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Spatial Equilibrium

July 20, 2021, 10:20 am
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Just a quick experiment with a small price-endogenous spatial equilibrium model.

In this problem, we consider commodities (goods) produced and consumed in different regions. We can trade between regions.

This price-endogenous spatial model will find both equilibrium prices, supplies, demand quantities, and trade patterns. Our model will have a connection with the transportation model. The difference is that we formulate the model as a system of complementarity constraints (no objective), with the duals explicitly in the model as variables. Or, in other words, we solve the KKT conditions.

I use a small data set from [3], and see if we can reproduce things.


Demand and supply functions


In the simplest form, we assume we have estimated linear supply and demand curves as follows: \[\begin{align}&\color{darkred}{Q}^{\color{darkred}d}_{c,r} = \color{darkblue}a_{c,r} +   \color{darkblue}b_{c,r}\color{darkred}{P}_{c,r}\\ &\color{darkred}{Q}^{\color{darkred}s}_{c,r} = \color{darkblue}\alpha_{c,r} +   \color{darkblue}\beta_{c,r}\color{darkred}{P}_{c,r}\end{align}\] The means, the demand and supply quantities depend on the price, The indices \(c,r\) indicate the commodity and region. 

Notes:

  • Demand/supply functions are typically of the form \(\color{darkred}Q=f(\color{darkred}P)\). Some models use the inverse demand and supply functions: \(\color{darkred}P=f^{-1}(\color{darkred}Q)\). Of course, when solved as a system of equations, this does not make a difference: we can use either form.
  • We can assume the demand function is downward sloping (higher prices mean less demand) and the supply function is upward sloping (higher prices mean more supply). Normal goods all follow this. There are only very rare cases where this is not the case. Examples of goods with upward-sloping demand curves are Giffen and Veblen goods [1].
  • The demand and supply functions can be a little bit more complicated by also considering the prices of other goods. E.g. the supply of beef may depend on the price of beef but also on the price of feed grains. For the latter, we would expect that if feed prices increase, the supply of beef decreases. We can write in general:\[\begin{align}&\color{darkred}{Q}^{\color{darkred}d}_{c,r} = \color{darkblue}a_{c,r} +   \sum_{c'}\color{darkblue}b_{c,r,c'}\color{darkred}{P}_{c',r}\\ &\color{darkred}{Q}^{\color{darkred}s}_{c,r} = \color{darkblue}\alpha_{c,r} +   \sum_{c'}\color{darkblue}\beta_{c,r,c'}\color{darkred}{P}_{c',r}\end{align}\] This is the form [3] is using.
  • A possible way to find values for the coefficients of these functions is to use an econometric analysis (regression). When we don't have enough data for that, another approach can be used called calibration.

Transportation cost


When shipping goods from region \(r\) to \(r'\), we assume there are transportation costs. A simple form is just to use constant unit shipping costs \(\color{darkblue}{tc}_{c,r,r'}\). There are some interesting cases to consider [2]:

  • \(\color{darkred}{P}_{c,r}+\color{darkblue}{tc}_{c,r,r'}\lt \color{darkred}{P}_{c,r'}\). This means there is a profit to be made by shipping more from \(r \rightarrow r'\). That means we are not in equilibrium: we will start shipping leading to other prices and quantities. Or stated differently: we are violating the zero-profit equilibrium condition.
  • \(\color{darkred}{P}_{c,r}+\color{darkblue}{tc}_{c,r,r'}\gt \color{darkred}{P}_{c,r'}\). This means that the price differential is not large enough to pay for the shipping cost. The result should be: no shipping from \(r \rightarrow r'\), or \(\color{darkred}x_{c,r,r'}=0\) where \(\color{darkred}x_{c,r,r'}\) indicate our trade flows.
  • \(\color{darkred}{P}_{c,r}+\color{darkblue}{tc}_{c,r,r'}= \color{darkred}{P}_{c,r'}\). This case allows for shipping: \(\color{darkred}x_{c,r,r'}\ge 0\).

Altogether we can now write: \[\color{darkred}{P}_{c,r}+\color{darkblue}{tc}_{c,r,r'}\ge \color{darkred}{P}_{c,r'} \perp \color{darkred}x_{c,r,r'} \ge 0\] This is notation for a complementarity condition. It means:
  • \(\color{darkred}x_{c,r,r'}\gt 0 \Rightarrow \color{darkred}{P}_{c,r}+\color{darkblue}{tc}_{c,r,r'} = \color{darkred}{P}_{c,r'} \) and
  • \( \color{darkred}{P}_{c,r}+\color{darkblue}{tc}_{c,r,r'} \gt \color{darkred}{P}_{c,r'} \Rightarrow \color{darkred}x_{c,r,r'}= 0\)
You may recognize this as complementarity from your linear programming textbook. In that context we would say \(\color{darkred}x_{c,r,r'}\) is the dual of the constraint \(\color{darkred}{P}_{c,r}+\color{darkblue}{tc}_{c,r,r'} \ge \color{darkred}{P}_{c,r'}\).  There we would say: if the dual \(\color{darkred}x_{c,r,r'}\gt 0 \), it is non-basic, and the constraint will be binding. And if the constraint is non-binding, the dual will be basic and \(\color{darkred}x_{c,r,r'} = 0\). Here we derived the same thing using economics.

The equilibrium conditions for the transportation cost are interesting and intuitive. 


Other equilibrium conditions


The other constraints are implemented as equality constraints. That is a bit of a shortcut: we can also implement them as proper equilibrium (complementarity) conditions. I believe the author of [1] meant these constraints always to be binding.

The complete model looks like:

Spatial Equilibrium Model
Sets
\(c,c'\)Commodities
\(r,r'\)Regions
Variables
\(\color{darkred}{Q}^{\color{darkred}s}_{c,r}\)Supply quantities
\(\color{darkred}{Q}^{\color{darkred}d}_{c,r}\)Demand quantities
\(\color{darkred}{P}_{c,r}\)Prices
\(\color{darkred}{x}_{c,r,r'}\)Shipped quantities
Equations
\[\color{darkred}{Q}^{\color{darkred}s}_{c,r} = \color{darkblue}\alpha_{c,r} + \sum_{c'}\color{darkblue}\beta_{c,r,c'}\color{darkred}{P}_{c',r}\]Supply equation
\[\color{darkred}{Q}^{\color{darkred}d}_{c,r} = \color{darkblue}a_{c,r} + \sum_{c'}\color{darkblue}b_{c,r,c'}\color{darkred}{P}_{c',r}\]Demand equation
\[\color{darkred}{Q}^{\color{darkred}s}_{c,r} + \sum_{r'}\color{darkred}x_{c,r',r}= \color{darkred}{Q}^{\color{darkred}d}_{c,r} + \sum_{r'}\color{darkred}x_{c,r,r'}\]Flow balance (production + import = consumption + export)
\[\color{darkred}{P}_{c,r} + \color{darkblue}{tc}_{c,r,r'} \ge \color{darkred}{P}_{c,r'} \perp \color{darkred}x_{r,r',c}\ge 0\]Transportation
\[\color{darkred}{x}_{c,r,r} = 0\]Exclude diagonal


This is not an optimization model but rather a (linear) complementarity model: a square system of equations with complementarity conditions. We need to solve it with an appropriate solver. These models can also be solved using Quadratic Programming (QP) solver [4].

Data


To try out this model, I used the small data set from [3]. 




I organized this as:

----     84 PARAMETER coeffcoefficients for demand and supply functions

INDEX 1 = Demand

                       Intercept       Wheat  FeedGrains        Beef

Wheat     .US          15364.000-4.960
Wheat     .EEC         14383.000-3.400
Wheat     .UKIreland    4427.000-1.290
Wheat     .Other       31731.000
FeedGrains.US         140556.000-1224.00048.130
FeedGrains.EEC         31694.000-157.00010.680
FeedGrains.UKIreland   12720.000-63.0005.820
Beef      .US          14190.000-6.950
Beef      .EEC          6830.000-3.290
Beef      .UKIreland    2178.000-1.420

INDEX 1 = Supply

                       Intercept       Wheat  FeedGrains        Beef

Wheat     .US          18520.000427.000-190.000
Wheat     .EEC         12655.000213.000-74.000
Wheat     .UKIreland    1441.00030.000-8.000
FeedGrains.US          98438.000-694.0002163.000
FeedGrains.EEC         16531.000-85.000207.000
FeedGrains.UKIreland    5446.000-35.00068.000
FeedGrains.Other        2479.000
Beef      .US           6536.000-84.0005.900
Beef      .EEC          3235.000-24.0002.900
Beef      .UKIreland     808.000-6.0001.000
Beef      .Other         584.000


----     84 PARAMETER transCostTransportation Cost Matrix

                              US         EEC   UKIreland       Other

Wheat     .US                          8.4388.03015.000
Wheat     .EEC             8.4380.42015.000
Wheat     .UKIreland       8.0300.42015.000
Wheat     .Other          15.00015.00015.000
FeedGrains.US                          8.4388.03015.000
FeedGrains.EEC             8.4380.42015.000
FeedGrains.UKIreland       8.0300.42015.000
FeedGrains.Other          15.00015.00015.000
Beef      .US                         84.80280.702100.000
Beef      .EEC            84.8024.221100.000
Beef      .UKIreland      80.7024.221100.000
Beef      .Other         100.000100.000100.000
Grain     .US                          8.4388.03015.000
Grain     .EEC                                     0.42015.000
Grain     .UKIreland                                          15.000


This leads to a square system of equations with 72 variables and 72 equations.

Reproducibility


The solution reported in [3] is a bit sloppy. It is somewhat infeasible. 



When I set my initial values to this solution, and solve with iteration limit 0 (that means: just check the initial values, don't solve), we see the following. First I look at the solution. There are infeasibilities related to the demand and supply functions:

---- EQU supplysupply function

                            LOWER          LEVEL          UPPER         MARGINAL

US       .Wheat          18520.000018517.780018520.000039048.0000 INFES
US       .FeedGrains     98438.000098451.150098438.0000143757.0000 INFES
US       .Beef            6536.00006535.33906536.00007854.0000 INFES
EEC      .Wheat          12655.000012653.900012655.000023152.0000 INFES
EEC      .FeedGrains     16531.000016531.510016531.000021370.0000 INFES
EEC      .Beef            3235.00003234.93103235.00004203.0000 INFES
UKIreland.Wheat           1441.00001441.28001441.00003059.0000 INFES
UKIreland.FeedGrains      5446.00005446.02005446.00006519.0000 INFES
UKIreland.Beef             808.0000808.1700808.00001256.0000 INFES
Other    .Wheat               .              .              .              .          
Other    .FeedGrains      2479.00002479.00002479.00002479.0000
Other    .Beef             584.0000584.0000584.0000584.0000

---- EQU demand  demand function

                            LOWER          LEVEL          UPPER         MARGINAL

US       .Wheat         -15364.0000-15364.1216-15364.000015032.0000 INFES
US       .FeedGrains   -140556.0000-140550.4133-140556.0000128448.0000 INFES
US       .Beef          -14190.0000-14189.7505-14190.00008438.0000 INFES
EEC      .Wheat         -14383.0000-14382.6640-14383.000014155.0000 INFES
EEC      .FeedGrains    -31694.0000-31693.2192-31694.000031768.0000 INFES
EEC      .Beef           -6830.0000-6830.3249-6830.00004347.0000 INFES
UKIreland.Wheat          -4427.0000-4427.9202-4427.00004341.0000 INFES
UKIreland.FeedGrains    -12720.0000-12719.5462-12720.000013909.0000 INFES
UKIreland.Beef           -2178.0000-2177.8378-2178.00001112.0000 INFES
Other    .Wheat         -31731.0000-31731.0000-31731.000031731.0000
Other    .FeedGrains          .              .              .              .          
Other    .Beef                .              .              .              .


We can also look at the equation listing to see the same:

  
---- supply  =E=  supply function

supply(US,Wheat)..  Qs(US,Wheat) - 427*P(US,Wheat) + 190*P(US,FeedGrains) =E= 18520 ;

      (LHS = 18517.78, INFES = 2.22000000000116 ****)

supply(US,FeedGrains)..  Qs(US,FeedGrains) + 694*P(US,Wheat) - 2163*P(US,FeedGrains) =E= 98438 ;

      (LHS = 98451.15, INFES = 13.1499999999942 ****)

supply(US,Beef)..  Qs(US,Beef) + 84*P(US,FeedGrains) - 5.9*P(US,Beef) =E= 6536 ;

      (LHS = 6535.339, INFES = 0.661000000000968 ****)

---- demand  =E=  demand function

demand(US,Wheat)..  - Qd(US,Wheat) - 4.96*P(US,Wheat) =E= -15364 ; (LHS = -15364.1216, INFES = 0.121600000000399 ****)

demand(US,FeedGrains)..  - Qd(US,FeedGrains) - 1224*P(US,FeedGrains) + 48.13*P(US,Beef) =E= -140556 ;

      (LHS = -140550.4133, INFES = 5.58670000001439 ****)

demand(US,Beef)..  - Qd(US,Beef) - 6.95*P(US,Beef) =E= -14190 ; (LHS = -14189.7505, INFES = 0.249499999999898 ****)



Let's verify the first supply equation. The regression equation looks like: \[S_{11}=18,520+427 P_{11}-190 P_{21}\] In the equation listing we see, this arrived in GAMS as: 

supply(US,Wheat)..  Qs(US,Wheat) - 427*P(US,Wheat) + 190*P(US,FeedGrains) =E= 18520 ;

This is the same, just reorganized a little bit. When we plug in the solution values:

Qs(US,Wheat)      39048
P(US,Wheat)       66.96
P(US,FeedGrains)  42.43

we get a value of 18517.78, which is a little bit infeasible (it should be 18520).


The infeasibilities are small, so the impact on the solution should also be small. 

----    197 PARAMETER mysol  our solution

                                US         EEC   UKIreland       Other  production       price

Wheat     .US            15032.00024015.78739047.68566.956
Wheat     .EEC                       14155.0001281.5187715.21323152.08066.956
Wheat     .UKIreland                              3059.0003058.56767.376
Wheat     .Other                                                                        81.956
Wheat     .consumption   15031.89814155.3494340.08531731.000
FeedGrains.US           128448.0007919.0377389.745143756.59742.435
FeedGrains.EEC                       21370.00021370.36850.873
FeedGrains.UKIreland                              6519.0006519.43250.465
FeedGrains.Other                      2479.0002479.00035.873
FeedGrains.consumption  128447.81531768.40413909.177
Beef      .US             7854.0007854.261827.588
Beef      .EEC                        4203.0004203.016754.814
Beef      .UKIreland                   143.6471112.0001255.805750.593
Beef      .Other           584.000584.000727.588
Beef      .consumption    8438.2614346.6631112.158


Complementarity in the solution


The results for equation \[\color{darkred}{P}_{c,r} - \color{darkred}{P}_{c,r'}   \ge -\color{darkblue}{tc}_{c,r,r'} \perp \color{darkred}x_{r,r',c}\ge 0\] are:

---- EQU trade  prices

                                      LOWER          LEVEL          UPPER         MARGINAL

Wheat     .US       .EEC              -8.4380          .            +INF             .          
Wheat     .US       .UKIreland        -8.0300-0.4200        +INF             .          
Wheat     .US       .Other           -15.0000-15.0000        +INF        24015.7872
Wheat     .EEC      .US               -8.4380          .            +INF             .          
Wheat     .EEC      .UKIreland        -0.4200-0.4200        +INF         1281.5180
Wheat     .EEC      .Other           -15.0000-15.0000        +INF         7715.2128
Wheat     .UKIreland.US               -8.03000.4200        +INF             .          
Wheat     .UKIreland.EEC              -0.42000.4200        +INF             .          
Wheat     .UKIreland.Other           -15.0000-14.5800        +INF             .          
Wheat     .Other    .US              -15.000015.0000        +INF             .          
Wheat     .Other    .EEC             -15.000015.0000        +INF             .          
Wheat     .Other    .UKIreland       -15.000014.5800        +INF             .          
FeedGrains.US       .EEC              -8.4380-8.4380        +INF         7919.0367
FeedGrains.US       .UKIreland        -8.0300-8.0300        +INF         7389.7450
FeedGrains.US       .Other           -15.00006.5620        +INF             .          
FeedGrains.EEC      .US               -8.43808.4380        +INF             .          
FeedGrains.EEC      .UKIreland        -0.42000.4080        +INF             .          
FeedGrains.EEC      .Other           -15.000015.0000        +INF             .          
FeedGrains.UKIreland.US               -8.03008.0300        +INF             .          
FeedGrains.UKIreland.EEC              -0.4200-0.4080        +INF             .          
FeedGrains.UKIreland.Other           -15.000014.5920        +INF             .          
FeedGrains.Other    .US              -15.0000-6.5620        +INF             .          
FeedGrains.Other    .EEC             -15.0000-15.0000        +INF         2479.0000
FeedGrains.Other    .UKIreland       -15.0000-14.5920        +INF             .          
Beef      .US       .EEC             -84.802072.7746        +INF             .          
Beef      .US       .UKIreland       -80.702076.9956        +INF             .          
Beef      .US       .Other          -100.0000100.0000        +INF             .          
Beef      .EEC      .US              -84.8020-72.7746        +INF             .          
Beef      .EEC      .UKIreland        -4.22104.2210        +INF             .          
Beef      .EEC      .Other          -100.000027.2254        +INF             .          
Beef      .UKIreland.US              -80.7020-76.9956        +INF             .          
Beef      .UKIreland.EEC              -4.2210-4.2210        +INF          143.6465
Beef      .UKIreland.Other          -100.000023.0044        +INF             .          
Beef      .Other    .US             -100.0000-100.0000        +INF          584.0000
Beef      .Other    .EEC            -100.0000-27.2254        +INF             .          
Beef      .Other    .UKIreland      -100.0000-23.0044        +INF             .

Note that the column MARGINAL prints the levels of the variable \(\color{darkred}x_{r,r',c}\). Indeed we can see that shipping occurs (\(\color{darkred}x_{r,r',c}\gt 0\)) only when the constraint is binding. And the reverse: for all cases where the constraint level is not at its lower bound, the variable \(\color{darkred}x_{r,r',c}=0\). This report is just like for an LP, except that the dual (or marginal) is now an explicit model variable.


Conclusion


Here we solved the problem from [3] as a square system of equations. The solution reported in [3] is slightly infeasible. I suspect this is due to using software and hardware with more limited precision in 1966 than we have access to these days. I further suspect the model was solved as a QP by integrating the problem. That was the main approach before complementarity solvers became available.

This simple equilibrium model shows how can we find an equilibrium solution that solves simultaneously for prices, demand and supply quantities, and trade patterns. These types of models are used quite a lot. 

It is a good example of modeling complementarity explicitly: if the price differential between supply and demand in two regions is too small, we don't trade.

References


  1. Giffen goods, https://www.investopedia.com/terms/g/giffen-good.asp
  2. Samuelson, Paul A., "Spatial Price Equilibrium and Linear Programming," Am. Econ. Rev. 42:283-303, June 1952
  3. D. Lee Bawden, "A Spatial Price Equilibrium Model of International Trade", American Journal of Agricultural Economics, Volume 48, November 1966, Pages 862-874
  4. Solving Linear Complementarity Problems without an LCP Solver, https://yetanothermathprogrammingconsultant.blogspot.com/2021/05/solving-linear-complementarity-problems.html


Appendix. GAMS Model

$ontext

   Spatial Equilibrium

   Quick and dirty model
   (Only "true" complementarity for trade, the rest are equalities)


   Data from:

   D. Lee Bawden
   A Spatial Price Equilibrium Model of International Trade
   American Journal of Agricultural Economics
   Volume 48, November 1966, Pages 862-874


$offtext


*---------------------------------------------------------------
* Data
*---------------------------------------------------------------


set
   dummy 'for better display'/Intercept/
   c 'commodities'/Wheat,FeedGrains,Beef/
   r 'regions'/US,EEC,UKIreland,Other/
   g(c) 'grains'/Wheat,FeedGrains/
;

* demand/supply as (linear) function of prices
table coeff(*,c,r,*) 'coefficients for demand and supply functions'

                            Intercept   Wheat    FeedGrains    Beef
Demand.Wheat.US              15364      -4.96
Demand.Wheat.EEC             14383      -3.40
Demand.Wheat.UKIreland        4427      -1.29
Demand.Wheat.Other           31731

Demand.FeedGrains.US        140556               -1224        48.13
Demand.FeedGrains.EEC        31694                -157        10.68
Demand.FeedGrains.UKIreland  12720                 -63         5.82
Demand.FeedGrains.Other

Demand.Beef.US               14190                            -6.95
Demand.Beef.EEC               6830                            -3.29
Demand.Beef.UKIreland         2178                            -1.42
Demand.Beef.Other

Supply.Wheat.US              18520        427     -190
Supply.Wheat.EEC             12655        213      -74
Supply.Wheat.UKIreland        1441         30       -8
Supply.Wheat.Other

Supply.FeedGrains.US         98438       -694     2163
Supply.FeedGrains.EEC        16531        -85      207
Supply.FeedGrains.UKIreland   5446        -35       68
Supply.FeedGrains.Other       2479

Supply.Beef.US                6536                 -84          5.9
Supply.Beef.EEC               3235                 -24          2.9
Supply.Beef.UKIreland          808                  -6          1.0
Supply.Beef.Other              584
;

alias(r,rr);

table transCost(*,r,rr) 'Transportation Cost Matrix'
                      US     EEC    UKIreland    Other
Beef.US                     84.802   80.702     100.000
Beef.EEC                              4.221     100.000
Beef.UKIreland                                  100.000
Beef.Other

Grain.US                     8.438    8.030      15.000
Grain.EEC                             0.420      15.000
Grain.UKIreland                                  15.000
Grain.Other
;


alias(c,cc);

* set the transportation cost for the individual grains
* and use
transCost(g,r,rr) = transCost('Grain',r,rr);
transCost(c,r,rr)$(transCost(c,r,rr)=0) = transCost(c,rr,r);

display  coeff,transCost;


*-----------------------------------------------------------------------
* spatial equilibrium
*-----------------------------------------------------------------------

positivevariables
   Qd(c,r)      'demand quantities at demand regions'
   Qs(c,r)      'supply quantities at supply regions'
   P(c,r)       'price'
   X(c,r,rr)    'quantities traded/shipped'
;

equations
   supply(c,r)     'supply function'
   demand(c,r)     'demand function'
   flows(c,r)      'trade flows'
   trade(c,r,rr)   'prices'
;


supply(c,r)..
   Qs(c,r) =e= coeff('supply',c,r,'intercept')
                + sum(cc,coeff('supply',c,r,cc)*P(cc,r));

demand(c,r)..
     coeff('demand',c,r,'intercept')
     + sum(cc,coeff('demand',c,r,cc)*P(cc,r)) =e= Qd(c,r);

flows(c,r)..
      Qs(c,r) + sum(rr, X(c,rr,r)) =e= Qd(c,r) + sum(rr, X(c,r,rr));

* this is true complementarity equation:
*       trade(r,rr,c) >= 0 compl. X(c,r,rr) >= 0
* only one can be non-binding
trade(c,r,rr)..
   P(c,r) + transCost(c,r,rr) =g= P(c,rr);

* "shipping" means here: trade between different regions
* so fix x=0 if we deal with the same region
x.fx(c,r,r) = 0;


model lcp /supply.Qs, demand.Qd, flows.P, trade.x/;


*-----------------------------------------------------------------------
* check solution from Bawden
*-----------------------------------------------------------------------


table sol(c,*,*)  'solution from Bawden'


                            US     EEC    UKIreland    Other    production  price

Wheat.US                   15032                       24016      39048     66.96
Wheat.EEC                         14155    1282         7715      23152     66.96
Wheat.UKIreland                            3059                    3059     67.38
Wheat.Other                                                                 81.96
Wheat.consumption          15032  14155    4341        31731

FeedGrains.US             128448   7919    7390                  143757     42.43
FeedGrains.EEC                    21370                           21370     50.87
FeedGrains.UKIreland                       6519                    6519     50.46
FeedGrains.Other                   2479                            2479     35.87
FeedGrains.Consumption    128448  31768   13909                  174125

Beef.US                     7854                                   7854    827.59
Beef.EEC                           4203                            4203    754.81
Beef.UKIreland                      144    1112                    1256    750.59
Beef.Other                   584                                    584    727.59
Beef.Consumption            8438   4347    1112

;

display sol;

* set initial values
x.l(c,r,rr)$(not sameas(r,rr)) = sol(c,r,rr);
Qs.l(c,r) = sol(c,r,'production');
Qd.l(c,r) = sol(c,'consumption',r);
P.l(c,r) = sol(c,r,'price');

* iterlim = 0 so solver will only check initial solution
* inspect the listing file
option limrow=9999,iterlim=0;
solve lcp using mcp;

*-----------------------------------------------------------------------
* compute equilibrium
*-----------------------------------------------------------------------

option iterlim=99999;
solve lcp using mcp;


*-----------------------------------------------------------------------
* reporting
*-----------------------------------------------------------------------

parameter mysol(c,*,*)  'our solution';
mysol(c,r,rr) = x.l(c,r,rr);
mysol(c,r,r) = Qs.l(c,r) - sum(rr$(not sameas(r,rr)),x.l(c,r,rr));
mysol(c,r,'production') = Qs.l(c,r);
mysol(c,'consumption',r) = Qd.l(c,r);
mysol(c,r,'price') = P.l(c,r);
display mysol;


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