TRrei - / COMPLEMENTARITY PROBLEMS AND THEIR REPRESENTATION IN GAMS by Johanee Bisschop and AZ::ander Meers Technical Note No.11 Research Project 671-58 July 1979 Development Research Center World Bank 1818 H Street, N.W. Washington, D.C. 20433 Preliminary and Confidential: Not for quotation or attribution without prior clearance from the authors. . iews expressed are those of the authors and do not necessarily reflect those of the World Bank. Abstract This note uses existing knowledge in classifying complementarity problems. It then develops a compact GAMS notation for models with complementarity conditions, and shows how the information contained in this notation can be used by the system for automatic detection of complementarity problems and special subclasses thereof. An application from the world steel industry is used to illustrate the concepts. Keywords: Complementarity Problems Quadratic Programing 1. Introduction Equilibrium models in economics take on several mathematical representations varying from square systems of linear simultaneous equations to square systems of nonlinear simultaneous equations, from linear optimization models to nonlinear optimization models. One specific representation that seems to be a natural choice is the complementarity problem (CP) (see e.g. [1], [2], and 151). The general mathematical statement for this problem is as follows: T y = F(x), x 1 0, y ! 0 , x T Y 0 (CP). In this statement, x and y are mxl vectors of variables (unknowns in the model). Since x > 0 and y 0, the condl-lon x y= 0 iplies the complementarity conditions x iy = 0 for each pair (xi,y ). These conditions occur quite naturally in the characterization of competitive market equilibria. For instance, when y expresses the excess supply of an item and xi is the corresponding price of the item, then the equation x y = 0 models the assumption that an item in excess supply must have a zero price. Another example is when x expresses some activity level and yi expresses a negative profit (a loss) associated with that activity, then the equation x y, 0 models the assumption that no activity that makes a negative profit is operated at a positive level. There are examples of complimentarity problems when the system F(x) = 0 has a nonnegative solution, which, by construction, solves also the complementarity problem. One example is the WISE model (version A) in the appendix. Why, one wonders, is it even worthwhile then to set 2 up the complementarity formulation? The answer in this instance is L"at a complementarity formulation is only worthwhile if one wants to make changes to such a model, and these changes involve the introduction of inequalities of the form F (x) > 0 , i = m + 1, ..., m + k. In many cases the augmented system F(x) = 0 does not have a solution, or produces a solution with negative components, and one is faced with a combinatorial search over 2k possible problem formulations. In these cases, zhe complimentarity formulation allows one to find one single representation of the model. In ad-1tion, one can use the complentarity equations to build "control" intc the model in the sense that specific variables car. become active when certain inequalities become binding. An example is version B of the WISE model in the appendix. Windfall profits are possible if and only if a prespecified bound on available capacity becomes binding. In this note we would like to take the importance of coaple- mentarity formulations in economic equilibrium problems as given, and concentrate on some technical aspects that are relevant for the development of GAMS. In the next section we will classify complementarity problems and relate them to other programming models. In section 3 we will describe the GAIMS representation of complementarity problems, while section 4 will investigate the process of automatic recognition of this class of problems. 3 2. A Classification of Complementarity Problems The nonlinear complementarity problem mentioned in the introduction can be slightly rewritten as y - F(x) (NCP) y x 0 ii x > 0 , y > 0 for i ,2,...,m. In this form it is a square system of 2n nonlinear equations with added nonnegativity constraints. Let the occurance or non-occurance of variables in equations be reflected via an incidence matrix of zeros and ones respectively. Then the incidence matrix of the system without the nonnegativity constraints has a special structure induced by the complementarity corditions x iy - 0. Figure 1 displays this structure. 11 0 1 11 0 11 0 0 0 111 Figure 1: The Incidence Matrix for the NCP As we shall see in the next section, the algebraic representation of this special structure becomes much more compact when we keep all but the north-east block implicitly. An important subclass of the general nonlinear complementarity problem is the linear complementarity problem (LCP). Whenever F(x) is a 4 linear system, the NCP is an LCP, even though the nonlinear complementarity conditions are still t A linear complementarity problem, there- fore, is any mathematical structure of the form y = q + Mx y x - 0 (LCP) xi > 0, yi > 0 i - 1,2,..., m As most readers are familiar with the general class of nonlinear programming problems, it is of interest to relate this class of problems to the NCP. Intuitivity one would expect the optimization problem to be a more general problem since its constraint set can include constraints of the form found in the NCP. It is true that the constraint set of the optimizatiou pro- blem can include constraints of the form found in a NCP, but not the entire set determining a square NCP. The square NCP defined in this section turns out to be a more general class of problems contradicting our intuition. We shall show that any nonlinear programming problem has an equivalent representation as a NCP, and that the converse is not true. Consider the general nonlinear programming problem Hinimize z = f(x) s.t. g(x) > 0 (NLP) x > 0 where g is a mxl vector valued function, and x is a mxl vector of nknowns. One can write the Lagrangian and the Kuhn-Tucker (necessary) optimality conditions for this problem as follows. 5 T T L(x,X,p) = f(x) - Xk g(x) - VTx with optimality conditions i) 7 Tf(x) - TG(x) - - T 0 (G(x) ( ax (ii) x. g(x) - 0 x > 0, g(x) > 0 (iii) T 0 U 1 0, x > 0. Let v = g(x). Then we can rewrite the entire set of conditions (i) through (iii) as follows: v = g(x) v > 0, x > 0 T V = Vf(x) -G (W) X > 0, > 0 with X i = 0, 1 = 1,2,..., m and jx = 0 , j = 1,2,..., m. This problem has exac1ty the same mathematical form as the NCP described at the beginning of this section. One can therefore conclude that any nonlinear programming problem has an equivalent representation as a nonlinear compleentarity problem. It is of interest to note that both the linear programming problem and the quadratic programming problem are equivalent to a linear complementarity problem. There are, however, linear complemen- tarity problem that do not correspond to either a linear program or a 6 quadratic program. The WISE model in the appendix is an example of such a model. In order to show that a quadratic programming problem (and therefore also a linear programming problem) corresponds to a linear complementarity problem, consider the following mathematical statements. Minimize z c x + - x Qx (Q symmetric) s.t. Ax > b (QP) x > 0 One can write the Lagrangian and the Kzhn-Tucker optimality conditions for this problem as follows: T 1 T T T L(x,X,u) - c x + - x Qx - I (Ax - b) - uITx 2 with optimality conditions T T T T ) + xT - 0 (ii) X T(Ax - b) =0 X > 0, (Ax -b) > 0 (iii) Tx = 0 U 1 0, x > 0 Let v = Ax - b. Then we can rewrite the entire set of conditions (i) through (iii) as follows: v = Ax - b v> 0, x > 0 11= c+Qx - AT X> 0, v> 0 7 with X.v. = 0, i - 1,2,..., m, and u-x. 0, j = 1,2,..., m. This problem has exactly the same =athe=atical form as the previously described LCP with parameters -b 0 A q [c and 4 AT 1] Given this general background on coplientarity problems, the next section will develop a compact notation for this class of problems. 3. The GAMS Representation of Co,plementarity Constraints Any GAMS model representation we have built thusfar has followed the strict rule that both equation names and variable names must be unique, and that the intersection of the two sets of names must be empty. We still adhere to these conventions for ordinary programming models, but allow a natural extension for the class of complementarity problems. As we hinted at in section 2. the special structure inherent in complementarity problems can be carried on implicitly. While building constraints in a complementarity problem, one always models an equation and its associated complementarity condition as a pair. We again refer to the WISE model in the appendix as an example. Consider the capacity utilization cunstraint set (4.1) in this model. y . CS - SS > 0 t tt- utilizable capacity supply of steel in millions of tons in million of Itons (Y . CS - SS) . (PP) 0 t t t t underutilized capacityJ demand price for the use of capacity 8 There are two things one should observe. The surplus variable which has always been present in the formal mathematical statement of the complementarity problem, is not explicitly defined here. As a result, each complementarity condition contains one term which is essentially a repetition of the inequality it is associated with. This duplication of effort is a burden on the model builder. On the other hand, the determina- tion of names in real-life models is also a burden on the model builder, which makes the formulation without the surplus variable attractive. (In GAMS we have the convention that the value of any implicitly defined slack/surplus variable is automatically associated with the equation name after the model has been solved.) We would like to propose the following set of conventions, which will prevent any duplication of effort and still not introduce explicit surplus variables in a GAMS representation with complementarity constraints. The GAMS compiler will accept a variable name as an equation name only if the following two conditions are satisfied. First of all, that equation name cannot be used in the equation itself to denote also a variable (i.e. that name cannot play its dual role in the same equation). Secondly, such an equation must be either of type =L= ("less than or equal to") or type =G= ("greater than or equal to"). henever a variable name is used as an equation name, and the above two conditions are satisfied, the system will generate internally the complementarity condition using the equation name as the (implied) associated complementarity variable. The GAMS represencation for the above equal -on set can then be written as 9 PP(T).. GAMMA(T) * CS(T) - SS(T) =G= 0 utilizable capacity u ply of steel] in million of to'. in millions of tens , or, as PP(T).. SS(T) -L- GAMMA(T) * CS(T) Note that by merely extending the semantics of the modeling language, we have obtained a most compact representation of compleentarity problems. At the same time, the semantic rules make the recognition of complimentarity constraints an unambiguous task which can be performed at th,. idel definition stage. In the next section we will examine the autouatic recognition of complementarity problems a little further, and use the WISE model of the appendix as an illustration. 4. The Automatic Recognition of Complementarity Problems In the previous section we have described the -amantic rules that determine when a GAMS equation has an associated complementarity condition. In this section we want to establish the necessary and sufficient conditions that will determine not only if a model .statement corresponds to a complementarity problem as stated in section 2, but also if a detected complementarity problem corresponds to a linear programming or a quadratic prograrming problem. If in the GAMS representa;.ion of a model the number of equations and the number of variables are the sane, and all equations are of type =L= or type =G=, then the underlying model is a complementarity problem if and only if each equation name is also a variable name (recall that 10 each equation name can only be used once to denote an equation in GAX1. If in a GAKS representation of a model the number of equations and the number of variables are the same, and there are exactly k equations of type =E= , then the underlying model is a complementarity problem if and only if there are exactly k variables with names not on the equation name list, such that each of these k variables can be substituted out uf the model using the k equations of type E. If the number of equations is greater than (less than) the number of variables, the system is possibly overspecified (underspecified), but cannot be viewed as a complementarity problem. In -7any applications, the equations of type =E= are definitions of intermediate and nonextremal variables. A nonextremal variable is (by definiton) a sum of monnegative terms, implying that its associated nonnegativity constraint is redundant. Such a variable can be substituted out without it becoming negative when its value is computed after the model has been solved. The WISE model in the appendix has two equations of type -E- , and they define two intermediate and nonextremal variables. We will use this model as an example. Leaving nit parameter definitions, we can write the following GAMS statement for model. version A. SET T ACTUAL TIME/ 1980, 1985, 1990 / SET TT VNTAGE TLME; TT(T, = YES; SET TTT CORRESPONDENCE BETWEEN ACTUAL TDME AN"D .1,NTAGE TIME / 1980.(1980, 1)85, 1990), 1985.(1985, 1990), 1990.1990/; EQUATION PP CAPACITY UTILIZATION CONSTRAINT FOR TIE PRODUCER OF STEEL SS NO EXCESS PROFIT CCNSTRAINT FOR TE PRODUCER OF STEEL PSD DEFINITION OF TEE SUPPIY PRI-E OF STEEL DS NO PROFIT CONSTRAINT FOR THE CONSUMER OF STEEL PDD DEFINITION OF TE DEA.AND PRICE OF STEEL 11 PI CAPACITY SUPPLY CONSTRAINT FOR TIE INVESTOR IC NO EXCESS PROFIT CONSTRAINT FOR THE INVESTOR PM NO EXCESS DEMIND CONSTRAINT ON STEEL MARKET CS NC EXCESS DEMAND CONSTRAINT ON STEEL CAPACITY EXPANSION MARKET; VARIABLE PD DE%AND PRICE OF STEEL PS SUPPLY PRICE OF STEEL PM MARKET ClEARING PRICE OF STEEL PP DE-MAND PRICE FOR THE USE OF PRODUCTION CAPACITY PI SUPPLY PRICE FOR THE USE OF PRODUCTION CAPACITY DS DLWAND FCR STEEL SS SUPPLY OF STEEL CS PRODUCTION CAPACITY OF STEL IC r-VESIfENT IN PRODUCTION CAPACITY; * A LISTING OF TaE GAMS EQUATIONS WITHOUT ANY EXPLANATIONS, PP(T).. GAMMA(T) * CS(T) =G= SS(T) SS(T).. FM(T) =L= PS(T) PSD(T).. PS(T) =E= OPCOST(T) + FP(T) DS(T).. PM(T) =G= PD(T) ; PDD(T).. PD(T) =E= LA3DA(T) - OMEGA(T) * DS(T) ; PI(T).. CSNOT + SUM(TT $ TTT(TT,T), IC(TT)) -G= CS(T) IC(T).. IVCOST(T) =G 3L-M(TT $ TTT(T,TT), SIGMA(T,TT) * PI(TT)) FM(T).. SS(T) -G= DS(T) ; CS(T).. PI(T) =G= PP(T) ; 12 In this model the number of equation names and variable names are the same and is equal to 9. The number of equations of type =E= is 2, and there are exactly 2 variables with names that do not appear in the equation name list, namely PS and PD. Both these variables are defined via the type =E- equations, and are nonextremal variables since OPCOST, LAMBDA and (-OMEGA) are all non-negative quan'ities. As we have seen, the useage of variable names as equation names to express complementarity constraints provides the GAMS system with information to detect if a particular model statement is indeed a complemantarity problem. As the system can also check if a model is linear, it can identify linear complementarity problems. For such problems it is of interest from the viewpoint of algorithm selection to see if they correspond to an underlying linear programming or quadratic progra-ning problem, and to see if this can be detected automatically. rhenever a nontrivial 2x2 partitioning of the M matrix in a linear compleentarity problem is skew-symmetric, then there is an underlying optimization problem. It is a quadratic programming problem if only one of the diagonal blocks is zero, and a linear programming problem if both diagonal blocks are zero. In any GAMS representation of a linear complementarity problem all equations of type =E= must first be substituted out. Then the matrix R can be formed by ordering rows and columns such that the ith row name is equal to the ith column name. It is straight forward to verify not only that R much have a symmetric incidence matrix, but also that only symnetric row and column permutaticns are requirf!d to obtain the block skew- symetric H matr-x if this exists. A simple matching of rows and columns can be usL-d to verify the symmetry of R, and to identify the 13 required row/column permutations to construct a block skew-symmetric matrix M. As there are time lags and leads in version A of the WISE model, it is not encugh to examine the structure of its incidence matrix for one typical time period only. Figure 2, therefore, portrays the incideace matrix of this model for all time periods with all equations of type -E= substituted out. This matrix is symmetric, but a matching of numerical row and column entries in rows/columns IC(l), IC(2), and IC(3) show that their values do not match since SIGMA (T,TT) is not equal to 1 for all pairs (T,TT). We can therefore conclude that version A of the WISE model is an example of a linear complementarity problem that does not correspond to an underlying optimization problem. т�ЧоЧ дS1М о47 то хта7vЧ И о�1г. гг uantlTi . т , 1 (С)S� т т (г)6J т Т (т)SJ Т Т (С)Ид � т (Z)Ид � г cnNd т (С)�1 т т сг)�� г т т (т)�г Т Т г t • (С)1д Т Т т (г)1д Т• т , (т)кд Т т (С)sa i т (г)sa Т т (т)Ба i t (С)66 Т Т (Z)SS i т (т)ss Т Т (С)дд Т т (г)дд Т т (Т)дд С Z Т С l• Т С Z т С Z т С Z т С Z Т Е Z Т S S 8 Ч Н Ч � � � I �I I S S д 6 S 8 д д д � � � д д д I 1 I д д д а а а 8 S 6 д д д -9т- References Cottle, R. and G. Dantzig, "Complementary pivot theory of mathematical progrnmmiag", Linear Algebra and Its Applications. 1 (1968), 103-125. Hansen, T. and A. Manne, "Equilibrium and linear complementarity an economy with institutional constraints an prices". IIASA Research Memorandum P-M-74-25, Laxenburg, Austria, (1974). Hashimoto, H., "Prospects for the world iron and steel economy: the WISE model", Economic Analysis and Projections Department, World Bank, Washington, D.C.,U.S.A. (1979). Keyzer, M., C. Lemarechal and R. Mdfflin, "Computing economic equilibria through nonsmooth optimization", IIASA Resea ch Memorandum RM-78-13, Laxenburg, Austria, (1978). Keyzer, M. "Analysis of a National Model with Domestic Price Policies and Quota on International Trade", IIASA Research Me randi= RM-77-19, Laxenburg, Austria, (1977). Appendix The following appendix is a rewritten version of section 4 of the proposed Staff Working Paper by Mr. Hideo Hashimoto of the Economic Analysis and Projections Department, World Bank, entitled "Prospects for the World Iron and Steel Economy: The WISE Model". It is used throughout this note as an illustration of the concepts relevant to the GAMS representation of co:plimentarity problems. A.1 4. Matheatical Formulation of the WISE Model In this section we will develop a world iron. and steel economy ,odel CVISE model) that projects the market situations of the iron and steel industry in future bench-mark years. In summary, the model will project prices, demand and supoly quantities of steel products, and future investments in steel production facilities. The steel industry is depicted at three distinct years, namely 1980, 1985 and 1990, thereby slmarizing investment behavior over the periods 1975-1980, 1980-1985 and 1985-1990 respectively. Considering the com- plexity surrounding the industry's investment decision, as was discussed in Section 2, we will develop two models, Model A and Model B. In Model A we assume that iron and steel producers implement investment in production facilities if and only if the expected present value of any returns exceeds the marginal cost of financing the new facilities. Model B is essentially the same as Model A, except that we assume the investment plans until 1985 to be the ones currently conceived by the industry. This modification allows us to examine future market behavior on the basis of existing capacity expansion plans. 4.1 Hodel A. Investments on the Basis of Rational ExDectations Before writing any mathematical equations, we would like to introduce the mcde! in order to facilitate its understanding. For conceptual reasons we will distinguish three different economic agents and two different markets in this model. The distinction is based on the role that each agent plays, even though in reality these roles may be assumed by the same person or institution. We have the following set of agents and markets. A.2 Agent 1: the producer of steel Agent 2: the consumer of steel Agent 3: the investor in steel production capacity and Market 1: the market for steel (with agents 1 and 2) Market 2: the market for additional steel production capacity (with agents 1 and 3). Whenever we refer to steel in this model we refer to steel products expressed in terms of their crude steel equivalent. The following assumptions reflect the underlying profit maximation behavior of the economic agents, and characterize the concept of market equili- brium used in this model. They are important for the understanding and evaluation of the model. i) No agent can make excess profits, i.e., profits in excess of the "accepted" markup. ii) There is no excess demand on any market. iii) Every item in excess supply has zero price. iv) No activity that makes a negative (excess) profit is operated at a positive level. One should decide if these assumptions are reasonable for the case of the world iron and steel industry. We would like to make the fol.ling ccments. The steel industry does contain several large companies, but is still considered an internationally competitive industry. This i:plies some degree of free market entry and exit.in support of the first assumption. The "accepted" markup incorporated in the model can, of course, be adjusted parame- I A.3 trically to reflect diverging views with respect to excess profit. Assumptions two and three are generally acceptable when an adjusting price mechanism is present. The last assumption prevents persistent losses for any agent which is a plausible assumption in the light that the model is a long-run projection model making five-year leaps. The relevant equations of the model are most naturally presented by modeling each agent and each market separately. When describing the variables (unkowns) in the model, it helps, on conceptual grounds, to distinguish between decision variables and adjustment variables. A decision variable is any variable under the direct control of a specific economic agent (e.g,, the supply variable), while an adjustment variable cannot be decided on by any one single agent. An adjustment variable is determined only as the result of two or more agents interacting on a market (e.g., the market equilibrium price of a good). This distinction will be made following the description of the variables in the model. The first letter of each variable name corresponds to a keyword for quick identifi- cation. The subscript t denotes time, and refers to the bench mark years 1980 (t-1), 1985 (t-2) and 1990 (t-3). Each bench mark year t represents a 5-year span, starting 2 1/2 years back and going 2 1/2 years forward. We will 1 1 sometimes use the cryptic notation t - - and t + - 2 2 Keyword Name Description Prices PDt Demand vrice of steel ($/ton) PS Supply pyice of steel ($/ton) t- ?X t Market clearing price of steel (S/ton) ?P Demand price (rent) offered by the t producer of steel for the use of steel production capacity ($/ton) A.4 Keyword Name Description PIt Supply price (rent) charged by investor for the use of steel production capacity ($/ton) - Demand DSt Demand for steel (millions of tons) Supply SSt Supply of steel (millions of tons) Capacity CS Production capacity of steel in operation (millions of tons' Investment IC Investment in additional production capacity of steel between the bench mark years t-l and t, where t - 1975 (millions of tons) o The variables SSt, CS and PSt are decision variables for producer of steel. The variables DSt and PDt are decision variables for the consumer of steel. The variable IC is a decision variable for the investor in additional steel t production capacity. The remaining variables PMt, PPt, and PIt are adjust- ment variables, establishing market equilibrium. We are now ready to develop the mathematical equations determining the model, describing each agent and market separately. The Producer of Steel (1) Capacity utilization constraint When deciding the level of SSt, the supply of steel, the producer faces a production capacity feasibility const):aint. We assume an upper limit on the industry-wide capacity utilization rate. If operational capacity is underutilized, then, by assumption, the rental value (demand prpce) offered for the use of production capacity will be zero. A. 5 Y *CS - SS > 0 t t t - Lutilizable capacity supply of steel in in lions of tons millinc of tons (4.1) Yt* CS - SS * PP = 0 [underutilized ca idemand price for the aty] use of capacity The parameter yt is the upper limit on the industry-wide capacity utilization rate. (2) No excess Drofits constraint The producer of steel cannot make excess profits. The supply price zepresents the costs. When the producer is faced with a persistent loss, the supply of steel should be zero. We assime that the operating costs are specified exogeneously, and include all input costs (those for iron ore, coking coal, steel scrap, energy and labor) plus an "accepted" markup. PS > PM t t [supply price of F Market price of steel] steel in S/tonJ L in S/ton (4.2) (Ps - ) ( ss-) 0 t t operating loss] supply of steel where, A.6 PS TG + PP t t t F 1 F 1 price offered for] supply :rice of operating cost trice offeapacity steel in S/t$L in/ton in s/on c I I in S/ton The Consumer of Sceel (3) No profit constraint The consumer of steel cannot make a profit. When faced with a positive gap between the market price and the demand price, the demand for a-teel hould be zero. 'We assume that the demand functions were estimated outside the model. FM > PD [narket price of demand price of steel in $/ton] Lsteel in S/ton (4.3) (RM - PD ) (DS) 0 excess market demand price of of steel j steel where PD - X - DS t t t t [demand price of a linear function steel in $/ton j of demand A.7 The parameters X and t are the intercept and coefficient in the estimated lir-ar demand function. The Investor in Additional Production Canacity (4) CaDacity supply constraint By assumption, there cannot be any excess,demand for the use of production capacity. On the other hand, any persistent oversupply of additional capacity by the investor should render a supply price of zero. We assume that the initial world prodaction capacity is known. t + IC > CS o01 - t initial capacity accumulate new capacity in lin capacity in operation in millions of tons millions of tons millions of tons] (4.4) t CS + I IC -C ) * pi) 0 r=l oversupply of supply price of additional capacity capacity The parameter CS is the initial world production capacity at t-1975. 0 (5) No excess profits constraint The investor in additional steel production capacity cannot make excess profits. When the investor is faced with a persistent loss, the investment activity in additional steel capacity should be zero. We assume that the inve.=en costs for the investor are exogeneoisly, and include construction costs plus equipment. A.8 3 VC > a P1 r tt fixed investment cost in discounted earnings in c/ta for additional capa- $/tea from the invest- city during the period ment in production t-1 and t capacity 3 IVC t - t ) * ( IC ) 0 T"t r investment loss 1 investment The parameter at- is a discount rate expressing at ti.e t - 1/2 the value of one unit c,f return made at tme r + 1/2, T t The Market for Steel (6) go excess demand constraint By assumption, there is no excess de=and f steel. On the other hand, if the producer of steel creates a persistent excess .rpply, the price of steel will be zero. SS >DS t supply of steel demAnd for sceel in millions of in millous of L tons tons (4.6) (SSt -DS ) ( ) 0 oversupply o rice of Isteel steel A.9 The Market for Additional Steel Production Capacity (7) No excess demand co:gtraint By assumption, there is no excess demand for the use of steel production capacity. This is expressed by requiring the supply price for the use of capacity to be.-,reater than the demand price. If on the other hand the supply price exceeds the demand price, we assume that the use of capacity V 11 be zero. PI _ Pt pply price for the d::and price for the use of capacity in use cf capacity in $/ton $/ton (4.7) (PIt p) * (CS) 0 excess supply price production for the use of capa- capacity L city in use Equation sets (1) through (7) coprise Model A. 4.2 Model B. Limited Capacity Exansion Until 1985 Model B is essentially the same as Model A, except that we assume the investment plans until 1985 to be the ones currently conceived by the industry. A- current plans seem to indicate a shortage of future available capacity, we have modeled them by imposiug a simple upper bound on the production A.10 capacity of steel in operation at time t=1985 (i.e., the variable CS ). Adding only this constraint to Model A may render it infeasible. One can verify this by tracing the effect of an artificially constrained supply (see equation set 4.1) throughout the model. That is why the following modification to Model A have been made to obtain our Model B. a) The total production capacity of steel in 1985 is limited from above. b) The produce: :5 steel can make a positive windfall profit in 1985 if and only if the limiting bound on available capacity is in effect. c) We assume the windfall profit made in 1985 will be offered to the investoc in ateel production capacity for future expansion (in this model the period 1985-1990). Let the variable PW2 be the windfall profit incurred by the producer of steel in $/ton at t-2-1985, and let the parameter CS2 be the exogeneously specified bound on available steel production capacity at t-2. Then the above modifica- tions a) through c) can be expressed using the following mathematical notation. a) CS CS Lprouction capacity upper limit on avail- in operation at able capacicity at t-1985 t-1985 b) (CS2 - CS 2) = 0 [ capacity surplus windfall profits for L n 1985 producer in 1985 A.11 3 c) IVC > E a PI t-1, 2 t t t tT T T=t IVC 3 a33 (PW + PI ) (t-3) fixed investment discounted stream of L coats [ benefits from investmentsj If -ae takes Model A, adds the equations under a) and b), and replaces the equation in the set (4.5) with the equation under c), one obtains our Model B. For reporting purpos.- --a want to follow an industry-wide custom and express prices in terms of dollars per ton of finished steel products. To obtain these, we convert the model prices per ton of crude steel equivalent using a yield based on the performance of a new mill (the marginal producer). Similarly, quantities can be expressed in terms of millions of tons of finished steel products. In this instance we use a conversion rate (yield) based on the performance of the average mill (the average producer). In addition, the amount of steel production in the- model is translated into the industry's requirements of inputs (iron ore, cuking coal, steel scrap, energy and labor) through the industry's average irput-output coefficients. These coefficients were exogeneously estimated by taking account of the vintage component of capacity in each projection year. A detailed discussion of the estimation of input-output coefficients is given in Appendix 3.