GAMS Reader -
Solution for the exercises
II.1. Results for Introductory level
Results
from 2.1.1.
Your desktop should look similar to this:
Results from 2.1.2.
Results from 2.1.3.
After running the model, the GAMS-IDE should look similar to
this[1]:
Results from 2.1.4.
In our simple example, the report on the variables should be as follows :
LOWER LEVEL UPPER MARGINAL
---- VAR
X1 -INF 3.0000
+INF .
---- VAR
X2 -INF 5.0000 +INF .
---- VAR Y -INF 8.0000 +INF .
X1 Explanatory text on the meaning of X1
X2 Explanatory text on the meaning of X2
Y Explanatory text on the meaning of Y
Therefore, the optimal values for variables X1, X2 and Y are 3, 5 and 8 respectively. Note that none of the variables has an lower or upper bound. INF means infinity, so the variables can take infinitely large and small values. The marginal value of all variables is zero, which confirms that the solution found is optimal.
Results from 2.1.5.
Removing the semi-colon from equation QX1 results in the following window:
At the end of the listing file the list of
errors are explained:
“409 Unrecognizable item - skip to find a new statement; looking for a ';' or a key word to get started again” is displayed.
Results from 2.1.6.
The problem is UNBOUNDED, because you can keep lowering the value of Y by lowering X1. Remember that X1 should be less than A, not equal to. So X1 can get any value below 3. Consequently, Y can get any value below 8. The minimum is then minus infinity; GAMS reports this as unbounded.
The listing file gives the following solve summary:
**** SOLVER STATUS 1 NORMAL COMPLETION
**** MODEL STATUS 3 UNBOUNDED
Results from 2.1.7.
The results for the variables when the tax rate (tax) equals 30 is:
LOWER LEVEL UPPER MARGINAL
VAR
DAMAGE -INF 6000.0000
+INF .
VAR ST_RATE -INF 0.5000 +INF .
VAR CATTLE -INF 500.0000 +INF .
VAR OBJ -INF 4000.0000 +INF .
For tax=20, obj equals 6000; for tax=10, obj equals 6000 and for tax=0, obj equals 4000; the optimal value of tax is 15, when obj equals 6250.
Make sure that you use the explanatory text.
Results from 2.1.8.
The values of X1 and Y should be identical to the earlier exercises: 3 and 8, respectively.
Did you think of removing the equation declaration and the equation definition for QX2?
The full model for emissions can look like this:
PARAMETERS
coef Emission coefficient CO2
OTHER Emissions of other greenhouse gasses;
Coef=0.03;
OTHER=5;
VARIABLES
CO2 Emissions of CO2
PRD Production quantity
EMIS Total emissions of greenhouse gasses;
EQUATIONS
QPRD Equation for economic production
QCO2 Equation for CO2 emissions
QEMIS Equation for total climate emissions;
QPRD.. PRD =G= 100;
QCO2.. CO2 =E= coef*PRD;
QEMIS.. EMIS =E= CO2+OTHER;
MODEL CLIMATE /ALL/;
SOLVE CLIMATE USING DNLP MINIMIZING EMIS;
With solution:
LOWER LEVEL UPPER MARGINAL
---- VAR EMIS -INF 8.000 +INF .
---- VAR CO2 -INF 3.000 +INF .
---- VAR PRD -INF 100.000 +INF .
II.2. Results for Intermediate level
Results
from 2.2.1
At the end of the listing file, the following lines have been added by GAMS:
VARIABLE EMIS.L = 8.000 Total emissions of greenhouse gasses
PARAMETER OTHER = 5.000 Emissions of other greenhouse gasses
Results from Exercise 2.2.2. Commenting out a single line
The line with the comment does not change the results.
Results from Exercise 2.2.3. Using SCALARS
SCALARS
coef emission coefficient /0.03/
OTHER other emissions /5/;
The result does not change.
Results from Exercise 2.2.4. Solving more than one model
As the value of OTHER has increased from 5 to 7, we expect that the value of RES2 equals RES1+2. The display statement gives:
PARAMETER RES1 = 8.000
PARAMETER RES2 = 10.000
The other values in the model have not changed (PRD and CO2 are unaffected).
Remember to check for each solve whether GAMS has found an optimal solution!
Results from 2.2.5
The display statement changes to:
---- 31 PARAMETER RES
1st solve 8.000, 2nd solve 10.000
Results from 2.2.6
In the solution listing, the solution for CO2 is not influenced, but PRD is changed:
LOWER LEVEL UPPER MARGINAL
---- VAR CO2 -INF 3.0000 +INF .
CO2 Emissions of CO2
---- VAR PRD Production quantity
LOWER LEVEL UPPER MARGINAL
1 -INF 10.0000
+INF .
2 -INF 50.0000
+INF .
3 -INF 40.0000 +INF .
Results from Exercise 2.2.7. Summing over an index
No changes in the results.
Results from 2.2.8
For each year 2000 to 2004 the same values result in the model:
---- 45 PARAMETER RES
2000 2001 2002 2003 2004
1st solve 8.000 8.000 8.000 8.000 8.000
2nd solve 10.000 10.000 10.000 10.000 10.000
The new variable TOTEMIS equals 40 in the first solve and 50 in the second solve.
Results from 2.2.9
The values of variable PRD changes, as does the parameter RES (and for instance also CO2 and CONCENT).
---- VAR PRD Production
quantity
LOWER
LEVEL UPPER MARGINAL
1.2000
-INF 10.0000 +INF .
1.2001
-INF 11.0000 +INF .
1.2002
-INF 12.0000 +INF .
1.2003
-INF 13.0000 +INF .
1.2004
-INF 14.0000 +INF .
2.2000
-INF 50.0000 +INF .
2.2001
-INF 52.0000 +INF .
2.2002
-INF 54.0000 +INF .
2.2003
-INF 56.0000 +INF .
2.2004
-INF 58.0000 +INF .
3.2000
-INF 40.0000 +INF .
3.2001 -INF 42.0000
+INF .
3.2002
-INF 44.0000 +INF .
3.2003
-INF 46.0000 +INF .
3.2004 -INF 48.0000 +INF .
---- 55 PARAMETER RES
2000 2001 2002
2003 2004
1st solve 8.000 8.150 8.300
8.450 8.600
2nd solve 10.000 10.150 10.300
10.450 10.600
Results from 2.2.10
The model results do not change, but you can see the difference in the solution for PRD:
---- VAR PRD Production
quantity
LOWER LEVEL UPPER
MARGINAL
1.2000
. 10.0000 +INF .
1.2001 . 11.0000
+INF .
1.2002
. 12.0000 +INF .
1.2003
. 13.0000 +INF .
1.2004
. 14.0000 +INF .
2.2000
. 50.0000
+INF .
2.2001
. 52.0000 +INF .
2.2002
. 54.0000 +INF .
2.2003
. 56.0000 +INF .
2.2004
. 58.0000 +INF .
3.2000
. 40.0000 +INF .
3.2001
. 42.0000 +INF .
3.2002
. 44.0000 +INF .
3.2003
. 46.0000
+INF .
3.2004
. 48.0000 +INF .
The lower bounds are no longer “-INF”, but a dot is given. This dot indicates zero.
Results from 2.2.11
If your model behaves normally, the starting values do not influence the results. So in this simple example, nothing changes.
Results from 2.2.12
Note that the lower and upper bound go on PRD (the variable) and not on PRD_DATA (the parameter); you cannot put bounds on parameters, as their values are already fixed.
If you put the upper value on PRD(“2”,T) at a level below the value in PRD_DATA(“2”,T), then the model becomes infeasible. The model does not make any sense anymore: the value cannot be smaller than 55 and larger than PRD_DATA(“2”,T) at the same time. So the upper bound on PRD(“2”,T) should be removed.
The solution listing for PRD for the first simulation contains:
---- VAR PRD Production quantity
LOWER LEVEL UPPER MARGINAL
1.2000 12.0000 12.0000
+INF 0.0300
1.2001 12.0000 12.0000
+INF 0.0300
1.2002 12.0000 12.0000
+INF .
1.2003 12.0000 13.0000
+INF .
1.2004 12.0000 14.0000
+INF .
2.2000 . 50.0000
+INF .
2.2001 . 52.0000
+INF .
2.2002 . 54.0000
+INF .
2.2003 . 56.0000 +INF .
2.2004 . 58.0000
+INF .
3.2000 . 40.0000
+INF .
3.2001 . 42.0000
+INF .
3.2002 . 44.0000 +INF .
3.2003 . 46.0000
+INF .
3.2004 . 48.0000
+INF .
And the display statement at the end delivers:
---- 63 PARAMETER RES
2000 2001 2002 2003 2004
1st solve 8.060 8.180 8.300 8.450 8.600
2nd solve 10.060 10.180 10.300 10.450 10.600
So the lower bound on PRD(“1”,T) affects the results for 2000 and 2001. Annual emissions, as reported in parameter RES are 0.06 and 0.03 higher than without the bounds.
Results from Exercise 2.2.13. Fixing variables
Fixing the level of PRD(“3”,”2000”) to 45 in the first solve will affect production in 2000 and consequently also emissions (and concentration).
---- VAR PRD Production quantity
LOWER LEVEL UPPER MARGINAL
1.2000 12.0000 12.0000
+INF 0.0300
1.2001 12.0000 12.0000 +INF 0.0300
1.2002 12.0000 12.0000
+INF .
1.2003 12.0000 13.0000
+INF .
1.2004 12.0000 14.0000
+INF .
2.2000 . 50.0000 +INF .
2.2001 . 52.0000
+INF .
2.2002 . 54.0000
+INF .
2.2003 . 56.0000
+INF .
2.2004 . 58.0000
+INF .
3.2000 45.0000 45.0000
45.0000 0.0300
3.2001 . 42.0000
+INF .
3.2002 . 44.0000
+INF .
3.2003 . 46.0000 +INF .
3.2004 . 48.0000 +INF .
---- 66 PARAMETER RES
2000 2001 2002 2003 2004
1st solve 8.210 8.180 8.300 8.450 8.600
2nd solve 10.060 10.180 10.300 10.450 10.600
Results from 2.2.14
Nothing changes, you just wrote the code in a different way.
Results from 2.2.15
From the solution listing of the first solve, it shows:
---- VAR DAM Damages from climate change in terms of production losses
LOWER LEVEL UPPER MARGINAL
1.2000 -INF 0.0500
+INF .
1.2001 -INF 0.0500
+INF .
1.2002 -INF 0.0500
+INF .
1.2003 -INF 0.0500
+INF .
1.2004 -INF 0.0500
+INF .
2.2000 -INF 0.0500
+INF .
2.2001 -INF 0.0500
+INF .
2.2002 -INF 0.0500
+INF .
2.2003 -INF 0.0500
+INF .
2.2004 -INF 0.0500
+INF .
3.2000 -INF 0.0500 +INF .
3.2001 -INF 0.0500
+INF .
3.2002 -INF 0.0500
+INF .
3.2003 -INF 0.0500
+INF .
3.2004 -INF 0.0500 +INF .
The display statement shows:
---- 68 PARAMETER RES
2000 2001 2002 2003 2004
1st solve 8.135 8.039 8.153 8.277 8.420
2nd solve 9.925 10.039 10.153 10.277 10.420
As a result of the damages, production goes down; this will lead to lower emissions and concentrations. Emissions are not exactly 5 percent lower than in the previous exercise, as the bounds on the variables also influence the results.
Using the conditional statement on the damage function, damages will be:
---- VAR DAM Damages from climate change in terms of production losses
LOWER LEVEL UPPER MARGINAL
1.2000 -INF 0.0500
+INF .
1.2001 -INF 0.0500
+INF .
1.2002 -INF 0.0500
+INF .
1.2003 -INF 0.0500
+INF .
1.2004 -INF 0.0500 +INF .
2.2000 -INF 0.0500
+INF .
2.2001 -INF 0.1000
+INF .
2.2002 -INF 0.1000
+INF .
2.2003 -INF 0.1000 +INF .
2.2004 -INF 0.1000
+INF .
3.2000 -INF 0.0500
+INF .
3.2001 -INF 0.0500
+INF .
3.2002 -INF 0.0500
+INF .
3.2003 -INF 0.0500
+INF .
3.2004 -INF 0.0500 +INF .
---- 68 PARAMETER RES
2000 2001 2002 2003 2004
1st solve 8.135 7.961 8.072 8.193 8.333
2nd solve 9.925 9.961 10.072 10.193 10.333
The higher damages will strengthen the effects on emissions.
Results from 2.2.16
Sector-specific emission coefficients:
---- 71 PARAMETER RES
2000 2001 2002 2003 2004
1st solve 8.145 7.960 8.047 8.136 8.229
2nd solve 9.970 9.960 10.047 10.136 10.229
Remember that the sector specific coef is also part of the equation CO2. COEF should be moved inside the SUM.
Changing the conditional statement for damages leads to:
---- VAR DAM Damages from climate change in terms of production losses
LOWER LEVEL UPPER MARGINAL
1.2000 -INF 0.0500
+INF .
1.2001 -INF 0.0500
+INF .
1.2002 -INF 0.0500
+INF .
1.2003 -INF 0.0500
+INF .
1.2004 -INF 0.0500
+INF .
2.2000 -INF 0.0500
+INF .
2.2001 -INF 0.0500
+INF .
2.2002 -INF 0.1000
+INF .
2.2003 -INF 0.1000
+INF .
2.2004 -INF 0.1000
+INF .
3.2000 -INF 0.0500
+INF .
3.2001 -INF 0.0500
+INF .
3.2002 -INF 0.0500
+INF .
3.2003 -INF 0.0500
+INF .
3.2004 -INF 0.0500 +INF .
---- 71 PARAMETER RES
2000 2001 2002 2003 2004
1st solve 8.145 8.063 8.047 8.136 8.229
2nd solve 9.970 10.063 10.047 10.136 10.229
Results
from 2.2.17
The model status:
L O O P S SOL 1st solve
S O L V E S U M M A R Y
MODEL CLIMATE OBJECTIVE CONCENT
TYPE DNLP DIRECTION MINIMIZE
SOLVER MINOS5 FROM LINE 62
**** SOLVER STATUS 1 NORMAL COMPLETION
**** MODEL STATUS 2 LOCALLY OPTIMAL
**** OBJECTIVE VALUE 39.6216
RESOURCE USAGE, LIMIT 0.109 1000.000
ITERATION COUNT, LIMIT 15 10000
EVALUATION ERRORS 0 0
The results:
---- 71 PARAMETER RES
2000 2001 2002 2003 2004
1st solve 8.050 7.915 7.805 7.884 7.967
2nd solve 9.827 9.915 9.805 9.884 9.967
Results from 2.2.18
---- 28 PARAMETER INDEX Straight line from zero to one
2001 0.250, 2002 0.500, 2003 0.750, 2004 1.000
Note that the solution listing does not display zero values; hence INDEX(“2000”) is not displayed.
Results from 2.2.19
In all cases, the optimal value for alpha is 0.8, which is equal to the upper bound. This means that GAMS gives the lowest possible weight to OTHER emissions. As these exceed the values of CO2 emissions, this was to be expected: GAMS tries to find the lowest combined emissions and adjusts the value of alpha to that purpose.
Solving Exercise 2.1.7. with free variable:
LOWER LEVEL UPPER MARGINAL
---- VAR TAX -INF 15.000 +INF .
---- VAR
DAMAGE -INF 37500.000 +INF .
---- VAR ST_RATE -INF 1.250 +INF .
---- VAR CATTLE -INF 1250.000 +INF EPS
---- VAR OBJ -INF 6250.000 +INF .
The
optimal tax rate is 15.
---- 63
PARAMETER check Auxiliary parameter to
check matching totals
( ALL
0.000 )
---- 63 PARAMETER smalldataset The aggregated dataset (usable input)
agri industry services
cons Col_tot
agri 5.000 45.000 50.000
industry
15.000 190.000 30.000 115.000 350.000
services
10.000 40.000 370.000 280.000 700.000
primary_in
20.000 120.000 300.000 440.000
Row_tot
50.000 350.000 700.000 440.000
Results
from 2.2.21
Did you remember to change the set SOL? The command OTHER(T)=7 in the loop should be changed to OTHER(T)=OTHER(T)+2.
---- 70 PARAMETER RES
2000 2001 2002 2003 2004
1st solve 8.050 7.915 7.805 7.884 7.967
2nd solve 9.827 9.915 9.805 9.884 9.967
3rd solve 11.827 11.915 11.805 11.884 11.967
The model focuses on the build-up of greenhouse gasses stemming from production. One could use the model to analyse the relationship between production quantities per sector and the building-up of greenhouse gas emissions. The inclusion of damages makes this model more realistic, as higher production values lead to higher damages, which in turn will destroy some produced goods. The relationship between damages and concentrations is that damages reduce production and hence reduce emissions of CO2. Consequently, concentrations of greenhouse gasses also go down. But there are a few problems: in this model, production plays only a negative role as it causes emissions, but in reality, production will increase consumption and hence utility and is thus a ‘good’ thing. Then, if production is ‘good’, damages are ‘bad’, while they are only ‘good’ in this model. But either good or bad, damages will decrease emissions.
You can make the model more realistic by including emission reductions through abatement, by including and maximising consumption subject to a present concentration maximum, et cetera.