Steady-state¶

The deterministic state of a model corresponds to steady-state values \(\overline{m}\) of the exogenous process. States and controls satisfy:

\(\overline{s} = g\left(\overline{m}, \overline{s}, \overline{x}, \overline{m} \right)\)

\(0 = \left[ f\left(\overline{m}, \overline{s}, \overline{x}, \overline{m}, \overline{s}, \overline{x} \right) \right]\)

where \(g\) is the state transition function, and \(f\) is the arbitrage equation. Note that the shocks, \(\epsilon\), are held at their deterministic mean.

The steady state function consists in solving the system of arbitrage equations for the steady state values of the controls, \(\overline{x}\), which can then be used along with the transition function to find the steady state values of the state variables, \(\overline{s}\).

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def residuals(model: Model, calib=None) -> Dict[str, List[float]]:

    if calib is None:
        calib = model.calibration

    res = dict()

    m = calib["exogenous"]
    s = calib["states"]
    x = calib["controls"]
    p = calib["parameters"]
    f = model.functions["arbitrage"]
    g = model.functions["transition"]

    res["transition"] = g(m, s, x, m, p) - s
    res["arbitrage"] = f(m, s, x, m, s, x, p)

    return res