ECON5005 · Quantitative Tools for Economics
Difference Equations
This chapter is where the static models of ECON5005 Quantitative Tools for Economics at the University of Sydney start to move through time. A difference equation (a recurrence relation) links a variable at period t to its own value at period t−1, so a single starting value and the rule generate the whole sequence. You learn to solve the first-order linear case in closed form as a complementary function plus a particular solution, and to judge its stability from the size and sign of the growth factor. It is the Week 13 capstone and a standard final-exam topic in economic dynamics.
What this chapter covers
- 01Read a first-order linear difference equation Y_t = b·Y_(t-1) + c and identify b, c and the initial value Y_0
- 02Solve the homogeneous case Y_t = b·Y_(t-1) as geometric growth: Y_t = Y_0·b^t
- 03Find the particular solution (equilibrium / fixed point) Y* = c/(1-b) for b ≠ 1
- 04Build the complementary function A·b^t and pin the constant from Y_0 with A = Y_0 - Y*
- 05Write the full general solution Y_t = (Y_0 - c/(1-b))·b^t + c/(1-b)
- 06Handle the special case b = 1: no equilibrium, arithmetic drift Y_t = Y_0 + c·t
- 07Apply the stability test: the equilibrium is stable exactly when |b| < 1, divergent when |b| > 1
- 08Tell the path shape apart: b > 0 converges monotonically, b < 0 oscillates (damped if |b| < 1)
- 09Interpret the long-run value economically and describe convergence or divergence in words
Solve a first-order difference equation and classify its stability
- +1Identify the constants from the standard form Y_t = b·Y_(t-1) + c: here b = 0.8, c = 60 and Y_0 = 500.
- +2Equilibrium (particular solution): set Y_t = Y_(t-1) = Y*, so Y* = c/(1-b) = 60/(1-0.8) = 60/0.2 = 300.
- +2Complementary function and constant: A = Y_0 - Y* = 500 - 300 = 200, so the general solution is Y_t = 200·(0.8)^t + 300.
- +2Stability: |b| = 0.8 < 1, so the equilibrium is stable and Y_t → 300; because b > 0 the approach is monotone (no oscillation) — the path falls from 500 toward 300.
- +1Check Y_1: from the formula Y_1 = 200·(0.8) + 300 = 160 + 300 = 460; from the recurrence Y_1 = 0.8·(500) + 60 = 460 — they agree.
Key terms
- Difference equation (recurrence relation)
- An equation linking a variable at time t to its earlier values, e.g. Y_t = b·Y_(t-1) + c. Together with an initial condition Y_0 it determines the entire sequence period by period.
- First-order / linear
- First-order means only one lag appears (Y_(t-1), not Y_(t-2)); linear means Y_(t-1) enters to the first power with constant coefficients. This is the case solved in closed form in this chapter.
- Homogeneous solution
- The solution of the pure case Y_t = b·Y_(t-1) with c = 0. Iterating gives geometric growth Y_t = Y_0·b^t.
- Particular solution (equilibrium / fixed point)
- A constant sequence that satisfies the equation, found by setting Y_t = Y_(t-1) = Y*: Y* = c/(1-b), valid when b ≠ 1. It is the long-run value the system can rest at.
- Complementary function
- The term A·b^t that solves the homogeneous part. It measures the gap between the variable and its equilibrium; the constant A is pinned from the initial condition as A = Y_0 - Y*.
- General solution
- Complementary function plus particular solution: Y_t = (Y_0 - c/(1-b))·b^t + c/(1-b). It reproduces Y_0 at t = 0 and satisfies the recurrence for all t.
- Stability
- Whether the system returns to equilibrium. The gap A·b^t dies out and Y_t → Y* exactly when |b| < 1 (stable); if |b| > 1 the gap grows and the path diverges (unstable).
- Oscillation
- The alternating, above-then-below pattern that appears when b < 0, because the powers b^t flip sign each period. It is damped (shrinking) if |b| < 1 and explosive (growing) if |b| > 1.
Difference Equations FAQ
Is this topic examined in ECON5005, and how does it appear?
Yes. Difference equations are the Week 13 material on economic dynamics, so they belong to the final exam rather than the mid-semester test (which covers the earlier algebra-through-systems weeks). They usually appear as a self-contained multi-part question: a part to write down the equilibrium Y* = c/(1-b), a part for the general solution Y_t = (Y_0 - Y*)·b^t + Y*, and a part naming the stability and the long-run value. The final is a 2-hour in-person written exam in the formal exam period; confirm your exact date, and whether it is open- or closed-book, on your Canvas page.
How do I tell whether the equilibrium is stable or unstable?
Look at the growth factor b. The gap from equilibrium is A·b^t, so it shrinks and the system converges when the absolute value |b| < 1 (stable), and it grows and the system diverges when |b| > 1 (unstable). The sign of b is a separate question about the path shape: b > 0 gives a smooth one-sided approach, while b < 0 makes the sequence oscillate above and below the equilibrium each period. So b = 0.8 is stable and monotone, b = -0.4 is stable but oscillating (damped), and b = -1.5 is unstable and oscillating (explosive). Watch the special case b = 1: there is no finite equilibrium and the path is the arithmetic drift Y_t = Y_0 + c·t.
Can AI help me with difference equations in ECON5005?
Yes, as a study aid. Sia, the AskSia AI tutor, can explain the complementary-function-plus-particular-solution method step by step, walk through a similar recurrence so you can see how to find the equilibrium and pin the constant, check where a sign slipped in 1-b when b is negative, and quiz you on the stability condition |b| < 1 and when a path oscillates. It is there to help you understand and practise the technique, not to hand in answers for you. Build the skill with it, then complete the actual assessed quizzes and exam yourself, in line with the University of Sydney's academic-integrity rules.
Studying with AI? Sia — free AI economics tutor works through ECON5005 step by step.
Exam move
Drill one routine until it is automatic: read off b, c and Y_0; compute the equilibrium Y* = c/(1-b); pin the constant A = Y_0 - Y*; write Y_t = A·b^t + Y*; then judge stability from |b| and the path shape from the sign of b. Practise on a mix of positive and negative b so both the monotone and the oscillating cases feel familiar, and always finish with the long-run number and a one-line description rather than stopping at A·b^t. Keep three checks in your pocket: at t = 0 the formula must give back Y_0, one step forward must match the recurrence, and b = 1 has no equilibrium (use Y_0 + c·t instead). Because long questions are graded on method, write the equilibrium, then the complementary function, then the constant, then the stability verdict — every visible step earns partial credit.
