ECON6002 · Macroeconomic Analysis
Growth Accounting & the Cass-Ramsey Model
Growth accounting and the Cass-Ramsey model form Topic 4 of ECON6002 Macroeconomic Analysis at the University of Sydney, welding two ideas together. The accounting half decomposes output per worker into physical capital, human capital and a productivity residual — total factor productivity (TFP), the famous Solow residual, or "the measure of our ignorance" — and shows that the amplifier α/(1−α) decides whether saving or TFP explains cross-country income gaps. The Cass-Ramsey half takes the Solow-Swan skeleton and endogenises the saving rate by letting an infinite-horizon household optimise, delivering a consumption Euler equation, the modified golden rule f′(k*) = ρ + θg + δ, a (k, c) phase diagram with its saddle path, and the key result that optimal saving guarantees dynamic efficiency (k*_CR < k_gold) where Solow's fixed saving rate could not. It is examined by the closed-book in-semester test (Topics 1–6), so you must be able to derive, not just recall, each result.
What this chapter covers
- 011. Growth vs development accounting — decomposing a growth RATE over time versus an income LEVEL across countries
- 022. The Hall-Jones decomposition — ln(Y/L) = [α/(1−α)]ln(K/Y) + ln(H/L) + ln A
- 033. TFP / the Solow residual — why A is a residual, not an input, and dominates income gaps
- 044. The amplifier α/(1−α) — how the capital share turns a saving gap into a modest or an enormous income gap
- 055. Cass-Ramsey setup — endogenising saving with an infinite-horizon optimising household and CRRA utility
- 066. The consumption Euler equation — MU today = discounted, return-weighted MU tomorrow, with θ = inverse IES
- 077. The phase diagram — the k̇=0 hump, the ċ=0 vertical locus, the steady state and the saddle path
- 088. The modified golden rule & dynamic efficiency — f′(k*) = ρ + θg + δ and why k*_CR sits below k_gold
Development accounting — can saving explain the income gap?
- +1Write the ratio. Since the two countries share the same (n+g+δ), it cancels, leaving y*_rich / y*_poor = (s_rich / s_poor)^(α/(1−α)).
- +1Compute the saving ratio: s_rich / s_poor = 0.27 / 0.03 = 9.
- +1(a) At α = 1/3 the exponent is α/(1−α) = (1/3)/(2/3) = 0.5, so the income ratio = 9^0.5 = 3. A ninefold saving gap yields only a threefold income gap.
- +1Interpret (a): real cross-country income gaps run to 20–30×, far above 3×, so capital/saving alone cannot explain them — the TFP residual A must carry most of the difference.
- +1(b) At the broad α = 2/3 the exponent is (2/3)/(1/3) = 2, so the ratio = 9^2 = 81. The same saving gap now generates an 81× income gap.
- +1Comment: the elasticity α/(1−α) is the amplifier. Counting human capital as part of "capital" raises α (Mankiw-Romer-Weil), so accumulation can rationalise the large observed dispersion that the α ≈ 1/3 view attributes to TFP.
Key terms
- Growth accounting
- Decomposition of a country's output GROWTH RATE over time into contributions from capital growth, labour growth and the residual growth of TFP (A). It answers "what drove this economy's growth?" — a rate-based exercise, distinct from development accounting.
- Development accounting
- Decomposition of the LEVEL of income per worker across countries into physical capital, human capital and TFP. It answers "why is one country richer than another?" and is the level-based counterpart to growth accounting.
- TFP / Solow residual (A)
- Total factor productivity — the part of output that measured capital and human capital cannot explain, backed out as a residual and famously called "the measure of our ignorance." Empirically it is the largest source of cross-country income gaps, so it is described as a residual, never a directly measured input.
- Human-capital-adjusted labour (H)
- Raw labour scaled up by the Mincerian returns to schooling, so a more-educated worker counts as more effective labour. Including H is what lets development accounting separate the human-capital contribution from the TFP residual.
- Cass-Ramsey model
- The Solow-Swan model with the saving rate endogenised: an infinite-horizon household optimises consumption, so saving becomes a choice rather than a fixed parameter. Solow's long-run conclusions survive, and the model is the bridge from Solow to the RBC/DSGE frameworks.
- CRRA utility & θ (inverse IES)
- Constant-relative-risk-aversion lifetime utility U = Σ βᵗ c_t^(1−θ)/(1−θ), where θ is the inverse intertemporal elasticity of substitution — the curvature parameter that Topic 2 wrote as σ. Watch the Greek switch: θ here is not the discount rate ρ.
- Modified golden rule
- The Cass-Ramsey steady-state condition f′(k*) = ρ + θg + δ. Because impatience raises the required marginal product of capital above the golden-rule level n+g+δ, the optimal capital stock sits below the golden-rule capital.
- Dynamic efficiency
- The property that an economy has not over-accumulated capital. Cass-Ramsey guarantees it — k*_CR < k_gold — because optimising households never save past the point where the marginal return falls below their discount rate; Solow's exogenous saving rate can instead push k above the golden rule into inefficiency.
Growth Accounting & the Cass-Ramsey Model FAQ
Can AI help me with growth accounting and the Cass-Ramsey model?
Yes — ask Sia to walk through any growth-accounting or Cass-Ramsey problem or concept step by step, the way University of Sydney tests it. It is an AI tutor that explains the derivation — the development-accounting decomposition, the Euler equation, the modified golden rule — so you learn how to reproduce it in the closed-book test, rather than an answer service.
What is the difference between growth accounting and development accounting?
Growth accounting decomposes a country's growth RATE over time; development accounting decomposes an income LEVEL across countries. The algebra rhymes but the question differs, so state which exercise you are doing in your opening line — reporting a level ratio as a growth contribution (or vice versa) loses the interpretation marks.
Why is the Solow residual called "the measure of our ignorance"?
Because TFP (A) is not observed directly — it is backed out as whatever measured capital and human capital cannot explain, i.e. a residual. Since that residual empirically accounts for the bulk of cross-country income gaps, we are effectively naming the thing we cannot yet explain, which is why it is described as a residual rather than a measured input.
How does the Cass-Ramsey model differ from the Solow-Swan model?
Solow assumes a constant, exogenous saving rate; Cass-Ramsey endogenises saving by letting an infinite-horizon household optimise consumption. Solow's long-run conclusions survive, but the optimising version adds a consumption Euler equation and guarantees dynamic efficiency (no over-accumulation), and it is the bridge to the RBC and DSGE models later in the unit.
What is the modified golden rule, and why is Ramsey capital below the golden rule?
The modified golden rule is the steady-state condition f′(k*) = ρ + θg + δ. Compared with the golden rule f′(k_gold) = n + g + δ, impatience (ρ) and consumption-smoothing (θg) raise the required marginal product, so f′ is higher and — since f′ falls in k — the capital stock is lower: k*_CR < k_gold. In one line: impatience keeps Ramsey capital below the golden rule, so over-accumulation is impossible.
Is Topic 4 examined on the final exam?
No. Growth accounting and Cass-Ramsey are examined by the 30% closed-book in-semester test, which covers Topics 1–6; the final exam covers only the New Keynesian material in Topics 7–10. The growth topics are foundational for later chapters, but you will be directly assessed on them in the in-semester test — check your unit outline for the test week and weights.
Studying with AI? Sia — free AI economics tutor works through ECON6002 step by step.
Exam move
Treat Topic 4 as one identity plus one optimisation. For the accounting half, learn the Hall-Jones decomposition ln(Y/L) = [α/(1−α)]ln(K/Y) + ln(H/L) + ln A cold, and drill the amplifier α/(1−α) until you can instantly turn a saving or capital-output gap into an income gap and attribute the leftover to the TFP residual — always naming A as a residual, not an input. For the Cass-Ramsey half, be able to set up the CRRA household problem, write the consumption Euler equation, set it stationary to reach the modified golden rule f′(k*) = ρ + θg + δ, and compare with the golden rule n+g+δ to sign k*_CR < k_gold; memorise the sentence "impatience keeps Ramsey capital below the golden rule, so over-accumulation is impossible." Because the in-semester test is closed-book with no formula sheet, rehearse each derivation until you can reproduce it without notes, sketch the (k, c) phase diagram with its k̇=0 hump, ċ=0 vertical locus and saddle path, and use the five-equations-for-five-endogenous-variables count as a self-check. Watch the Greek switch (θ = inverse IES, not σ, and not the discount rate ρ), and ask Sia to generate similar practice questions and check each step of your working as you revise.
