Questions: Total Factor Productivity and the Sources of Growth

Question 1

Country A and Country B have identical savings rates, labor force growth rates, and initial capital stocks. After 50 years, Country A is twice as wealthy per capita. Growth accounting would attribute this divergence primarily to:

• A. Country A having accumulated more capital through higher investment efficiency
• B. Country A having higher TFP growth — differences in technology, institutions, and efficiency that factor accumulation alone cannot explain
• C. Country A having a younger population, contributing more labor hours per capita
• D. Country A experiencing lower capital depreciation rates, preserving more of its stock

Answer: (B)

With identical savings rates and labor force growth, the Solow model predicts both countries converge to the same steady-state income level — persistent capital accumulation differences cannot explain the divergence. TFP (the Solow residual) captures everything that is not capital or labor: technology, institutions, organizational efficiency, infrastructure quality. The empirical finding is that most long-run per-capita income divergence across countries traces to TFP differences, not factor accumulation differences. Options C and D might contribute to TFP through indirect channels but are not the growth accounting category.

Question 2

Using the Cobb-Douglas framework Y = AK^0.3 L^0.7, output grows 3% per year, capital grows 4%, and labor grows 1%. What is TFP growth?

• A. 3% — TFP growth equals total output growth when using a production function approach
• B. 1.9% — TFP equals the combined contribution of capital and labor to output growth
• C. 1.5% — computed as output growth minus the unweighted average of input growth rates
• D. 1.1% — the residual after subtracting weighted factor contributions: 3% minus 0.3(4%) minus 0.7(1%)

Answer: (D)

Growth accounting decomposes output growth as ΔY/Y = α(ΔK/K) + (1-α)(ΔL/L) + ΔA/A. Plugging in: 3% = 0.3(4%) + 0.7(1%) + ΔA/A = 1.2% + 0.7% + ΔA/A = 1.9% + ΔA/A. Therefore ΔA/A = 3% - 1.9% = 1.1%. Option A conflates TFP growth with total output growth. Option B gives the factors' total contribution (1.9%) — what inputs explain — which is the opposite of TFP. Option C ignores income shares, incorrectly treating capital and labor contributions as equally weighted.

Question 3

In the Solow model, doubling a country's capital stock per worker will approximately double its output per worker, because capital and output scale proportionally.

• T. True
• F. False

Answer: False

The Cobb-Douglas production function has diminishing returns to capital: output per worker scales as k^α, where α is capital's income share (typically ~0.3). Doubling capital per worker multiplies output per worker by 2^0.3 ≈ 1.23 — a 23% increase, not a doubling. This is precisely why capital deepening cannot sustain long-run growth: each additional unit of capital adds less and less output. Only TFP growth — shifts in the production function itself — can sustain growth indefinitely without hitting diminishing returns.

Question 4

TFP is called a 'residual' because it cannot be measured directly — it is inferred from the output growth that factor accumulation alone cannot explain.

• T. True
• F. False

Answer: True

We observe output growth and can measure capital and labor growth along with their income shares. After attributing growth proportionally to factor inputs, whatever remains is the Solow residual — TFP growth. TFP is 'measured by subtraction,' not by directly observing technology improvements. This is sometimes called 'a measure of our ignorance' because it bundles together everything we cannot attribute to factors: technology, institutions, efficiency, organizational practices, and human capital quality.

Question 5

Why can capital deepening not sustain long-run per-capita growth, and what does TFP represent that capital accumulation cannot capture?

Model answer: Capital deepening faces diminishing returns: each additional unit of capital added to a fixed labor force raises output by less than the previous unit, following the concave shape of the production function. Eventually, an economy reaches a steady state where additional investment only offsets depreciation and population growth, with no net increase in capital per worker. TFP represents shifts in the production function itself — better technology, more efficient organization, improved institutions, higher-quality human capital — that raise the output achievable from any given stock of inputs. Unlike capital, TFP improvements do not diminish: a better algorithm or stronger institutions raise productivity for every unit of capital and labor simultaneously.

The policy implication is that long-run growth depends on TFP drivers — R&D, education, institutional reform, technology adoption — not only on incentivizing capital accumulation. Higher savings rates produce a transition to a higher steady state, but only sustained TFP growth produces ongoing improvement beyond that steady state.