ECON6002 · Macroeconomic Analysis
The Two-Period Model
The Two-Period Model is Topic 2 of ECON6002 Macroeconomic Analysis at the University of Sydney, and the chapter where time first enters the course. A household chooses consumption now (C₁) and later (C₂), saving or borrowing at the net real rate r and discounting future utility by β, which delivers the three results reused to the end of the unit: the intertemporal budget constraint, the consumption Euler equation, and the intertemporal elasticity of substitution (IES = 1/σ). Because a higher interest rate makes future consumption cheaper, the model is also where the substitution, income, and wealth effects of a rate change are pulled apart. It is examined by the 30% closed-book in-semester test (Topics 1–6), which carries no formula sheet.
What this chapter covers
- 011. The two-period problem — maximise u(C₁) + β·u(C₂) subject to C₁ + S = Y₁ and C₂ = Y₂ + (1+r)S, with discount factor β ∈ (0,1)
- 022. The intertemporal budget constraint (IBC) — eliminate saving S to get C₁ + C₂/(1+r) = Y₁ + Y₂/(1+r) ≡ W, lifetime wealth in period-1 dollars
- 033. Relative price of future consumption — normalise the price of C₁ to 1, so C₂ costs 1/(1+r) today; a higher r makes the future cheaper
- 044. The consumption Euler equation — u′(C₁) = β(1+r)·u′(C₂): marginal utility today equals discounted, return-weighted marginal utility tomorrow
- 055. Log and CRRA utility — u′ = 1/C gives the linear tilt C₂ = β(1+r)C₁; CRRA u′ = C^(−σ) gives the growth rule C₂/C₁ = [β(1+r)]^(1/σ)
- 066. The intertemporal elasticity of substitution — IES = d ln(C₂/C₁)/d ln(1+r) = 1/σ, measuring how strongly the household re-times consumption when r moves
- 077. Substitution, income, and wealth effects — decomposing a change in r; the sign of dC₁/dr is negative if IES > 1, zero if log (IES = 1), positive if IES < 1
- 088. Consumption smoothing and applications — saver vs borrower from the income tilt, the housing-wealth (collateral) channel, and λ as the marginal value of lifetime wealth
Solve a log-utility household and classify saver or borrower
- +1Lifetime wealth from the IBC: W = Y₁ + Y₂/(1+r) = 70 + 22/1.1 = 70 + 20 = 90.
- +1Log-utility Euler equation: u′(C) = 1/C, so C₂ = β(1+r)C₁ = 0.8 × 1.1 × C₁ = 0.88 C₁.
- +1Substitute the Euler equation into the IBC: C₁ + (0.88 C₁)/1.1 = C₁(1 + β) = 1.8 C₁ = 90.
- +1Solve for period-1 consumption: C₁ = W/(1+β) = 90/1.8 = 50.
- +1Period-2 consumption: C₂ = 0.88 × 50 = 44 (check the IBC: 50 + 44/1.1 = 50 + 40 = 90 ✓).
- +1Saving: S = Y₁ − C₁ = 70 − 50 = 20 > 0, so the household saves — period-1 income is high relative to Y₂, so it moves resources forward to smooth consumption.
Key terms
- Two-period model
- The smallest dynamic model of a household that chooses consumption in period 1 and period 2, saving or borrowing at rate r and discounting the future by β; the base for every later intertemporal model in the course.
- Intertemporal budget constraint (IBC)
- The single lifetime constraint C₁ + C₂/(1+r) = Y₁ + Y₂/(1+r) = W, obtained by eliminating saving; the present value of consumption equals the present value of income.
- Consumption Euler equation
- The intertemporal optimality condition u′(C₁) = β(1+r)·u′(C₂): the household is indifferent at the margin between spending a dollar today and saving it for tomorrow.
- Discount factor β
- A number in (0,1) measuring patience; the household weights next period's utility by β, so a smaller β means a more impatient household that consumes more today.
- CRRA / power utility
- The utility form u(C) = C^(1−σ)/(1−σ) with constant relative risk aversion σ; marginal utility is C^(−σ), and log utility is the special case σ = 1.
- Intertemporal elasticity of substitution (IES)
- Equal to 1/σ, it measures how strongly the household re-times consumption across periods when the interest rate (the relative price of future consumption) changes.
- Wealth effect
- Present only when Y₂ > 0: a higher r cuts the present value Y₂/(1+r) of future income, shrinking lifetime wealth W and so reducing current consumption C₁.
- Consumption smoothing
- Spreading consumption evenly across periods to equalise marginal utility; the reason a household with high current income but low future income saves.
The Two-Period Model FAQ
Is the two-period model on the ECON6002 exam?
Yes. Topic 2 is examined by the 30% in-semester test (midterm), which is closed-book with no formula sheet and covers Topics 1–6. The final exam only tests the New Keynesian topics (7–10), so the two-period model is a midterm asset. Practice comes from Tutorials 1–6.
What is the consumption Euler equation and why does it matter?
It is the first-order condition u′(C₁) = β(1+r)·u′(C₂), which says the marginal utility given up by consuming a dollar less today equals the discounted, return-weighted marginal utility gained tomorrow. It matters because the same equation (with small tweaks) reappears in the Cass–Ramsey, Real Business Cycle, and New Keynesian models, so mastering its derivation here pays off all semester.
What is the difference between σ and the intertemporal elasticity of substitution?
σ is the coefficient of relative risk aversion (the curvature of CRRA utility); the IES is its reciprocal, 1/σ. The course deliberately switches Greek letters in later topics (Topic 4 writes the inverse IES as θ), so always read the definition — confusing σ with 1/σ flips the predicted sign of the interest-rate effect on consumption.
When does a household save rather than borrow in the two-period model?
Compare current income Y₁ with the present value of future income Y₂/(1+r). If period-1 income is high relative to period-2 income, the household saves (S > 0) to smooth consumption; if future income is high, it borrows against it (S < 0). With log utility the split is clean because C₁ = W/(1+β) regardless of r.
Can AI help me with the two-period model?
Yes — ask Sia to walk through any two-period model problem or concept step by step, the way University of Sydney tests it. Sia is an AI tutor that explains the derivation of the Euler equation, the IBC, and the substitution/income/wealth decomposition, so you learn the method rather than just seeing a final number.
What is the most common mistake on two-period model questions?
Four recur: dropping the β in the Euler equation, writing (1+r) on the wrong side, confusing σ with the IES (1/σ), and ignoring the wealth effect when Y₂ > 0. The clean sign rule for dC₁/dr — negative if IES > 1, zero if log, positive if IES < 1 — only holds when Y₂ = 0, so check that the wealth effect is absent before quoting it.
Studying with AI? Sia — free AI economics tutor works through ECON6002 step by step.
Exam move
Learn the derivation, not the formula, because the midterm is closed-book with no formula sheet. Practise the four-step routine until it is automatic: (1) write the intertemporal budget constraint C₁ + C₂/(1+r) = Y₁ + Y₂/(1+r) and state that the price of future consumption is 1/(1+r); (2) derive the Euler equation from the Lagrangian first-order conditions rather than quoting it, since both the derivation and the naming carry marks; (3) combine the Euler equation with the IBC to solve for C₁, C₂ and S, then classify the household as a saver or borrower from the sign of S; and (4) for any interest-rate question, name the substitution, income and wealth effects and use the IES to sign the net response, checking whether a wealth effect (Y₂ > 0) is present. Work Tutorials 1–6 under timed, closed-book conditions, and rehearse the interpretation of the Lagrange multiplier λ as the marginal utility of lifetime wealth — a reliable short-answer mark.
