Economic Growth: formula sheet

Key formulas

Cobb-Douglas production function

Y = A × K^α × L^(1-α)

Assumes constant returns to scale. α is capital's share of output, 1-α is labor's share.

Growth accounting equation

ΔY/Y = ΔA/A + α(ΔK/K) + (1-α)(ΔL/L)

ΔA/A is TFP growth, found as the residual. Use growth rates as decimals or percentages consistently.

Per-worker production function

Y/L = A × (K/L)^α

Output per worker depends on capital per worker and technology.

Labor productivity growth

Growth in Y/L ≈ ΔA/A + α × growth in (K/L)

Also approximately equals growth in Y minus growth in L (or hours).

Potential GDP growth (labor force approach)

Growth in potential GDP = growth in labor input + growth in labor productivity

Labor input is the labor force or total hours worked.

Real GDP growth rate

g = (Real GDP this year ÷ Real GDP last year) − 1

Use real, not nominal, GDP. Nominal GDP growth is approximately real growth plus inflation.

Real GDP per capita growth

Per capita growth ≈ real GDP growth − population growth

The subtraction is an approximation. For exact work, divide real GDP by population in each year and compare.

Cobb-Douglas production function

Y = A × K^α × L^(1 − α)

Assumes constant returns to scale. α is capital's share of income.

Growth accounting equation

ΔY/Y = ΔA/A + α × ΔK/K + (1 − α) × ΔL/L

Growth rates are in percent. Output growth equals TFP growth plus weighted input growth.

Solow residual (TFP growth)

ΔA/A = ΔY/Y − α × ΔK/K − (1 − α) × ΔL/L

Rearranged form. It is the unexplained growth after capital and labor.

Labor productivity growth

Δ(Y/L)/(Y/L) = ΔA/A + α × [ΔK/K − ΔL/L]

Output per worker growth equals TFP growth plus α times growth in capital per worker. This form relies on the Cobb-Douglas constant-returns assumption, meaning the exponents α and 1 − α sum to 1.

Neoclassical production function

Y = A × F(K, L)

Y is output, A is total factor productivity (technology), K is capital, L is labour. Constant returns to scale in K and L together.

Per-worker form

y = Y ÷ L = A × f(k), where k = K ÷ L

Diminishing returns: f(k) rises at a decreasing rate as k rises.

Steady-state condition

s × y = (δ + n) × k (technology constant). With technology growth: s × y = (δ + n + g) × k

s is saving rate, δ is depreciation rate, n is labour force growth. The first form holds when A is constant. When technology grows, measure k and y per effective worker. Then g is the growth rate of labour-augmenting technology, which equals θ ÷ (1 − α) in a Cobb-Douglas function. Investment must cover depreciation, new workers and the rise in technology. Capital and output per effective worker are then constant, while output per worker grows at g.

Steady-state growth of total output

ΔY ÷ Y = θ ÷ (1 − α) + n

θ is the growth rate of technology (TFP) and α is capital's share of output in a Cobb-Douglas function. The first term is the growth of output per worker.

Steady-state growth of output per worker

Δy ÷ y = θ ÷ (1 − α)

Depends only on technology growth and capital share. It does not depend on the saving rate.

Endogenous (AK) model

Y = A × K

Constant returns to capital (broadly defined). Output per worker can grow indefinitely, and a higher saving rate raises the long-run growth rate.

Absolute convergence

Poor economies grow faster than rich ones; all converge to the same per-capita income

Holds only if economies share the same traits. Weak empirical support across all countries.

Conditional convergence

Each economy converges to its own steady state; growth rate rises with distance below that steady state

Steady state depends on savings, population growth, human capital, institutions and technology.

Convergence club

Countries with similar characteristics converge within a group; groups differ and do not converge to each other

Moving between clubs needs changes in fundamentals such as institutions or human capital.

Neoclassical catch-up logic

Lower capital per worker → higher marginal product of capital → faster growth

Relies on diminishing marginal returns and technology that can be copied.

Quick revision

• Potential GDP growth comes from growth in labour input and growth in labour productivity.
• Growth accounting: ΔY/Y = ΔA/A + α × ΔK/K + (1 − α) × ΔL/L.
• Per-worker form: Δ(Y/L)/(Y/L) = ΔA/A + α × Δ(K/L)/(K/L).
• TFP growth is the residual: output growth minus the weighted growth of capital and labour.
• α is capital's share of national income and (1 − α) is labour's share.
• Classical theory: growth is temporary because population growth erodes gains in per-capita income.
• Neoclassical theory: diminishing marginal returns to capital, so long-run per-capita growth comes from technology.
• Neoclassical steady state: capital per worker is stable and growth in output per worker equals θ/(1 − α), where θ is TFP growth. Growth in total output equals θ/(1 − α) plus labour force (population) growth.
• Endogenous growth: investment in human capital and R&D can give non-diminishing returns, so growth is sustained from within.
• Absolute convergence: poorer economies catch up to richer ones regardless of country characteristics. Conditional convergence: economies with the same savings rate, population growth and production function converge to the same steady state and income level; economies with different characteristics converge to different steady-state levels, so they need not catch up with richer economies. Convergence is relative to each economy's own steady state, and a poorer economy grows faster the further it is below its own steady state.
• Club convergence: economies within a group converge to the group's level, but groups differ.
• Capital deepening raises labour productivity but with diminishing returns; technology progress does not.

Common mistakes

• Adding capital and labor growth without weighting them by income shares. Fix: Always multiply capital growth by α and labor growth by (1-α) before adding.
• Using α as labor's share. Fix: α belongs to capital. Labor gets 1 - α.
• Applying α to labor and 1 − α to capital. Fix: Remember that α is always capital's share. The exponent on K is α and the exponent on L is 1 − α. Check the vignette for which share is given.
• Treating TFP growth as an observed input. Fix: TFP is the residual. Compute it as output growth minus the weighted growth of capital and labor unless the question gives it.
• Saying a higher saving rate raises the long-run growth rate in the Solow model. Fix: In Solow, saving changes the steady-state level of output per worker only. Long-run per-worker growth comes from technology.
• Forgetting that technology is exogenous in the neoclassical model. Fix: If the model explains technology by choices inside the economy, it is endogenous. If technology is simply given, it is neoclassical.
• Saying conditional convergence means poor countries never catch up. Fix: Conditional convergence says growth is faster the further an economy is below its own steady state. Countries with similar traits head to similar levels.
• Treating absolute convergence as well supported by the data. Fix: Remember it needs the strong assumption that economies share the same traits. Evidence across all countries is weak.

Exam tips

• Write α next to the numbers first. Most errors come from swapping the shares.
• Check whether the question asks for total output growth or per-worker growth.
• Expect a qualitative question on which factor can sustain growth. The answer is technology, not capital alone.
• If TFP is not given, assume you must compute it as the residual.
• Watch for hours worked versus number of workers when computing productivity.
• Look for the share given in the vignette. If it is labor's share, convert to α before you calculate.
• When the stem says per capita, per worker or total, circle that word before you touch the numbers.
• Use the per worker form to check your answer: TFP plus α times capital deepening should match.