Langrange multipliers/shadow prices

[steve_1983]

Langrange multipliers are an integral part of dsge models. Theoretically, they are always positive. But in dynare model section, we do not introduce anything to restrict the values of these multipliers to positive values. Does that imply that while we can ensure that their steady state values are positive, they can squander away into the negative territory when the model is away from the steady state? I have often see the langrange multipliers moving from negative to positive in the IRFs, hence the question.

[jpfeifer]

That comes from doing a polynomial approximation. See e.g.

Negative numbers

That is because a first-order approximation approximates exp(x) with a linear function exp(x_ss)*(x-x_ss), where x_ss is the steady state. Of course, a linear function will always hit negative values with some probability (as long as its slope is not 0). Using a second-order approximation means fitting a parabola to exp(x), thus preserving some of the curvature. This may avoid having negative values if the vertex is above 0 (but for extreme shocks does not preserve monotonicity of exp(x)). Th…