Department of Agricultural and Applied Economics

Transcription

1 Staff Paper Series Staff Paper P75-9 June 1975 Note: MODELING OF NONLINEAR FUNCTIONS INTO A LINEAR PROGRAMMING FORMAT Terry L Roe Department of Agricultural and Applied Economics University of Minnesota Institute of Agriculture St Paul, Minnesota 55108

2 Staff Paper P75-9 June 1975 Note: MODELING OF NONLINEAR FUNCTIONS INTO A LINEAR PROGRAMMING FORMAT Terry L Roe Staff Papers are published without formal review within the Department of Agricultural and Applied Economics

3 Terry L Roe May 16, 1975 Note: MODELING OF NONLINEAR FUNCTIONS INTO A LINEAR PROGIUMMING FORMAT I INTRODUCTION The linearization of nonlinear relationships into a linear programming framework generally increases the number of row equations in the linear programming model in direct proportion to the number of linear approximations to the nonlinear relationship It is well known that the computer time required for the solution of a linear programming problem increases substantially faster with the addition of row equations to the problem than with the addition of column equations Thus a need existed to develop an alternative method for the specification of nonlinear relationships into a linear programming framework which would decrease the number of rows required to specify these relationships John Duloy and Roger Norton developed a specification which basically 1/ satisfied this need In their specification of CHAC,z nonlinear product demand functions were incorporated into a linear programming framework so that only two row equations were required for each nonlinear function Thus, n number of linear approximations to a nonlinear function requires the addition of ricolumns equations to the model (as did the previous specification) but only two row equations This specification technique is also used in the Tunisian sector analysis model The economic specification of these nonlinear relationships vary, however, depending on whether the function being linearized is a

4 -2- production function, supply function or a product demand function The purpose of this note, therefore, is to respond to requests made by students, requests made by researchers working on a macro-model in the Fibers and Grains Branch of ERS and to provide our Tunisian colleagues with insights into modifications of the Duloy-Norton approach which must be made depending on the relationships to be specified in a linear programming format II OBJECTIVES -- PROCEDURE The specific objectives of this note are to provide insights into the linear programming formulation of: 1 product demand functions under conditions of a (a) non-competitive market and (b) a competitive market situation, 2 factor demand functions under (a) competitive and (b) noncompetitive factor market conditions, and 3 nonlinear supply functions The plan of the paper is to reproduce, in more detail, the Duloy-Norton formulation for the product demand case This leads into the linear programming specification of the factor demand relationships and finally the specification of the supply problem A sample problem format is used to present each of the above cases III PRODUCT DEMAND SPECIFICATION The monopolistic case The monopolistic case is presented as a point of departure because it is somewhat more straightforward then the competitive market case

5 -3- We begin by assuming linearity of demand and the absence of cross-price and income elasticities Let the demand functions facing the monopolist be expressed as: (10) P= f(q) =a-bq The total revenue (TR) function facing the monopolist then, is: (11) TR= g(q) = QP= aq-bq2 Suppose that the input-output characteristics of the monopolist can be expressed in the following linear programming formulation: (12) A~ <B _ where A is an n x m matrix of coefficients g is a m column vector of activities representing alternative ways of producing product Q and, ~ is an rlcomponent vector of available resources used in the production of Q The objective of the monopolist can be stated as to find {Q, p, ~} such that subject to (12) z =QP- ~~ a max The linear programming specification of this problem requires linear approximationsof (11) These linear approximations can be obtained as follows:

6 -4- (13) TR* = aq j j -bq: j =1,J where j denotes the j-th level of Qor, in terms of the linear programming specification below, the j-th linear approximation of (11) Since equation (11) can be expressed as TR = Jg (Q)dQ= J(a - 2bQ)dQ it can also be approximated in discrete terms as TR* J ~jg (dj)(qj - Qj-l)= ~j(a - 2bdj)(Qj - Qj_l) where, in the quadratic form case (1,1), ~j = (Qj Qj-1)/2~ The linear programming problem is; find {X j! Q } such that (2,0) Z* = Zj TR $Xj-~~ amax subject to (12) and (21) Z,Q X, JjJ - j j 5 ~ which states that the quality of {Q~} produced must equal or exceed the quantity demanded J {QjXj}, and the convexity constraint (22) 2,X < 1 Jj- Notice that together with the concavity 01 (11) this constraint Permits no more than two {X } to appear at positive levels in the optimal solution j and that O < X < 1 -j- Problem (20) is also expressed in Tableau form in Table 1 while (13) is expressed graphically in Figure 1

7 -5- d tn!= c 0 l-l V1 vl VI VI l-i I I, al Cn I : t- 3 d N l-l * vl x al N m (n aj H d -d

8 -6- (1 1 FIGURE 1 Depiction of Equation (13)

9 -7- The basic specification of the linear programming tableau is similar for the remaining cases For the competitive product market case, only the procedure used to compute the {TR~} change This is considered in the following section The Competitive Market Case In a competitive market, individual firms take prices, as expressed by (10), as given In this case, the appropriate function for the industry producing Q is the integral of (10), ie, (30) TR = ff(q)dq = J(a - bq)cjq = aq - 5bQ2, This is,also equal to the area under the demand function (10) If (12) is specified to depict the production characteristics of many firms, then the maximization of the objective function: (31) ZO= ZjTRjXj-~q subject to (12), (21) and (22) where TR: = Zjf(~j)(Qj - Qj_l) = Ij(a - 5b~j)(Qj - Qj_l) is equivalent to the maximization of the sum of consumers and producers 5/ surplus In the case of (31),consumer surplus (CS) is Cs = f(a - bq)dq - PQ = 5b[]2

10 -8- and producefs surplus (PS) is Ps =PQ- G Q = aq -bq2-@ which, when summed, is the objective function z =CS-I-PS= aq - 5bQ2 -&~ expressed in discrete terms by equation (31) The only difference between the tableau for this problem and that depicted in Table I is that the {TR~} values in Table I are replaced by the {TR~} values of (31) Factor demand specifications are considered in the next section where it is shown that the basic tableau design is similar to the tableau in Table 1 Iv FACTOR DEMAND SPECIFICATION The Simple Case Two problems are considered in this section This leads into the final section where a programming format nonlinear supply function is cast into a linear based on a problem considered in this section For purposes of demonstrating the specification of factor demand, suppose we are interested in determining (Y) and utilization of fertilizer (N, P) clearing (1) the product market for Y at fertilizer market where N* and P* denote the level of production of which would be consistent with a given Price PY and (2) a the total quantity of nitrogen and phosphorous available at a given price P, P These assumptions hp are made initially inorder to minimize the ambiguity of the linear

11 -9- programming specification The procedure followed is to state the nonlinear programming formulation and then develop the linear programming formulation so that the conditions for an optimum of the nonlinear formulation are consistent with the linear formulation Let the production function (40) reflect the input-output characteristics of the agricultural sector al a2 (40) Y=AN p where, for purposes of exposition, a3, A=eaL, denotes the efficiency of the technology and the total land area (L) allocated to the production of Y is given, (al, a,, a,) are positive coefficients which sum to a value JL 2 equal to or less than Specifically, the nonlinear programming problem can be stated as to find {N, P} such that unity (41) Z=PYAN al P 2 - PnN - PPP a max)subject to N < N*, P < P* where N*, p* are known total available supplies N and P respectively While the solution to this problem can be easily found without the use of programming, it quickly becomes a non trivial problem is additional constraints are considered and/or if several alternative regions, and crops are added to the problem

12 -1o- The linear approximation$of (40) are defined as follows al a2 EY = Zj AN P il J ij ij ij = Z,Y = Zj A N:: Pa2 i I J ij ij = th th where j denotes the j linear approximation of (40) for the i ratio of N:P constant The That is, for every i, N:P is being allocated in a fixed or factor ratio linearization of (40) can be shown more clearly in Figure 2a for the situation where i = 1,2 and j = 1,2 The corresponding isoquant diagram is depicted in Figure 2b where it can be seen that i = 1,2 are essentially two alternative expansion paths The linear programming O1lOws: Find ij P}, specification of (41) can be stated as such that * (50) z* = i j ~ijxij - PVN - PPP a max,subject to (51) i j ijxij 5 N (52) i j ijxij 5 p (53) N _ < N*, P<P* - and the convexity constraint (54) i j ij ~ 1

13 Y P i=2 \ ~ \ I \l \l 4 /[, N (2 a) P (2 b) FIGURE 2 The Linearized Specification of (40)

14 where * al 2 Rij =PY y ij = A ij ij and Py is a constant Constraints (51) and (52) associate the i,jth level of output Y lj with the corresponding levels of N ij and ij nput he convexity constraint (54) restricts the solution to only one point on the production surface given Che concavity of (40) The tableau corresponding to (5,0) is expressed in Table 2 The Extended Case The modification of (50) to include K regions and L varieties or crops, as in the case of the Tunisian fertilizer distribution model, is 6/ stated below and in Table 3 Find (Xijkl, Nkl, Pkl) such that subject to i j llx1l -N;l:o i j llx1l - ;1~0 (61) 0 i j KLXKL- XL~ izjpklxkl-piu- < 0,

15 -13- Table 2 Linear Programming Tableau for Problem (40)S * * Rllxll RIJ%J * * RIIX1l 1#IJ - qn - pp 11 lj 11 IJ -1 <0 11 lj 11 P IJ -1<0 1, <1 1 c N* 1-<P* Va;iables (TRij, Pr, are {X, N, P}, and PN ijp * P*] are given p ij ij N

16 -14- The convexity constraints for each k, 1; (62) and constraintson the total allocation of fertilizer: (63) kzlp:l~p * Problem (60) is stated in tableau form in Table 3 The modification of (60) to include a product demand curve is simply an extension of the model presented in the previous section For the competitive market case, the total revenue elements of (60) are * * Rij = ij-l a - 5b?ij)(Yij - Yij-l) and * * Rij = ~ij-1 -t-(a- 2b?ij)(Yij - Yij-l) for the monopolistic case where 1! ij = (Yij ij-l )/2 The linear programming specification of a nonlinear total cost function is briefly considered in the next section

17 o 0 l-i o VI VI VI VI o VI 1 & VI VI -x 1% VI,!,x O& 1 I (4,- < I d Ozz c! F4 w l-l 4 I I w w V7 w? I!4 aj? v-i w al hi m * -u (ḳ w-l w c z l-l 0 w em w be 5 E! (d N M : 9-I Cl

18 -15- V, SUPPLY FIJNCTIONSPECIFICATION The linear programming specification of a nonlinear supply function follows the basic procedures outlined above The following simple problem is used to demonstrate this specification Let: Pa-bY = Y denote the demand for product Y in a competitive market where Y=AN al ~a2 denotes the aggregate production function of the industry Let Pn, P P denote factor prices faced by producers of Y in a competitive factor market where (70) Pn=kN, Pp=cP, In this case the problem is to find N, P such that (80) Z= TR-TCn -TCP amax where, as in the competitive product market case considered above al a2 (81) TR= (a- 5bY)(AN P ) and TCn = (k/($ d 1 1) NL

19 (82) TCP = (c/(62l))p The linear approximation of TR is, as before, TRU ij = Wa - 5b?ij)(Yij - Yij-l) and, for the special case of a quadratic form,??ij (Yij Yij-1)/2 where al Y ij = ij a2 ij such that The linear approximations of (82) are 6~1 TC* nj = (k/(dll)) Nj, 621 TC = (c/(62-t-l))p PJ j The linear programming formulation of (80) is to find {X ij nj Xpj) (90) Z* = ZiZj TR~jXij - ZjTC~j - ZjTC~j a maximum subject to:fertilizer allocation constraints, <0 izj ijxij - injxnj - izj ijxij - jpjxpj 50

20 -17- a convexity constraint which constrains the solution to a point on the surface of the production function, and izj ij 5 1 convexity constraints which require function corresponding to the total that only the jth step of the cost level of N and P used appears in the solution z x < 1, J nj ZX < 1 j pj The linear programming tableau for this problem appears in Table 4 VI CONCLUSIONS This note attempts to briefly expand and clarify the Douley-Norton method for specifying nonlinear relationships into a linear programming framework Using a sample problem format, product and factor demand, and factor supply function$are specified to accomplish this objective It should be noted that, in the case of product demand for example, cross-price and income effects are either not included in the analysis or their effects are aggregated into the constant term of the demand function Therefore, they do not enter the analysis endogenously Quadratic and separatable programming have been used to deal with this aspect of the problem

21 -13- Table4 Linear Programming Tableau for Problem (90) * * ZTR*X ZjTR* J }5 l~j I,jxI,j - jtcfijxfij- ltcpjxpj ZN J l,j IN - ZN J I,j JJ <0 L,P J l,j ZP J I,j - ZP Jj <o Z41 Zil J J <1 <1 Zj1 <1 Variables are {Xij, Xnj, Xpj} and {TR~j~Nij~ ij~ T C;j, T@ pj j P } are given j

22 -19- Duloy, John and Norton, Roger, CHAC, A Programming Model of Mexican Agriculture, Multi-Level Planning: Case Studies in Mexico, edited by Louis Goreux and Alan Marine,North-Holland Publishing Co, 1973 CHAC is the name Duloy and Norton use to refer to the Mexican agricultural programming model ythe notation ~ E is used to indicate that the summation occurs overj =1, J andj~r $ = 1, I unless otherwise indicated In this case, TR~ = aqs - bq~ = Z; ;=s(a - WQs Qs-11/2)(Qs - Qs_l) lsee, Samuelson, Spatial Price Equilibrium and Linear Programming, AER Vol 42 andlor Takayama T and G Judge, Equilibrium Among Spatially Separated Markets, ECONOMETRICA, October, The i, j subscripts are omitted from the TRO and X symbols in equation (60) for convenience