### Local sensitivity analysis at optimum

Posted:

**Tue Sep 05, 2017 5:56 pm**Dear all,

I would like to study the impact of a small variation of the optimal variables in an equlibrium problem.

My question is related to this older thread Jacobian at optimal condition. It is explained there how to obtain the Jacobian matrix using the Convert utility of GAMS (at the example of the weapons assignment problem).

As I am interested in a regional equilibrium, I started with the example speq1 presented here by Erwin Kalvelagen. The example includes three regions for which the supply and demand price is calculated. Thanks to the approach described for the weapons problem, I can also get the Jacobian matrix after solving the spatial equilibrium problem with Conopt. As far as I can see, this matrix (as well as the column listing) presents the sensitivities for all variables included in each equation. For instance, for the supply s and the supply price p, it shows dp1/ds1, dp2/ds2 and dp3/ds3.

However, is there a way to estimate the impact of changing variables not directly included in a specific equation? For instance, the change of the supply price in region 2 after an increase of the demand in region 1? Basically, I am hoping for something like directional derivatives (f1/x1, f1/x2, ..) but I am not sure if my idea makes sense for this problem.

Any suggestion would be really appreciated.

Best regards

I would like to study the impact of a small variation of the optimal variables in an equlibrium problem.

My question is related to this older thread Jacobian at optimal condition. It is explained there how to obtain the Jacobian matrix using the Convert utility of GAMS (at the example of the weapons assignment problem).

As I am interested in a regional equilibrium, I started with the example speq1 presented here by Erwin Kalvelagen. The example includes three regions for which the supply and demand price is calculated. Thanks to the approach described for the weapons problem, I can also get the Jacobian matrix after solving the spatial equilibrium problem with Conopt. As far as I can see, this matrix (as well as the column listing) presents the sensitivities for all variables included in each equation. For instance, for the supply s and the supply price p, it shows dp1/ds1, dp2/ds2 and dp3/ds3.

However, is there a way to estimate the impact of changing variables not directly included in a specific equation? For instance, the change of the supply price in region 2 after an increase of the demand in region 1? Basically, I am hoping for something like directional derivatives (f1/x1, f1/x2, ..) but I am not sure if my idea makes sense for this problem.

Any suggestion would be really appreciated.

Best regards