Update Auxiliary variables authored by Normann Rion's avatar Normann Rion
......@@ -112,7 +112,7 @@ Here is a proposed addition to the Dynare auxiliary variables wiki, describing *
## Type 15: Heterogeneous model multipliers (since Dynare 7)
This kind of auxiliary variable has `M_.aux_vars(i).type = 15` and is introduced to represent Lagrange multipliers associated with complementarity constraints in heterogeneous-agent models. These arise when an agent's decision is constrained by inequality bounds, such as borrowing constraints, portfolio limits, or non-negativity requirements.
Considerins heterogeneity dimension `j`, This kind of auxiliary variable has `M_.heterogeneity(j).aux_vars(i).type = 15` and is introduced to represent Lagrange multipliers associated with complementarity constraints in heterogeneous-agent models. These arise when an agent's decision is constrained by inequality bounds, such as borrowing constraints, portfolio limits, or non-negativity requirements.
When a policy variable `x` is subject to bounds `x ∈ [x_min, x_max]`, Dynare unfolds the complementarity condition into a pair of auxiliary variables and equations:
......@@ -137,8 +137,8 @@ u′(cₜ) = β · (1 + rₜ₊₁) · u′(cₜ₊₁) − MULT_L_a + MULT_U_a
where both the lower and upper multipliers appear with appropriate signs.
* `M_.aux_vars(i).endo_index` gives the position of the multiplier in `M_.endo_names`.
* `M_.aux_vars(i).eq_nbr` gives the index of the equation where the complementarity condition applies.
* `M_.heterogeneity(j).aux_vars(i).endo_index` gives the position of the multiplier in `M_.endo_names`.
* `M_.heterogeneity(j).aux_vars(i).eq_nbr` gives the index of the equation where the complementarity condition applies.
For example, suppose a household chooses `aₜ₊₁ ∈ [a_min, a_max]`. The preprocessor introduces:
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