Hi GAMS community!

I have a question about shadow prices on binding constraints.

In the model below, I am trying to simulate an entity’s decision on how to comply with a greenhouse gas emission limit law.

There are seven years in which this model is run. The entity can either choose to purchase one unit of offset at a cost of 3, or pay a penalty of 100 for exceeding the legal GHG emission limit, denoted by “emissionsCap.” Offsets expire yearly, so every subsequent year, entities must purchase new offsets. Each year, the entity emits 10 units (annualEmissions(t)). The cap (emissionsCap(t)) starts at 11 units and decreases by 1 unit per year. I’ve set the maximum number of offsets the entity can use to 3. So, I expect the entity to buy 0 offsets in year 1, but offsets in years 2, 3, and 4, and then pay a penalty in years 6 and 7.

When I calculate shadow prices on emissionsCap(t), I get [0, 0, 3, 3, **100,** 100, 100]. However, I expect in year 5 to get a shadow price of 3, rather than 100, since we are taking the shadow price by relaxing the constraint. When I relax the offset usage limit to 3.5 units, the model does as I expect, namely I get shadow prices of [0, 0, 3, 3, **3,** 100, 100].

Can someone help me understand what is going on here?

Here is my code below:

```
Sets:
t = [1, 2, 3, 4, 5, 6, 7];
Parameters:
Cost = 3;
Penalty = 100;
annualEmissions(t) = [10, 10, 10, 10, 10, 10, 10];
emissionsCap(t) = [11, 10, 9, 8, 7, 6, 5];
Variables:
Z;
Positive Variables:
offsetsOwned(t) Offsets owned in year t
offsetsLost(t) Expiration of offsets in time t
offsetsBought(t) Offsets bought in year t
overEmissions(t) Amount of emissions over the emissions cap fined $100/tCO2e
netEmissions(t) Entity emissions net of compliance options used
;
Equations:
obj.. Z =e= sum(t, Cost * offsetsOwned(t) + overEmissions(t) * penalty);
emissionCap(t).. overEmissions(t) + emissionsCap (t) =g= netEmissions(t);
netEmissionsDefinition(t).. netEmissions( t) =e= annualEmissions(t) - offsetsOwned(t);
complianceConstraint(t).. offsetsOwned(t) =e= offsetsOwned(t-1) + offsetsBought(t) – offsetsLost(t);
retirementLimit(t).. offsetsLost(t) =l= offsetsOwned(t-1);
uptake(t).. offsetsOwned(t) =l= 3;
Model sim / all /;
Solve sim using LP minimizing Z;
```