9. The Monopsonist. π = c x [g s + s Gs] (9.1) subject to. Ax s (9.2) x 0, s 0. The economic interpretation of this model proceeds as in previous

Transcription

1 9. The Monopsonist An economic agent who is the sole buer of all the available suppl of an input is called a monopsonist. We will discusss two categories of monopsonists: the pure monopsonist and the perfectl discriminating monopsonist. Pure Monopsonist In analog to the monopolist treatment, we will discuss the pure monopsonist s behavior assuming that she will be the sole buer of a vector of inputs. Let p s = g + Gs be such a vector of inverse linear suppl functions, where s is a (m 1) vector of quantities of inputs purchased b (supplied to) the monopsonist, the matrix G is a (m m) matrix of price slopes in the input suppl functions, the vector g contains intercept coefficients and p s is a (m 1) vector of input prices. We assume that the matrix G is smmetric and positive definite. In analog to the monopolist, the monopsonist owns the input suppl functions. In order to concentrate on the pure monopsonist s behavior, we assume that this economic agent is a price taker on the output markets and produces his outputs b means of a linear technolog. The decision of the pure monopsonist is to find the optimal quantities of inputs to purchase on the market and to find the optimal quantit of outputs to produce in such a wa to maximize profit. Total revenue of the price taker entrepreneur is defined as TR = c x, where c is a (n 1) vector of market prices for the outputs which are represented b the vector x, conformable to the vector c. Total cost of the pure monopsonist is defined as TC = p ss = g s + s Gs. Hence, the primal specification of the price taker and the pure monopsonist can be stated as the maximization of profit a linear technolog: Primal max x,s π = c x [g s + s Gs] (9.1) Ax s (9.2) x 0, s 0. The economic interpretation of this model proceeds as in previous

2 126 The Monopsonist chapters. The primal constraints (9.2) represent the input quantit equilibrium conditions according to which the demand of inputs, Ax, must be less than or equal to the input suppl, s. The primal objective function represents the pure monopsonist s profit defined as total revenue minus total cost. A diagrammatic illustration of the discussion related to model [(9.1), (9.2] is given in figure 9.1 $ g + 2Gs p s g + Gs g s s Figure 9.1. The Pure Monopsonist. The pure monopsonist equilibrium is given b the equalit between the marginal factor cost, g +2Gs, and the shadow price of inputs,, called also the vector of marginal revenue products of the inputs. The dual specification of the pure monopsonist s problem, as stated in [(9.1), (9.2)] is achieved via KKT theor. For this purpose, the Lagrangean function corresponding to the primal problem [(9.1), (9.2)] is specified as L(x, s, ) =c x g s s Gs + [s Ax] (9.3) where the vector represents Lagrange multipliers or, equivalentl, shadow prices of the limiting inputs. The KKT conditions derived from (9.3) are L x = c A 0 (9.4) x L x = x [c A ]=0 (9.5)

3 The Monopsonist 127 L s = g 2Gs + 0 (9.6) s L s = s [ g 2Gs + ] =0 (9.7) L = s Ax 0 (9.8) L = [s Ax] =0 (9.9) together with the nonnegativit of all vectors, x 0, s 0, 0. KKT conditions (9.4) and (9.6) constitute the dual constraints of the pure monopsonist s problem. Constraint (9.4) states that the marginal cost of producing outputs x must be greater than or equal to the corresponding marginal revenue, c. Constraint (9.6) states that the marginal expenditure on inputs (traditionall called also the marginal factor cost), g +2Gs, must be greater than or equal to the marginal valuation of those inputs,, called also the marginal revenue product or (in the case of a price taker on the output markets) the value marginal product. For the pure monopsonist s behavior, the marginal factor cost (MFC) is greater than the price for the inputs paid b her which is p s = g + Gs. The statement of the dual specification of the pure monopsonist s problem, as stated in the primal [(9.1), (9.2)], takes advantage of the information found in KKT conditions (9.5) and (9.7) where c x = x A and s = g s+2s Gs, respectivel. B using these equations in the Lagrangean function (9.3), the dual specification of the pure monopsonist s problem results in Dual min TCMO = s, s Gs (9.10) A c (9.11) g +2Gs (9.12) s 0, 0. The objective function (9.10) specifies that an entrepreneur wishing to take over the monopsonist s firm will attempt to minimize the total cost of market options, (TCMO), represented b the producer surplus of the pure monopsonist. The quantit s Gs is the pure monopsonist s profit for the entrepreneur who owns the firm, as can be deduced b equating the primal and the dual objective functions (equivalentl, b using KKT conditions (9.5), (9.7), and (9.9)) to produce π = c x [g s + s Gs] =s Gs. (9.13)

4 128 The Monopsonist The linear complementarit structure corresponding to this specification of the price taker on the output markets and the pure monopsonist s problem is stated as M = 0 0 A 0 2G I, q = A I 0 c g 0, z = x s (9.14) which represents the KKT conditions [(9.4)-(9.9)] in compact form. The matrix M is rather sparse, with several null matrices and two non informative identit matrices. One of the objectives of the analsis in this chapter is to reduce the number of null and non informative matrices in the M matrix of the LC problem. Perfectl Discriminating Monopsonist While the pure monopsonist must pa the same price for all the input units that she purchases, the perfectl discriminating monopsonist is capable of charging a different price for each unit of input that is purchased on the market. Hence, her total expenditure on inputs is the integral under the input suppl functions. As in the previous section, the vector of input suppl functions is assumed to be p s = g + Gs, where s isa(m 1) vector of quantities of inputs purchased b (supplied to) the monopsonist, the matrix G isa(m m) matrix of price slopes in the input suppl functions, and p s is a (m 1) vector of input prices. We assume that the matrix G is smmetric and positive definite. Total input cost for the perfectl discriminating monopsonist, therefore, is computed as TC = s 0 (g + Gs) ds = g s + 1 2s Gs. (9.15) The necessar condition for the existence of the integral as stated in (9.15) is that the matrix G b smmetric, as we have assumed above. We are now read to state the primal problem of the price taker on the output market and of the perfectl discriminating monopsonist: Primal max x,s π = c x [g s + 1 2s Gs] (9.16) Ax s (9.17) x 0, s 0. Except for the factor 1 2, the primal problem [(9.16), (9.17)] has the same structure of the primal problem [(9.1), (9.2)] which deals with the

5 The Monopsonist 129 pure monopsonist s problem. The primal constraints (9.17) state that the demand for inputs, Ax, must be less than or equal tu the input suppl, s. Hence, b analog, the dual specification of the perfectl discriminating monopsonist ca be stated as Dual min s, TCMO = 1 2s Gs (9.18) A c (9.19) g + Gs (9.20) s 0, 0. The left-hand-side of the dual constraint (9.20) is the vector of input suppl functions. Hence, if an positive amount of inputs is purchased b the perfectl discriminating monopsonist, that is if s > 0, then the dual constraint (9.20) will be fulfilled as an equation (via complementar slackness conditions) with the result that p s = g + Gs =, s > 0. (9.21) Constraints (9.19) state the economic equilibrium conditions according to which the output marginal cost, A, must be greater than or equal to marginal revenue, c. A diagrammatic illustration of the discussion related to the dual pair of models (9.16) through (9.20) is given in figure 9.2 $ p s = g + Gs g + Gs/2 g Figure 9.2. The Perfectl Discriminating Monopsonist. The equlibrium of the perfectl discriminating monopsonist is established b the equalit between the marginal revenue product,, and the input s s

6 130 The Monopsonist suppl function, g + Gs. At that point, the input price, p s equals the marginal revenue product. The linear complementarit structure corresponding to this specification of the price taker and the perfectl discriminating monopsonist s problem is stated as M = 0 0 A 0 G I, q = c g, z = x s. (9.22) A I 0 0 The matrix M is as sparse as the matrix M in (9.14), with several null matrices and two non informative identit matrices. Perfectl Discriminating Monopsonist Riformulated It turns out that it is possible to achieve a considerable econom and elegance in the specification of the problem involving the behavior of the perfectl discriminating monopsonist. The econom involves the elimination of the vector of input quantities, s, from the primal and dual specifications of problems [(9.16), (9.17)] and [(9.18)-(9.20)], respectivel. The corresponding LC problem given in (9.22), therefore, will be reduced b the dimension of the s vector. Furthermore, the primal problem will contain primal and dual variables, as will soon be apparent, while the dual problem will contain onl dual variables. In this case, therefore, the structure of asmmetric quadratic programming (AQP), as discussed in chapter 5, can be used for the specification of the behavior of an economic agent who is a price taker on the output markets and a monopsonist on the input markets. Let us assume that a positive amount of all inputs will be purchased b the perfectl discriminating monopsonist and, thus, p s = g + Gs =, as stated in (9.21). That is, the dual variable vector of the technical constraints in the primal problem [(9.16), (9.17)] is equal to the input price vector p s. In other words, we can take = g +Gs to be the vector of input suppl functions facing the perfectl discriminating monopsonist. With the further assumption that the matrix G is smmetric positive definite, we can invert the sstem of input suppl functions to read s = G 1 g + G 1 (9.23) = b + E where E G 1 and b G 1 g. We recall that the inverse matrix G 1 exists because G is positive definite, b assumption. The present goal, therefore, is to transform the primal and the dual problems of the perfectl discriminating monopsonist as given in [(9.16),

7 The Monopsonist 131 (9.17)] and [(9.18)-(9.20)] into equivalent specifications that do not include the s vector of input quantities. We begin the transformation with the total cost function in (9.16) b replacing the vector s with its equivalent expression in (9.23): TC = g s + 1 2s Gs (9.24) = g [b + E]+ 1 2[b + E]E 1 [b + E] = g b + g E + 1 2b E 1 b + b E = b E 1 b b + 1 2bE 1 b + b E = 1 2b E 1 b E. Similarl, the dual objective function (9.18) can be rewritten as TCMO = 1 2s Gs (9.25) = 1 2b E 1 b + b E. The expression 1 2b E 1 b is a fixed (constant) quantit that does not enter in the optimization process. Hence, it is possible to restate the dual pair of quadratic programming problems [(9.16), (9.17)] and [(9.18)-(9.20)] describing the behavior of a price taker on the output markets and of a perfectl discriminating monopsonist on the input markets b the following and equivalent dual pair of problems: Primal max π = x, c x 1 2 E (9.26) Ax b + E (9.27) x 0, 0 and Dual min TCMO = b E (9.28) A c (9.29) 0. The linear complementarit structure corresponding to this specification of the price taker and the perfectl discriminating monopsonist s problem is stated as [ M = 0 A A E ], q = [ ] c, z = b [ ] x. (9.30) The M matrix has now onl one null matrix, indicating the direction for achieving the full information of the matrix M.

8 132 The Monopsonist Perfectl Discriminating Monopolist and Monopsonist b SQP In this section we combine the behavior of an economic agent who behaves as a perfectl discriminating monopolist on the output markets and as a perfectl discriminating monopsonist on the input markets using the smmetric quadratic programing structure (SQP) discussed in chapter 5. This smmetric structure achieves all the objectives of the economic analsis presented in previous sections without loss of information. Therefore, combining the primal specification of the perfectl discriminating monopolist stated in chapter 8, problem (8.13), with the primal specification of the perfectl discriminating monopsonist stated in [(9.26), (9.27)] we can write Primal max x, π = c x 1 2x Dx 1 2 E (9.31) Ax b + E (9.32) x 0, 0 and Dual min TCMO = x, b E + 1 2x Dx (9.33) A c Dx (9.34) 0, x 0. The economic interpretation of the dual pair of problems [(9.31), (9.32)] and [(9.33), (9.34)] follows established guidelines. The primal objective function (9.31) represents the profit of the given entrepreneur after adjusting it b the quantit 1 2b E 1 b which was left out of the optimization specification because it is a known fixed quantit. Thus, total revenue of the perfectl discriminating monopolist is (c x 1 2x Dx) while total cost of the inputs purchased b the perfectl discriminating monopsonist is ( 1 2b E 1 b E), as given b (9.24). The primal constraints (9.32) state that the demand for inputs, Ax, must be less than or equal to the quantit of inputs purchased on the market, that is s = b + E, as given b (9.23). The dual objective function (9.33) represents the combined total cost of market options for the perfectl discriminating monopolist, 1 2x Dx, and for the perfectl discriminating monopsonist, b E, after adjusting it b the same quantit 1 2b E 1 b as stated in (9.25). The dual constraints (9.34) state the familiar condition according to which the output marginal cost, A, must me greater than or equal to output price, p = c Dx, for the perfectl discriminating monopolist.

9 The Monopsonist 133 The linear complementarit structure corresponding to this SQP specification of the perfectl discriminating monopolist and the perfectl discriminating monopsonist s problem is stated as [ D A M = A E ], q = [ ] c, z = b [ ] x. (9.35) The matrix M has achieved full information with no null matrices. Pure Monopolist and Pure Monopsonist b SQP There remains to discuss the treatment of the pure monopolist and the pure monopsonist in a smmetric quadratic programming framework. To achieve this goal we must first recognize a technical aspect of quadratic programming: the relationship between the quadratic form in the objective function and its derivative, appearing in the appropriate constraint, must be in the ratio of 1 to 2. This fact is not surprising and, in fact, it is a natural consequence of the differentiation process. The pure monopsonist purchases the amount of inputs represented b the vector s on the input suppl function p s = g + Gs and pas p s for each unit of the purchased input vector. Thus, s = G 1 g + G 1 p s, assuming that G is smmetric and positive definite. According to KKT condition (9.6) (assuming s > 0), however, the marginal expenditure (marginal factor cost) of inputs at the pure monopsonist equilibrium,, is higher than the price paid b the pure monopsonist in the amount of = g +2Gs = p s + Gs (9.36) = p s + G[ G 1 g + G 1 p s ] = p s g + p s = g +2p s. Thus, the input price vector paid b the pure monopsonist is p s = g. (9.37) Finall, we can express the suppl function in terms of, the marginal factor cost, instead than the input price vector p s, b noticing that s = G 1 g + G 1 p s (9.38) = G 1 g + G 1 [ g] = 1 2G 1 g + 1 2G 1 = b + Ē

10 134 The Monopsonist where b = 1 2G 1 g and Ē = 1 2G 1. The total cost of the pure monopsonist is given in the objective function (9.1) as TC = g s + s Gs which can be transformed in terms of the vector b replacing s with its equivalent expression of (9.38) TC = g s + s Gs (9.39) = g [ b + Ē]+[ b + Ē] G[ b + Ē] = b G b Ē. Similarl, the total cost of market options of the pure monopsonist is given b the objective function (9.10) and it can be transformed in the dual variables as TCMO = s Gs (9.40) =[ b + Ē] G[ b + Ē] = b G b + b Ē. Notice that the quantit b G b is a constant (fixed) quantit that does not affect the optimization process. We are now read to state the dual pair of problems of an economic agent who behaves as a pure monopolist and as a pure monopsonist using the SQP framework. Primal max x, π = c x x Dx 1 2 Ē (9.41) Ax b + Ē (9.42) x 0, 0 and Dual min TCMO = b Ē + x Dx (9.43) x, A c 2Dx (9.44) 0, x 0 where, it should be recalled, b = 1 2G 1 g and Ē = 1 2G 1. The economic interpretation of the dual pair of problems [(9.41), (9.42)] and [(9.43), (9.44)] follows the same outline given for the perfectl discriminating monopolist and monopsonist.

11 The Monopsonist 135 The linear complementarit structure corresponding to this SQP specification of the pure monopolist and pure monopsonist s problem is stated as [ ] [ ] [ ] 2D A c x M = A Ē, q =, z =. (9.45) b The matrix M has the full information structure with no null matrices. Pure Monopolist and Pure Monopsonist with Asmmetric D and G Matrices When the matrices D and G of the output demand functions p = c Dx facing the pure monopolist and the input suppl functions p s = g + Gs facing the pure monopsonist are asmmetric (the empiricall more realistic case), it is still possible to formulate the problem of this economic agent as a dual pair of optimization specifications. Let us assume, that the matrices D and G are asmmetric positive semidefinite. Then, the primal problem is Primal max π = c x x Dx [g s + s Gs] (9.46) x,s Ax s (9.47) x 0, s 0 The Lagrangean function is L(x, s, ) =c x x Dx g s s Gs + [s Ax] (9.48) where the vector represents Lagrange multipliers or, equivalentl, shadow prices of the limiting inputs. The KKT conditions derived from (9.48) are L x = c (D + D )x A 0 (9.49) x L x = x [c (D + D )x A ]=0 (9.50) L s = g (G + G )s + 0 (9.51) s L s = s [ g (G + G )s + ] =0 (9.52) L = s Ax 0 (9.53) L = [s Ax] =0 (9.54)

12 136 The Monopsonist together with the nonnegativit of all vectors, x 0, s 0, 0. The marginal revenue of the pure monopolist is MR = c (D + D )x and the marginal factor cost of the pure monopsonist is MFC = g +(G + G )s. Hence, the dual problem corresponding to [(9.46), (9.47)] is Dual min x,s, TCMO = x Dx + s Gs (9.55) A c (D + D )x (9.56) g +(G + G )s (9.57) x 0, s 0, 0. The linear complementarit structure corresponding to this specification of the pure monopolist and the pure monopsonist s problem is stated as M = (D + D ) 0 A 0 (G + G ) I, q = c g, z = x s. (9.58) A I 0 0 The matrix M is rather sparse, with several null matrices and two non informative identit matrices. This specification of the pure monopolist and pure monopsonist (with asmmetric D and G matrices) can be reformulated as a SQP structure for achieving a reduction in dimension and full information of the matrix M. Borrowing from the LC problem stated in (9.45), the SQP specification corresponding to (9.58) can be stated as [ (D + D M = ) A 1 A 2(Ē + Ē ) ], q = [ ] c, z = b [ ] x (9.59) where the definition of the matrix Ē and vector b retain the previous specification, that is, b = 1 2G 1 g and Ē = 1 2G 1. Perfectl Discriminating Monopolist and Perfectl Discriminating Monopsonist with Asmmetric D and G Matrices In chapter 8 we have alread discussed the fact that the case of a perfectl discriminating monopolist facing output demand functions p = c Dx with an asmmetric D matrix cannot be fornulated as a dual pair of optimization problems because KKT theor leads to the wrong marginal revenue condition. For a similar reason, the case of a perfectl discriminating monopsonist associated with an asmmetric G (or E) matrix cannot be stated as a dual pair of optimization problems. The decision problem of this entrepreneur, however, can be formulated as an equilibrium problem

13 The Monopsonist 137 that takes on the following structure (using previous and appropriate KKT conditions, as guidelines) Equilibrium Problem Input Demand/Suppl Ax s (9.60) Quantit Complementar Slackness ( s Ax ) =0 Economic Equilibrium Output Market A c Dx Economic Complementar Slackness x ( A c + Dx ) =0 Economic Equilibrium Input Market g + Gs Economic Complementar Slackness s (g + Gs ) =0. The LC problem corresponding to (9.60) is given as M = D 0 A 0 G I, q = c g, z = x s. (9.61) A I 0 0 This equilibrium problem can be reformulated as a LC problem and solved b Lemke s pivot algorithm discussed in chapter 6. The matrix M is rather sparse, with several null matrices and two non informative identit matrices. This specification of the equilibrium problem for the perfectl discriminating monopolist and the perfectl discriminating monopsonist can be reformulated as an equivalent SQP structure for achieving a reduction in dimension and full information of the matrix M. Borrowing directl from the primal and dual constraints (9.32) and (9.34), the SQP equilibrium problem of a perfectl discriminating monopolist and a perfectl discriminating monopsonist facing demand functions for outputs and suppl functions for inputs with asmmetric D and E matrices can be stated as Smmetric Equilibrium Problem Input Demand/Suppl Ax b + E (9.62) Quantit Complementar Slackness ( b + E Ax ) =0 Economic Equilibrium Output Market A c Dx Economic Complementar Slackness x ( A c + Dx ) =0 with a LCP specifications as [ D A M = A E ], q = [ ] c, z = b [ ] x. (9.63) The LC problem (9.63) is structurall different from the LC problem (9.35) because the matrices D and E in (9.63) are asmmetric.

14 138 The Monopsonist Numerical Example 1: Price Taker and Pure Monopsonist In this section we illustrate the case of an entrepreneur who is a price taker on the output markets and a pure monopsonist on the input markets. This corresponds to the scenario discussed in the traditional model (9.1) and (9.2) that is reproduced below, for convenience. Primal max x,s π = c x [g s + s Gs] (9.1) Ax s (9.2) x 0, s 0. c = The required data are given as , g = 1 3 3,A= ,G= The solution of the problem was obtained b GAMS with the following numerical results: Pure Monopsonist Profit = Total Revenue = Total Cost = x 1 = x x = 2 =4.0895, s = s 1 = s x 3 = =2.4202, = 1 = = s x 4 = = = The above numerical solution of the price taker/pure monopsonist entrepreneur was achieved with a traditional texbook representation of the monopsonistic behavior. We wish to illustrate now how to achieve the same solution using a programming specification adapted for the price-taking entrepreneur from model (9.41) and (9.42), that is also reproduced here, for convenience. Primal max x, π = c x 1 2 Ē (9.64) Ax b + Ē (9.65) x 0, 0

15 The Monopsonist 139 The relationship between the given elements of problem [(9.1),(9.2)] and those of problem [(9.64),(9.65)] have been established in previous sections as b = 1 2 G 1 g and Ē = 1 2 G 1. The numerical espression of b and Ē are as follows: b = , Ē = The solution of problem [(9.64),(9.65)] is AQP Pure Monopsonist Profit = AQP Total Revenue = AQP Total Cost = x 1 = x x = 2 =4.0895, = 1 = x 3 = = x 4 = = which is identical to the solution achieved solving the traditional textbook problem given in [(9.1),(9.2)]. Notice that profit must be computed as Profit = b G b + b Ē, while total cost is TC = b G b Ē, as explained in (9.40) and (9.39), respectivel. Numerical Example 2: Pure Monopolist and Pure Monopsonist B SQP With Asmmetric D and E matrices This numerical example deals with an entrepreneur who is a pure monopolist on the output market and a pure monopsonist on the input market. This scenario has been discussed in relation with problem (9.41) and (9.42) which we reproduce here, for convenience. Primal max x, π = c x x Dx 1 2 Ē (9.41) Ax b + Ē (9.42) x 0, 0 The added feature is that we choose matrices D and E that are asmmetric. The definition of vector b and matrix Ē is identical to that in numerical

16 140 The Monopsonist example 1. The relevant data are 15 8 c =, 13 = ,A= Ē = D = The solution of the SQP problem is given as SQP Pure Monopsonist Profit = SQP Total Revenue = SQP Total Cost = x 1 = x x = 2 =1.0532, = 1 = x 3 = = x 4 = = Notice that, in this case, all final commodities are produced at positive levels, in contrast to the preceding numerical example dealing with a price taker on the output market. In order to recover the structure of the traditional textbook specification of the pure monopsonist, it is sufficient to compute: G =2Ē 1, g = Ē 1 b and s = b + Ē. Here are the results: G = 2 5 1, g = 1 3, s = s 1 = s 2 = s 3 =1.0244

17 The Monopsonist 141 Numerical Example 3: Price Taker and Perfectl Discriminating Monopsonist This numerical example assumes the structure discussed in model (9.16) and (9.17), reproduced here for convenience: Primal max x,s π = c x [g s + 1 2s Gs] (9.16) Ax s (9.17) x 0, s 0. It is ver similar to the problem discussed in example 1 and, indeed, we will use the same numerical information in order to assess some differences between the two tpes of economic behavior displaed b the monopsonistic entrepreneur. The required data are given as c = , g = 1 3 3,A= ,G= The solution of the problem was obtained b GAMS with the following numerical results: Perfectl Discriminating Monopsonist Profit = Total Revenue = Total Cost = x 1 = x x = 2 =8.1790, s = s 1 = s x 3 = =4.8405, = 1 = = s x 4 = = = We know that this problem has an equivalent reformulation in term of a smmetric specification as discussed in relation to problem (9.26) and (9.27), reproduce here for convenience: Primal max x, π = c x 1 2 E (9.26) Ax b + E (9.27) x 0, 0

18 142 The Monopsonist The relationship between the given elements of problem [(9.16),(9.17)] and those of problem [(9.26),(9.27)] have been established in previous sections as b = G 1 g and E = G 1. The numerical espression of b and E are as follows: b = , E = The solution of problem [(9.26),(9.27)] is AQP Perfectl Discriminating Monopsonist Profit = AQP Total Revenue = AQP Total Cost = x 1 = x x = 2 =8.1790, = 1 = x 3 = = x 4 = = which is identical to the solution achieved solving the traditional textbook problem given in [(9.16),(9.17)]. Notice that the output quantities, the total revenue, total profit and total cost of the perfectl discriminating monopsonist of example 3 is exactl the double of the same quantities resulted in numerical example 1 for the price taker and the pure monopsonist. Command File for GAMS: Numerical Example 3 We list a command file for the nonlinear package GAMS for solving the numerical problem presented in example 3. Asterisks in column 1 relate to comments. $TITLE A Traditional Textbook Perfectl Discriminating Monopsonist $OFFSYMLIST OFFSYMXREF OPTION LIMROW =0 OPTION LIMCOL =0 option iterlim = option reslim = OPTion nlp =minos5

19 The Monopsonist 143 option nlp =conopt3 option decimals =7 ; SETS j Output variables / x1, x2, x3, x4 / i Inputs / 1, 2, 3 / alias(i,k,kk) ; parameter c(j) /x1 15 x2 8 x3 13 x4 16/ parameter gi(i) / / Intercept of demand functions Intercept of input suppl Table A(i,j) Technical coefficient Matrix x1 x2 x3 x ; Table G(i,k) Slopes of suppl functions ; Scalar Scale Parameter to define Economic Agents / 0.5 /; Variables Monopsobj Objective function s name qx(j) Primal variables: Output quantities is(i) Primal variables: Input quantities ; positive variables qx, is ; Equations objequ Name of the objective function s equation techno(i) Name of the technical contraints ;

20 144 The Monopsonist objequ.. monopsobj =e= sum(j, c(j)qx(j) ) - ( sum(i,gi(i)is(i)) + scalesum((k,i), is(k)g(i,k)is(i)) ) ; techno(i).. sum(j, A(i,j)qx(j)) - is(i) =L= 0 ; Model puremonops /objequ, techno /; solve puremonops using nlp maximizing monopsobj ; compute profit, total cost of phsical resources, total revenue parameter profit, totcost, totrev ; totrev =sum(j, c(j)qx.l(j) ) ; profit =monopsobj.l ; totcost =( sum(i,gi(i)is.l(i) ) + scalesum((k,i), is.l(k)g(i,k)is.l(i)) ) ; Inverse of the matrix G parameter identit(i,k) Identit matrix; identit(i,k)$(ord(i) eq ord(k) ) =1 ; parameter zero(i) ; zero(i) =0 ; parameter compident(i,k) ; variables invx(i,k) Inverse of the G matrix obj Name of the objective function in the inverse problem ; equations eq(i,k) objfunc ; objfunc.. obj =e= sum((i,k), zero(i)invx(i,k)zero(k) ) ; eq(i,k).. sum(kk, G(i,kk)invx(kk,k) ) =e= identit(i,k) ; model inver / objfunc, eq /; solve inver using nlp minimizing obj;

21 The Monopsonist 145 compident(i,k) =sum(kk, G(i,kk)invx.l(kk,k) ) ; computation of bbar =-invgg, Ebar =-InvG parameter bbar(i) intercept of suppl function in sqp Ebar(i,k) matrix Ebar in sqp ; Ebar(i,k) =invx.l(i,k) ; bbar(i) =- sum(k, invx.l(i,k)gi(k) ) ; sqp for the monopsonistic problem variables sqpmonops Name of the objective function in the SQP model sqpx(j) Primal variables: x vector sqp(i) Primal variables: vector ; positive variables sqpx, sqp ; equations sqpobjeq Name of the objective function equation in SQP techeq(i) Name of the constraints ; sqpobjeq.. sqpmonops =e= sum(j, c(j)sqpx(j) ) - (1/2) sum((i,k), sqp(i)ebar(i,k)sqp(k) ) ; techeq(i).. sum(j, A(i,j)sqpx(j)) =L= bbar(i) + sum(k,ebar(i,k)sqp(k)) ; model sqpmonips / sqpobjeq, techeq / ; solve sqpmonips using nlp maximizing sqpmonops ; parameter bgb, TCMO, TC, Profsqp, totrevaqp ; bgb =sum((i,k), bbar(i)g(i,k)bbar(k) ) ; TC =- bgb/2 + (1/2) sum((i,k), sqp.l(i)ebar(i,k)sqp.l(k) ) ; TCMO =bgb/2 + sum(i, bbar(i)sqp.l(i)) + (1/2) sum((i,k), sqp.l(i)ebar(i,k)sqp.l(k) ) ; Profsqp =sqpmonops.l + bgb/2 ; totrevaqp =sum(j, c(j)sqpx.l(j) ) ; displa monopsobj.l, qx.l, is.l, techno.m ; displa profit, totcost, totrev ;

22 146 The Monopsonist displa identit, invx.l, compident ; displa bbar, ebar ; displa sqpmonops.l, Profsqp, TC, TCMO, totrevaqp, sqpx.l, sqp.l, techeq.m ;