An economy doubles its capital stock while holding labor and TFP constant. With α = 1/3 in the Cobb-Douglas function Y = AK^α L^(1−α), output increases by approximately:
A100% — doubling a major input doubles output
B26% — less than doubles, due to diminishing returns to capital
C67% — equal to (1 − α) times the percentage increase in capital
D50% — the average of the factor shares
With Y = AK^(1/3)L^(2/3), doubling K while holding A and L fixed multiplies Y by 2^(1/3) ≈ 1.26 — a 26% increase, not 100%. This is the diminishing returns property: the exponent α = 1/3 < 1 means each additional unit of capital contributes less than the previous one. Option A reflects the misconception that doubling one factor doubles output, which would only hold under constant returns to that factor alone.
Question 2 Multiple Choice
In the Solow growth model, what is the only source of sustained long-run per-capita output growth?
AContinuous capital accumulation through higher household saving rates
BPopulation growth, which expands the labor force
CGrowth in Total Factor Productivity (A)
DInternational trade that expands access to cheaper capital goods
Capital accumulation runs into diminishing returns: each new unit of capital raises output less than the last. Eventually the extra output from new capital exactly covers depreciation, and the capital stock stabilizes at a 'steady state' where per-capita growth halts. Population growth raises total output but not necessarily per-capita output. Only TFP growth (A) shifts the entire production function upward, lifting the steady state and enabling indefinitely sustained per-capita growth.
Question 3 True / False
In the Cobb-Douglas function Y = AK^α L^(1−α), the exponents α and (1−α) represent capital's and labor's shares of national income under competitive factor markets.
TTrue
FFalse
Answer: True
Under competitive factor markets, each factor is paid its marginal product. The marginal product of capital is ∂Y/∂K = αY/K, so capital's total income (K × MPK) = αY — a fraction α of total output. Similarly, labor's share is 1−α. This is a remarkable feature of the Cobb-Douglas form: the output elasticities and the income shares are identical. With α ≈ 1/3, capital earns roughly one-third and labor two-thirds of GDP, consistent with observed data in many developed economies.
Question 4 True / False
Because TFP (A) is such an important driver of growth, economists have developed a comprehensive theory of what causes it to change.
TTrue
FFalse
Answer: False
TFP is defined as the Solow residual — growth in output that cannot be explained by measured growth in capital and labor. Robert Solow himself described it as 'a measure of our ignorance.' Technology, institutions, education quality, management practices, and resource allocation efficiency all contribute to A, but we lack a unified theory of what drives TFP. This is precisely why growth accounting is humbling: 30–50% of historical growth in developed economies is attributed to something we cannot fully explain.
Question 5 Short Answer
Why can't an economy sustain long-run per-capita growth through capital accumulation alone?
Think about your answer, then reveal below.
Model answer: Capital exhibits diminishing marginal returns: because α < 1 in the Cobb-Douglas function, each additional unit of capital adds less output than the previous one. As the capital stock grows, the extra output per new unit of capital falls. Eventually, the additional output from one more unit of capital equals the depreciation on that unit, and net investment goes to zero. The capital stock stabilizes at a 'steady state,' and per-capita output growth stops. Only improvements in TFP (A) can shift the production function upward and sustain growth beyond this limit.
This result — the impossibility of indefinite capital-led growth — is the central insight of the Solow model. It redirects attention from saving rates to technological change: if you want to explain why some countries grow indefinitely richer, you must explain what drives their TFP. The production function defines what accumulation can and cannot do.