Section 3

Production and Capital Accumulation

Aggregate variables

$Y(t)$: Output. Total GDP.
$K(t)$: Capital. Total physical capital stock.
$L(t)$: Labour. Total workers (= population, fully employed).
$A(t)$: Technology. Total factor productivity (TFP); labour-augmenting.
$C(t)$: Consumption. Total household consumption.
$I(t)$: Investment. Total investment in new capital.

Per effective worker variables

$k$: $K / (AL)$ — the state variable of the model.
$y$: $Y / (AL) = f(k)$ .
$c$: $(1-s)y$ .
$i$: $sy$ .

Parameters

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

Production function

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

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

In intensive form — dividing through by $AL$ and using constant returns to scale:

y = f(k) = k^\alpha

Here $y$ is output per effective worker, $k$ is capital per effective worker, and $lpha$ is capital's share of income.

Factor market equilibrium

In a competitive economy, factors earn their marginal products:

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

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

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

\dot{K} = I - \delta 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

This is the heart of the Solow model: the law of motion for capital per effective worker $k \equiv K/(AL)$ .

Step 1
$y = k^\alpha$
Normalise by effective labour: define k ≡ K/(AL) and y ≡ Y/(AL). Dividing the Cobb-Douglas function by AL gives y = k^α.
Step 2
$\dot{k} = \frac{\dot{K}}{AL} - k(n + g)$
Differentiate $k = K/(AL)$ with respect to time. Since $d \ln(AL)/dt = g + n$ :
Step 3
$\dot{k} = s k^\alpha - (\delta+n+g)k$
Derive the capital motion equation by differentiating k = K/(AL) with respect to time using the quotient rule (dA/A = g, dL/L = n): k̇ = sk^α − (δ+n+g)k.
Step 4
$\dot{k} = s f(k) - (n + g + \delta)k$
Substitute $\dot{K} = sY - \delta K$ and recognise $Y/AL = k^\alpha$ . This single equation governs the entire dynamics of the model.
Step 5
$(\delta+n+g)k$
The term (δ+n+g)k is break-even investment: the share of output that must be invested to keep k constant as capital depreciates and the effective labour force grows. Any surplus raises k; any shortfall lowers it.