ECON1003 · Quantitative Methods In Economics
Linear Functions
The straight line is the workhorse of ECON1003: every linear model — demand, supply, cost, revenue, a budget — is one equation, y = mx + c, pinned down by its slope m and its intercept c. Read those two numbers and you can describe, plot and manipulate any linear relationship in the unit. This chapter builds the toolkit the midterm leans on: writing a line from a point and a slope, rearranging a scrambled equation into slope-intercept form before reading m and c, switching between a demand function Q = f(P) and its inverse P = f(Q), and finding break-even where total revenue crosses total cost. It closes with the one subtlety the exam exploits relentlessly — linear elasticity is not the slope — using the point formula ε = (dQ/dP)(P/Q), which carries forward into every later topic.
What this chapter covers
- 011.1 Equation, slope and intercept — y = mx + c
- 02Rearranging a scrambled line before reading m and c
- 031.2 Demand, supply and inverse functions
- 041.3 Cost, revenue, profit and break-even
- 051.4 Linear elasticity — and why it is not the slope
- 06The budget line and its slope as a price ratio
Worked example: from a scrambled line to slope, intercept and elasticity
- +1(a) Isolate P. 2P = 100 − 0.5Q ⇒ P = 50 − 0.25Q. So inverse demand has intercept 50 and slope −0.25.
- +1(b) Flip to Q = f(P). From 2P + 0.5Q = 100: 0.5Q = 100 − 2P ⇒ Q = 200 − 4P. Note the slope dQ/dP = −4 is the reciprocal of −0.25, not equal to it.
- +1(c) Quantity at P = $20. Q = 200 − 4(20) = 120.
- +1Apply the point formula. ε = (dQ/dP)(P/Q) = (−4)(20/120) = −0.667.
- +1Classify. |ε| = 0.67 < 1, so demand is inelastic at P = $20 — reading the slope −4 as the elasticity would be wrong.
Key terms
- Slope (m)
- The rate of change of a line: how much y changes when x rises by one unit, m = rise/run = (y₂ − y₁)/(x₂ − x₁). A horizontal line has m = 0; a vertical line has an undefined slope.
- Inverse function
- The same relationship solved for the other variable. Demand Q = a − bP and inverse demand P = (a/b) − (1/b)Q describe the same curve, but their slopes are reciprocals (−b vs −1/b), not equal — picking the wrong one quietly wrecks intercepts, elasticity and surplus.
- Break-even quantity
- The output at which profit is exactly zero, where the total-revenue line crosses the total-cost line. Below it the firm makes a loss; above it, a profit.
- Point elasticity
- The unit-free responsiveness of quantity to price at a stated point, ε = (dQ/dP)(P/Q), reported as an absolute value. On a straight-line demand the slope dQ/dP is constant but P/Q changes, so elasticity differs at every point.
- Budget line
- The set of bundles a consumer can just afford, p₁x₁ + p₂x₂ = M. Rearranged it is a straight line whose slope is minus the price ratio −p₁/p₂, the rate at which the market lets you trade one good for the other.
Linear Functions FAQ
Why can't I just read the slope off the equation as written?
Because lines are often handed to you scrambled, such as 2y + 4x − 4 = 0. You must isolate y first: 2y = 4 − 4x ⇒ y = 2 − 2x, so m = −2 and c = 2. Reading the '4' as the slope is the classic blunder — always rearrange to y = mx + c before reading m and c.
Is the slope of a demand curve the same as its elasticity?
No. The slope dQ/dP is a constant on a straight line, but elasticity ε = (dQ/dP)(P/Q) also depends on the P/Q ratio, which changes as you move along the curve. So the same line is elastic near the top (large P/Q) and inelastic near the bottom. 'Steeper means more inelastic' is only valid comparing two curves at the same point.
How do I tell whether a question wants Q = f(P) or P = f(Q)?
Decide which variable is on which axis first. The unit uses 'demand function' loosely. If you need quantity from price, use Q = f(P); if you need price from quantity (for example to plot it or to find revenue), use the inverse P = f(Q). Their slopes are reciprocals, so using the wrong form silently corrupts every later step.
Exam move
Make slope-intercept form automatic: whatever equation you are given, rearrange to y = mx + c before reading anything off it. Practise switching between a demand function and its inverse until it is reflex, and always state which variable is on which axis. For elasticity, separate the two numbers in your head — the slope dQ/dP is the coefficient on P and is constant, while elasticity multiplies it by P/Q at the point and changes along the curve. Drill the line → slope/intercept → elasticity chain on fresh numbers; it recurs in every later topic and the midterm rewards getting it clean and fast.
