ECON5005 · Quantitative Tools for Economics
Functions, Systems & Market Equilibrium
This chapter of the University of Sydney ECON5005 Quantitative Tools for Economics study guide covers the algebra of functions and linear models, the workhorse of the whole unit. You will learn to read a straight line through its slope and intercepts, solve simultaneous equations, find where a supply-and-demand market clears, and measure how sharply quantity responds to price using elasticity. It maps to Week 2 of the Semester 2, 2026 teaching schedule and is examined on both the in-person mid-semester test and the 2-hour final.
What this chapter covers
- 01Define a function and read the linear form y = mx + c off an equation or a graph
- 02Compute the slope m = Δy/Δx and the vertical (c) and horizontal (−c/m) intercepts
- 03Use the economics plotting convention: price on the vertical axis and inverse demand P(Q)
- 04Solve a 2×2 system by substitution or elimination
- 05Diagnose the three outcomes: unique solution, no solution, or infinitely many
- 06Find market equilibrium by setting quantity demanded equal to quantity supplied
- 07Back-substitute for the equilibrium price and quantity and reject invalid (negative) roots
- 08Calculate point price-elasticity of demand ε = (dQ/dP)·(P/Q)
- 09Classify demand as elastic, unit-elastic or inelastic from the magnitude of ε
Market equilibrium and elasticity from linear schedules
- +1Set demand equal to supply: 100 − 5P = 10P − 20.
- +1Collect terms: 100 + 20 = 10P + 5P, so 120 = 15P, giving P* = $8.
- +1Back-substitute for quantity: Q* = 100 − 5(8) = 60. Check on supply: 10(8) − 20 = 60 — they agree.
- +1Elasticity needs dQ/dP from the demand schedule Qd = 100 − 5P, so dQ/dP = −5. Apply ε = (dQ/dP)·(P/Q) = (−5)·(8/60).
- +1Evaluate: ε = −40/60 ≈ −0.67. Since |ε| = 0.67 < 1, demand is inelastic at the equilibrium — quantity responds less than proportionally to price.
Key terms
- Linear function
- A rule of the form y = mx + c whose graph is a straight line with slope m and vertical intercept c.
- Slope
- The rate of change m = Δy/Δx: how much the output moves per one-unit rise in the input. Positive slopes rise, negative slopes fall.
- Intercepts
- The vertical intercept c is y when x = 0; the horizontal intercept is x = −c/m, where the line crosses the x-axis (y = 0).
- Inverse demand
- Demand written with price as the subject, P(Q), so it can be drawn with price on the vertical axis — the economics graphing convention.
- System of equations
- Two or more equations solved together. A 2×2 linear system has a unique solution, no solution (parallel lines), or infinitely many (dependent lines).
- Market equilibrium
- The price P* and quantity Q* at which quantity demanded equals quantity supplied — found by setting Qd(P) = Qs(P) and keeping the non-negative root.
- Price-elasticity of demand
- The unit-free responsiveness of quantity to price at a point: ε = (dQ/dP)·(P/Q), normally negative for demand.
- Elastic vs inelastic
- Demand is elastic if |ε| > 1 (quantity responds more than price), unit-elastic if |ε| = 1, and inelastic if |ε| < 1.
Functions, Systems & Market Equilibrium FAQ
Why is price on the vertical axis when quantity is what responds?
It is a long-standing convention in economics. Even though we usually treat quantity as a function of price, supply-and-demand diagrams put price on the vertical axis, so a demand schedule Qd = a − bP is rearranged into inverse demand P = a/b − (1/b)Q before graphing. Keep the two slopes straight: dQ/dP comes from the schedule, while the drawn line shows dP/dQ, and they are reciprocals of each other.
How do I know whether a system has one, none, or infinitely many solutions?
Compare the lines. Different slopes means they cross once (a unique solution). The same slope but different intercepts means parallel lines that never meet (no solution). If one equation is just a multiple of the other, the lines coincide and there are infinitely many solutions — the algebraic signal is a zero determinant with a consistent right-hand side.
Can AI help me with functions and market equilibrium in ECON5005?
Yes — an AI tutor like Sia is useful for understanding the method. It can walk you step by step through setting Qd = Qs, solving for P* and Q*, checking the answer on both schedules, and explaining why a point elasticity is elastic or inelastic, using practice numbers so you learn the moves. Use it to build understanding and check your reasoning, not to hand in generated answers: ECON5005 is assessed by in-person written exams, so real practice solving problems yourself is what pays off, and you should follow the University's academic-integrity rules.
Studying with AI? Sia — free AI economics tutor works through ECON5005 step by step.
Exam move
Master the pattern, not just one problem. First get fluent reading any line — slope, both intercepts, and which axis each variable sits on — because every model here is built on it. Then drill the equilibrium routine until it is automatic: set quantity demanded equal to quantity supplied, solve for the price, substitute back for the quantity, and confirm the two schedules agree. Layer elasticity on top by always deriving dQ/dP from the demand schedule (never its reciprocal) before plugging into ε = (dQ/dP)·(P/Q). Practise on the in-person written format at roughly two minutes per mark, showing every line so you collect method marks even when a number slips, and reject any negative price or quantity with a one-line reason. Confirm the exact assessment weights and the open/closed-book and calculator rules on your Canvas assessment page.
