Solow

An economy whose capital per worker approaches its steady state through saving and depreciation.

• Name: solow
• Field: economics/macroeconomics
• Formalism: ordinary differential equations
• References: Solow 1956

Factory

@register_model(
    field='economics/macroeconomics',
    tags=('growth', 'ode'),
    references=('Solow 1956',),
    defaults={'until': 200.0, 'dt': 1.0},
)
def solow(k: float = 1.0, *, s: float = 0.2, alpha: float = 0.3, delta: float = 0.05) -> System:
    """Return an economy whose capital per worker approaches its steady state through saving and depreciation.

    Args:
        k:
            The initial capital per worker.

        s:
            The saving rate.

        alpha:
            The capital share of output.

        delta:
            The depreciation rate.

    Returns:
        system:
            The model, with the economy `economy`.

    Examples:
        >>> from skcomplex.models.economics.macroeconomics import solow
        >>> solow(s=0.3).interactions[0].s
        0.3
    """
    return System(
        elements=[EconomyElement('economy', k=k)],
        interactions=[SolowGrowthInteraction('growth', target='economy', s=s, alpha=alpha, delta=delta)],
    )

Components

class EconomyElement(Element):
    """Economy described by its capital per worker.

    Args:
        label:
            The label of the economy.

        visual:
            How the element is drawn, or `None` for `default_visual_`.

        k:
            The initial capital per worker.

    Examples:
        >>> from skcomplex.models.economics.macroeconomics import EconomyElement
        >>> EconomyElement('country', k=1.0).k
        1.0
    """

    def __init__(
        self,
        label: str,
        visual: VisualSpec | None = None,
        *,
        k: Annotated[float, StateSpec(domain=RealSet.closed_open(0, oo))] = 1.0,
    ) -> None:
        super().__init__(label, visual=visual)
        self.k = k

class SolowGrowthInteraction(RateInteraction):
    """Solow growth of the capital per worker `k` of a target, `dk/dt = s * k**alpha - delta * k`.

    A rate interaction on the state `k` whose rate is the saving of a Cobb-Douglas output minus depreciation.

    Args:
        label:
            The label of the interaction.

        target:
            The path of the economy.

        s:
            The saving rate.

        alpha:
            The output elasticity of capital.

        delta:
            The depreciation rate.

    Examples:
        >>> from skcomplex.models.economics.macroeconomics import SolowGrowthInteraction
        >>> SolowGrowthInteraction('solow', target='country', s=0.3).s
        0.3
    """

    def __init__(
        self,
        label: str,
        *,
        target: Annotated[str, RoleSpec()],
        s: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.2,
        alpha: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 0.3,
        delta: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 0.05,
    ) -> None:
        super().__init__(label, target=target, state='k')
        self.s = s
        self.alpha = alpha
        self.delta = delta

    def _get_rate(self, namespace: SimpleNamespace) -> Expression:
        """Return the rate `s * k**alpha - delta * k`.

        Args:
            namespace:
                The namespace of the symbols the rate can use, with the namespace of the target.

        Returns:
            rate:
                The rate of the capital per worker.
        """
        k = namespace.target.k
        rate: Expression = namespace.s * k**namespace.alpha - namespace.delta * k
        return rate

Equations

economy.k: diff(k_economy(t), t) == s_growth*k_economy(t)^alpha_growth - delta_growth*k_economy(t)