Glossary

Capital Accumulation

Capital accumulation is the net change in an economy's capital stock $K$ over time: gross investment $I$ minus the depreciation of existing capital $\delta K$ . In continuous time, the aggregate law of motion is $\dot K = I - \delta K = sY - \delta K$ , where $\delta$ is the depreciation rate (the fraction of the capital stock that wears out each period), $s$ is the saving rate (the fraction of output devoted to new investment), and $Y$ is aggregate output. The stock grows whenever gross investment exceeds replacement needs, shrinks whenever depreciation outpaces investment, and is stationary at $\dot K = 0$ .

In plain terms, capital accumulation is the process of building up the stock of productive assets — machines, factories, roads, software — faster than those assets wear out. It is the engine of growth in classical and neoclassical models: more capital per worker means more output per worker, at least until diminishing returns set in. It helps to distinguish two modes. Capital deepening occurs when the capital-labor ratio $k$ rises — each worker is equipped with more capital, pushing output per worker up along the production function. Capital widening occurs when the capital stock grows just fast enough to equip a growing workforce, leaving $k$ flat. Long-run growth in output per worker requires either sustained deepening or improvements in total factor productivity.

K_{t+1} = (1-\delta)K_t + I_t

capital-law-of-motion

Discrete-time capital law of motion: next period's capital stock equals the undepreciated portion of today's stock

(1-\delta)K_t

plus new investment

I_t

. In per-effective-worker terms the Solow model restates this as

\dot k = sf(k) - (n+g+\delta)k

, where

k

is capital per effective worker,

f(k)

is output per effective worker,

n

is the population growth rate, and

g

is the rate of labor-augmenting technical progress. The term

(n+g+\delta)k

is break-even investment — the amount needed to hold

k

constant against depreciation, population growth, and technology growth.

Gross vs. net investment: Gross investment $I$ is the total flow of new capital goods produced in a period — it includes both replacement of worn-out capital and additions to the stock. Net investment $I - \delta K$ is what actually changes the capital stock; it is positive when the economy is accumulating capital and negative when the stock is shrinking.
Depreciation: $\delta K$ is the capital that wears out, becomes obsolete, or is retired each period. The depreciation rate $\delta$ is typically modeled as a constant fraction of the existing stock, ranging from around 4–5% per year for structures to 15–20% for equipment and software in empirical calibrations.
Capital deepening: A rise in the capital-labor ratio $k = K/L$ (or capital per effective worker $k = K/(AL)$ in the augmented Solow model). Capital deepening raises output per worker along the production function $f(k)$ , but because of diminishing marginal returns, each additional unit of $k$ contributes less than the last.
Break-even investment: $(n + g + \delta)k$ — the investment per effective worker required to hold $k$ constant. Population growth at rate $n$ and technology growth at rate $g$ continuously dilute capital per effective worker, so new investment must cover depreciation $\delta k$ plus this dilution effect before $k$ can rise.