong move:

Not checking equilibrium quantity by substituting into both demand and supply

Why:

Small algebraic errors when solving for price often lead to incorrect quantity, and checking catches these mistakes before you move on

Correct move:

After finding equilibrium price, substitute into both equations to confirm you get the same quantity

Wrong move:

Using a positive coefficient for price in a direct demand equation

Why:

Demand has a negative relationship between price and quantity, so the slope coefficient must be negative

Correct move:

Always double-check the sign of the slope: negative for demand, positive for supply in direct form

Wrong move:

Assuming slope equals price elasticity of demand along a linear curve

Why:

Slope is constant along a linear demand curve, but elasticity changes as you move up or down the curve

Correct move:

Remember slope measures absolute change, while elasticity measures percentage change; do not equate the two

5. Quick Reference Cheatsheet

Concept	Formula	Key Notes
Direct linear form	$Q = a + b P$	Q = quantity, P = price, a = intercept, b = slope
Slope (econ convention)	$slope = \frac{Δ P}{Δ Q}$	P on vertical axis, Q on horizontal axis
Equilibrium condition	$Q_{d} = Q_{s}$	Set equal, solve for P, then solve for Q
Inverse form	Rearrange to make P the subject	Frequently required in exam questions

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 1
Calculate slope of linear supply curve
2021 · 2
Find equilibrium price algebraically

What's Next

The core quantitative skills covered in this sub-topic are applied across all units of IB Economics HL. You will use linear equations and equilibrium solving when analysing consumer and producer surplus, market intervention impacts, aggregate demand and supply models, and international trade calculations. Mastery of these basic methods is critical to accessing full marks on all quantitative questions in Papers 1 and 2, which account for a large share of your final grade. Next, you will build on these foundational skills to learn about price elasticity of demand and supply, where you will explore the difference between constant slope and changing elasticity along linear curves.