Lagrange multiplier sign for the binding inequality constraint

Dear all,

I have an inequality constraint (capital adequacy for the bank with endogenous dividends payoff policy) in the NK model. While solving for the steady state and solving the DSGE model I assume the constraint always binds. Things seem to work with no issues.

However, the Lagrange multiplier associated with this constraint is negative in the steady state (all other multipliers are positive).

  1. As far as I am concerned, there is no sign restriction for equality constraint, but if inequality constraint binds, the multiplier should be strictly positive. Hence, which case applies here: binding inequality constraint or equality constraint?

  2. Moreover, I have read about “numerical error” in the context of negative multipliers. Could someone please clarify what it means? Might it mean that the steady state I have is not the correct one?

Looking forward to any piece of advice!

If you satisfy the usual setup, the multiplier indeed needs to be positive. That’s what the Karush-Kuhn-Tucker theorem tells us. However, it’s easy to get a negative multiplier by mixing up the signs or wrongly expressing the constraint. See Karush–Kuhn–Tucker conditions - Wikipedia for the right setup.

Numerical error is highly unlikely unless you multiplier is really tiny.

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May I please clarify: assuming binding inequality constraint implies strictly positive multiplier?
Why then steady state is found and stochastic simulations are performed with no problems detected by Dynare?

Yes, for an inequality constraint of the form g(x)\leq 0 the multiplier should be positive in steady state if the constraint is binding .


So from Dynare performing simulations with no problems we can infer nothing about the appropriateness of the steady state? It still might be incorrect?

You only know that the steady state solves the entered equations. However, if your equations are correct and your model has multiple steady states, then you may have selected the wrong one.