Steady State Computation & Verification¶

In dynamic economic modeling, the deterministic steady state represents the long-run stationary equilibrium where all time indices collapse (\(y_{t-1} = y_t = y_{t+1} = \bar{y}\)) and all stochastic shocks are set to zero (\(\varepsilon_t = 0\)).

Solving for the steady state is a necessary prerequisite for first-order perturbation around the steady-state point.

Analytical Steady States ([~])¶

Whenever possible, defining an exact closed-form analytical steady state is fastest and most reliable:

# Analytical steady state for Neoclassical Growth Model
z[~] <- 0.0
k[~] <- ((1/β - (1-δ)) / α)**(1 / (α-1))
y[~] <- k[~]^α
i[~] <- δ * k[~]
c[~] <- y[~] - i[~]

Steady-state definitions:

• Can reference model parameters (α, β, δ).
• Can reference previously defined steady-state values (e.g., y[~] references k[~]).
• Are evaluated in topological order during model loading.

Verifying Steady States¶

Dyno provides tools to verify that declared steady states satisfy the dynamic system:

1. model.residuals¶

Evaluates the absolute difference \(|LHS - RHS|\) for each equation at the current steady-state point:

from dyno import DynoModel

model = DynoModel("examples/neo.dyno")

for i, res in enumerate(model.residuals, start=1):
    print(f"Eq {i} residual: {res:.2e}")

2. model.check()¶

Checks whether any residual exceeds the numerical tolerance (default: \(10^{-6}\)). If so, it raises a SteadyStateError:

from dyno.errors import SteadyStateError

try:
    model.check()
    print("Steady state is verified!")
except SteadyStateError as e:
    print("Residual mismatch detected:")
    print(e)

Numerical Steady State Finding (model.steady())¶

For complex models where an analytical steady state is intractable, Dyno provides an automated numerical root finder.

How it Works¶

1. Specify initial guesses in your .dyno file using [~]:

   # Initial guesses
   k[~] <- 5.0
   c[~] <- 1.0
   y[~] <- 1.0
   

2. Call model.steady() in Python:

# Solves f(y_bar, y_bar, y_bar) = 0 numerically
model = model.steady()

# Verify convergence
model.check()
print("Computed Steady State:", model.steady_state)

model.steady() uses SciPy's non-linear root-finding algorithms (hybr / Powell hybrid method) to solve the stationary system of equations simultaneously.

Steady State Diagnostics and Error Handling¶

If the numerical solver fails to converge, Dyno provides diagnostic information:

• Unmatched variables: Variables that appear in dynamic equations but lack initial guesses.
• Equation residuals: Which equations are furthest from equilibrium.
• Bounds or scale issues: When variables take invalid values (e.g., negative capital or zero consumption inside a logarithm).