Section 2

Assumptions

The model keeps its environment intentionally spare. Fewer moving parts means the mechanism of capital accumulation stays transparent. The twelve assumptions below define the production side and the broader macroeconomic setting.

Production assumptions

Code	Assumption	Meaning
A1	Single good	The economy produces one homogeneous good used for both consumption and investment.
A2	Neoclassical production function	$Y = F(K, AL)$ satisfies constant returns to scale, positive but diminishing marginal products, and Inada conditions.
A3	Constant returns to scale	Double all inputs $\Rightarrow$ double output: $F(\lambda K, \lambda L) = \lambda F(K, L)$ .
A4	Positive marginal products	$\partial F/\partial K > 0$ and $\partial F/\partial L > 0$ .
A5	Diminishing marginal products	$\partial^2 F/\partial K^2 < 0$ and $\partial^2 F/\partial L^2 < 0$ .
A6	Inada conditions	$\lim_{K \to 0} \partial F/\partial K = \infty$ and $\lim_{K \to \infty} \partial F/\partial K = 0$ . These guarantee a unique, stable steady state.

A1–A6 describe the production environment.

Economy-wide assumptions

Code	Assumption	Meaning
A7	Closed economy	No trade; all output is consumed or invested domestically.
A8	Exogenous savings rate	Households save a fixed fraction $s \in (0,1)$ of income — no intertemporal optimisation.
A9	Exogenous population growth	Labour grows at constant rate $n \geq 0$ .
A10	Exogenous technology	Technology grows at constant rate $g \geq 0$ , not explained within the model.
A11	Constant depreciation	Capital depreciates at constant rate $\delta \in (0,1)$ per period.
A12	Competitive markets	Factors are paid their marginal products; profits are zero in equilibrium.

A7–A12 describe the macroeconomic environment.

Modelling shorthand

Output is produced by a Cobb-Douglas production function $Y = K^\alpha (AL)^{1-\alpha}$ . This has constant returns to scale overall but diminishing returns to capital alone. The savings curve $sf(k) = sk^\alpha$ is therefore concave and crosses the linear break-even line $(n+g+\delta)k$ exactly once.

A constant fraction $s$ of income is saved and invested each period; the remainder $(1-s)$ is consumed. Savings equals investment one-for-one (closed economy, no government). This gives a clean closed-form steady state.

Capital depreciates at rate $\delta$ . Population grows at rate $n$ and labour-augmenting technology grows at rate $g$ — both set outside the model. Together $\delta + n + g$ forms the break-even investment rate.

Labour markets clear at every point in time, so all workers are employed at the prevailing wage. This lets us normalise by effective labour $AL$ and work entirely in per-effective-worker units.