Section 3

Production and Capital Accumulation

Everything in the Solow model flows from one equation: the production function. This section sets up that function, converts it to per-effective-worker units, and derives the single equation that governs the entire economy.

The production function

The model uses a Cobb–Douglas production function with labour-augmenting technology (Harrod-neutral). This is the only functional form consistent with balanced growth (Uzawa's theorem):

Y = F(K, AL) = K^\alpha (AL)^{1-\alpha}, \quad \alpha \in (0,1)

Here $Y$ is total output, $K$ is the capital stock, $L$ is the labour force, $A$ is the level of technology, and $\alpha$ is capital's share of income (approximately $\tfrac{1}{3}$ in most economies).

Intensive form

Constant returns to scale (A3) lets us divide through by $AL$ and write everything per unit of effective labour. Define $k \equiv K/(AL)$ and $y \equiv Y/(AL)$ . Then:

y = f(k) = k^\alpha

The intensive form. Every variable we need is now a function of

k

alone.

Production function $f(k) = k^{\alpha}$

With

\alpha = 1/3

the curve is steep near the origin (Inada: each unit of capital is very productive when capital is scarce) and flattens at large

k

(diminishing returns). The investment curve

sf(k)

has the same shape, just scaled down by

s

Factor market equilibrium

Competitive markets pay each factor its marginal product. Differentiating $f(k) = k^\alpha$ :

r = f'(k) = \alpha k^{\alpha - 1}

Rental rate of capital. Declines as

k

rises — diminishing returns.

w = f(k) - k f'(k) = (1-\alpha)k^\alpha

Real wage per unit of effective labour.

Goods market clearing

All output is either consumed or invested. The savings rate $s$ determines the split:

I = sY, \qquad C = (1-s)Y

Capital accumulation

Investment adds to capital; depreciation subtracts from it. With $I = sY$ :

\dot{K} = sY - \delta K

Population and technology growth

\frac{\dot{L}}{L} = n \quad \Rightarrow \quad L(t) = L_0 e^{nt}

\frac{\dot{A}}{A} = g \quad \Rightarrow \quad A(t) = A_0 e^{gt}

The fundamental equation of motion

We want $\dot{k}$ , the rate of change of $k \equiv K/(AL)$ . The derivation has four clean steps — each follows mechanically from the last.

Step 1
$k \equiv \frac{K}{AL}$
Define the state variable. Everything reduces to the evolution of $k$ .
Step 2
$\dot{k} = \frac{\dot{K}}{AL} - k\,(n + g)$
Differentiate $k = K/(AL)$ with respect to time. Since $\frac{d}{dt}\ln(AL) = n + g$ , the quotient rule gives $\dot{k} = \dot{K}/(AL) - k(n+g)$ .
Step 3
$\frac{\dot{K}}{AL} = \frac{sY - \delta K}{AL} = sk^\alpha - \delta k$
Substitute $\dot{K} = sY - \delta K$ from the capital-accumulation equation. Dividing by $AL$ gives $sY/AL = sk^\alpha$ and $\delta K/AL = \delta k$ .
Step 4
$\dot{k} = sf(k) - (n + g + \delta)\,k$
Collect steps 2 and 3. This single equation governs the entire dynamics of the model. The term $(n+g+\delta)k$ is break-even investment: the share of output needed to keep $k$ constant as capital depreciates and effective labour grows.

When $sf(k) > (n+g+\delta)k$ , capital per effective worker rises ( $\dot{k} > 0$ ). When $sf(k) < (n+g+\delta)k$ , it falls ( $\dot{k} < 0$ ). The economy always moves toward the crossing point — the steady state $k^*$ .

Parameter reference

$s$: Savings rate. $s \in (0,1)$ ; typical range 0.15–0.35.
$\delta$: Depreciation rate. $\delta \in (0,1)$ ; often $\approx 0.05$ – $0.10$ .
$n$: Population growth rate. $n \geq 0$ ; often $\approx 0.01$ – $0.02$ .
$g$: Technological growth rate. $g \geq 0$ ; often $\approx 0.01$ – $0.02$ .
$\alpha$: Capital's share of output. $\alpha \in (0,1)$ ; often $\approx 1/3$ .

Convergence path

The plot below shows $k(t)$ for a default set of parameters. The economy starts away from $k^*$ and converges to it over time. Use the phase diagram in Section 5 to see why the direction of movement is always toward $k^*$ .