University of California, Davis Department of Agricultural and Resource Economics ARE 252 Optimization with Economic Applications Lecture Notes 5 Quirino Paris The Monopolist................................................................. page 1 The Pure Monopolist with symmetric D matrix........................................... 1 The Perfectly Discriminating Monopolist with symmetric D matrix........................... 3 The Pure Monopolist with asymmetric D matrix.......................................... 5 The Perfectly Discriminating Monopolist with asymmetric D matrix.......................... 6 Solution of the Quadratic Programming Problem Linear Complementarity Problem (LCP)........ 7 The Dual of the Least-Squares Method................................................... 8 The Monopolist (See also chapter 8 of the Symmetric Programming textbook) Meaning of monopoly: single seller on the market. (Greek: mono = single; poly from poló = to sell) There are several types of monopolists. We will deal with two types: 1. The pure monopolist 2. The perfectly discriminating monopolist. The Pure Monopolist with symmetric D matrix A pure monopolist charges the same price for each unit of sold output. In general, a monopolist produces and sells more than one commodity. Hence, a monopolist develops and owns a system of demand functions. These demand functions are Marshallian (uncompensated) demand function that, in general, exhibit a non-symmetric matrix of price slopes. To make it easier (at the beginning), however, we assume that a pure monopolist owns the following system of inverse demand functions p = c Dx, where D is a symmetric positive definite matrix. We assume also that a vector b of limiting inputs is available to produce the vector of outputs x by means of a linear technology matrix A of dimensions (m n, m < n). We finally assume that the overall monopolist s objective is to maximize profit, but because of the linear technology, it is necessary to decompose profit into the two objectives of maximizing total revenue and minimizing total cost. These two sub-problems constitute the primal and the dual model, respectively. The typical monopoly diagram is presented in figure 1 Figure 1. The pure monopolist s diagram 1
Figure one describes the primal and the dual specifications of the pure monopolist s problem. We are ready now to state the primal problem: Primal: maxtr = p x = (c Dx) x = c x x Dx D x =[p ] x = [c ] x [x ] (1 n)(n 1) (1 n)(n 1) (1 n)[(n n)](n 1) D S Ax b x 0 We are interested in deriving the dual problem. This goal requires to derive the relevant KKT conditions and using the dual complementary slackness condition to simplify the Lagrange function which, then, will become the dual objective function. Lagrange function with relevant KKT conditions L = c x x Dx + y [b Ax] L = c 2Dx A y 0 x dual constraints L x = x [c 2Dx A y] = 0 x CSC Using the information of the CSC (complementary slackness conditions) in the form of c x = 2x Dx + x A y it is possible to simplify the Lagrange function by replacing c x L = 2x Dx + x Ay x Dx + y b y Ax = y b + x Dx = TC Therefore, the dual specification of the pure monopolist s problem is stated as mintc = TC pp + TCMO Dual = b y + x Dx MC MR A y c 2Dx c D x y A 2 (n m)(m 1) (n 1) (n n)(n 1) Dual constraints: A y c 2Dx. Alternatively, the dual constraints can be re-specified as A y {c Dx} Dx MC p Dx Hence, the quantity Dx is a measure of the pure monopolist s market power. 2
The reason why we cannot maximize directly the pure monopolist profit is that if we were to define maxπ = p x b y Ax b the corresponding Lagrange function would be losing the term b y. The correct and meaningful interpretation of any dual problem requires the introduction of a dual economic agent who watches the primal economic agent and bids to take over his firm. In the pure monopolist s problem, the dual economic agent wants to minimize total cost, TC, that includes the total cost of the physical plant, TC pp = b y, and the total cost of market options, TCMO = x Dx. Recall that demand functions represent options to sell and make profit. Therefore, the dual economic agent must reimburse the primal monopolist of the possibility of making profit if he were to operate the monopoly firm, that is Profit = x Dx. Clearly, the take-over monopolist wants to minimize the cost of purchasing the demand functions from the primal monopolist. This is actually what happens in the real economy when economic agents buy out firms. The profit of the pure monopolist is computed after the solution of the NPP. Recall that, at the optimum solution, the primal objective function value is equal to that of the dual objective function. Therefore TR = TC pp + TCMO c x x Dx = by + x Dx and since profit is defined as total revenue minus total cost of physical plant, we have π = TR TC pp = ( c x x Dx) b y = x Dx = TCMO. The primal monopolist maximizes profit while the dual monopolist (who wishes to take over the monopoly firm) minimizes the total cost of market options (that is the cost of purchasing the demand functions) and the total cost of the physical plant. The Perfectly Discriminating Monopolist with symmetric D matrix The perfectly discriminating monopolist charges a different price for each unit of commodity sold. This may not be a far fetched assumption. In the internet, several firms attempt to charge different prices to different purchasers. With the increasing collection of information about internet users, several analytics firms attempt to measure the willingness to pay of individual customers. This has to do with analytics, machine learning, AI. The usual assumptions about initial information: p = c Dx 1. Demand functions:, where the matrix D is symmetric positive definite 2. Linear technology matrix A with dimensions (mxn, m<n) 3. Limiting input supply, b. In the case of the perfectly discriminating monopolist, total revenue is the integral under the inverse demand functions. Figure 2 gives an idea of what happens: x * TR = (c Dx ) dx = c x * 2 1 (x * ) Dx * 0 3
Figure 2. The Perfectly discriminating monopolist This multiple integral is similar to integrating a linear function of only one variable. For example w Z = (17 5x)dx = 17w 2 1 5w 2 0 = 17w w5w 0 The primal model of the perfectly discriminating monopolist is stated as follows Primal maxtr = c x 1 2 x Dx D S Ax b x 0 To derive the dual model we start with the Lagrange function, compute the relevant KKT conditions, use the dual CSC to simplify the Lagrange function that becomes the dual objective function; Lagrange function L = c x 1 2 x Dx + y [b Ax] Relevant KKT conditions L = c Dx A y 0 dual constraints x L x = x [c Dx A y] = 0 CSC x Use the complementary slackness conditions in the form of c x = x Dx + x A y to replace c x in the Lagrange function and then simplify: L = c x 1 2 x Dx + y [b Ax] = x Dx + x A y 1 2 x Dx + y b y Ax = y b + 1 2 x Dx Therefore, the perfectly discriminating monopolist s dual model is stated as Dual mintc = TC pp + TCMO Of course, x 0, y 0. = b y + 2 1 x Dx MC MR A y c Dx = p 4
The economic interpretation of the dual problem is similar to that of the pure monopolist. A dual perfectly discriminating monopolist wants to minimize the total cost of buying out the monopoly firm. He will have to minimize the total cost of purchasing the physical plant and the total cost of purchasing the demand functions (TCMO). The dual constraints represent the familiar relation of marginal cost being greater than or equal to marginal revenue that, in this case, is equal to the price along the inverse demand function. Profit of the perfectly discriminating monopolist is computed by equating the primal and dual objective functions and rearranging terms according to the profit definition = total revenue minus the total cost of the physical plant: π = (c x 1 x Dx) b y = 1 x Dx = TCMO. 2 2 The Pure Monopolist with asymmetric D matrix We have stated previously that, in general, a system of Marshallian demand functions does not exhibit a symmetric matrix of price slopes. Therefore, we now assume that the system of inverse demand functions is specified as p = c Dx with an asymmetric positive definite D matrix. The remaining information is as above. The pure monopolist s total revenue is always p x and, therefore, the primal model is Primal maxtr = p x = c x x Dx D S Ax b x 0 The Lagrange function L = c x x Dx + y [b Ax] with relevant KKT conditions (remember that D is asymmetric) L = c (D + D )x A y 0 dual constraints x L x = x [c (D + D )x A y] = 0 CSC x Use the complementary slackness conditions in the form of c x = x (D + D )x + x A y to replace c x in the Lagrange function and then simplify: L = x (D + D )x + x Ay x Dx + y b y Ax = x Dx + x Dx x Dx + y b = by + x Dx = TC Note that x D x = x Dx because x D x is a scalar. Therefore, the dual problem of the pure monopolist with an asymmetric positive definite D matrix is mintc = TC pp + TCMO Dual = b y + x Dx 5
MC MR A y c (D + D )x = c Dx D x = p D x Of course, x 0, y 0. The economic interpretation is similar to that of the pure monopolist with a symmetric D matrix. The term D x is a measure of the monopolist s market power. The Perfectly Discriminating Monopolist with asymmetric D matrix The system of inverse demand functions looks as before p = c Dx but, now, the D matrix is asymmetric positive definite. This case causes a serious mathematical problem. In other words, the integral under this system of demand functions does not exist. There exists a theorem (Frobenius theorem) that states that the integral of a system of (differential) equations exists if and only if the matrix of slopes is symmetric. If the integral is undefined, total revenue cannot be defined. Hence there is no objective function to maximize. When the total revenue function does not exist also marginal revenue is undefined (no derivative of a function that is not defined). But the problem of the perfectly discriminating monopolist does indeed exist. He wants to know how many units of the various commodities should be produced. When this piece of information is known, the selling price for each unit of commodity will also be known. When discussing the Economic Equilibrium it was stated that there are economic problems that cannot be specified as an optimization problem. This case is an example of non optimization but of equilibrium, nevertheless. Therefore, the problem of the perfectly discriminating monopolist with an asymmetric D matrix must be solved by using directly the structure of the economic Equilibrium: D S Ax b (S D)P = 0 [b Ax] y = 0 P 0 y 0 MC P A y c Dx = p (MC P)Q = 0 [A y c + Dx] x = 0 Q 0 x 0 The solution of the above system of relations solves the problem of the perfectly discriminating monopolist with an asymmetric D matrix. What happens, therefore, is that it is not possible to define profit as there are no primal and dual objective functions to equate. The only financial measure is the amount of money that corresponds to total cost as the dual complementary slackness conditions illustrate: y Ax = c x x Dx TC = Bundle of money The money made by this perfectly discriminating monopolist cannot be divided exactly (by formula) between total revenue and profit because we do not have a measure of revenue. It falls upon the board of directors of this firm to divide the money between dividends and remuneration of productive factors. This is a trial and error operation. Often shareholders fire CEOs or board of directors. 6
Solution of Quadratic Programming Problems Linear Complementarity Problem (LCP) All the models discussed in previous sections represent quadratic programming specifications. Their solution requires the solution of the system of KKT conditions (or of the Economic Equilibrium). This means that the primal and dual constraints and the corresponding complementary slackness conditions must be handled simultaneously. There exist a very elegant framework that encompasses all the models that we discuss in this course. Such a framework has the name of Linear Complementarity Problem (LCP). The algorithm for solving the LCP is the most simple and the most elegant method for finding the solution (if it exists) of all the problems we discuss in this course. It is due to Carlton Edward Lemke. KKT conditions of the pure monopolist with a symmetric D matrix We begin by collecting the primal and the dual constraints into appropriate matrices (from page 2) with all constraints running in the same direction: 2Dx + A y c Ax b 2D A x c (1) A 0 y b The complementary slackness conditions are similarly stated in matrix notation x [2Dx + A y c] = 0 Now add the two equations y [ Ax + b] = 0 2Dx + A y c = 0 Ax + b 2D A c (2) x y x + x y = 0 A 0 y +b We have not solved anything, as yet. We just organized the KKT conditions in a convenient and compact framework by defining the following matrix and vectors: M = 2D A A 0 x y, q =, z = Now we can write the matrix equations in (1) and (2) to form the Linear Complementarity Problem: Given (M,q), find z 0 such that Mz + q 0 z Mz + z q = 0 This LCP is the problem to solve for the pure monopolist with a symmetric D matrix. KKT conditions of the perfectly discrimination monopolist with a symmetric D matrix We repeat the process outlined in the previous case by speeding up the writing. From page 4 c b x y 7
Dx + A y c Ax b x [ Dx + A y c] = 0 y [ Ax + b] = 0 Add the two complementary slackness conditions and define the M matrix and the q,z vectors: D A c x M =, q =, z = A 0 b y Compare the M matrices in the two types of monopolists. The remaining components are identical. KKT conditions for the pure monopolist with an asymmetric D matrix From page (D + D )x + A y c M = Ax b x [(D + D )x + A y c] = 0 y [ Ax + b] = 0 (D + D ) A A 0, c x q =, z = b y Economic Equilibrium relations for the perfectly discriminating monopolist with an asymmetric D matrix From page 6 Dx + A y c Ax b x [ Dx + A y c] = 0 y [ Ax + b] = 0 D A c x M =, q =, z = A 0 b y The M matrix looks similar ( identical ) to the matrix M in the case of a perfectly discriminating monopolist with a symmetric D matrix. But in this case the D matrix is asymmetric. Hence, the solution vectors x,y will be different. The Dual of the Least-Squares Method (See also chapter 5, pages 70-72 and pages 75-77) In general, statisticians and econometricians do not know that the Least-Squares method has a dual specification (try and ask them). That is, it is possible to obtain identical estimates of the parameters and the residuals of a regression function by maximizing something (rather than minimizing the sum of squared residuals) some linear constraints. 8
The linear regression model stated as in a traditional specification: Let y be a (T 1) vector of sample observations, X be a (T k) matrix of explanatory variables, u be a (T 1) vector of disturbances and β be a (k 1) vector of unknown parameters. We postulate the following regression function y = Xβ + u with E(u) = 0 and var(u) = σ 2 I T. The primal version of the Least-Squares (LS) approach is Primal minssr = 1 2 u u y = Xβ + u where SSR is the sum of squared residuals. This specification is a typical quadratic programming problem of the type that we have encountered discussing the monopolist, for example. Hence, we are able to derive the dual specification of it by forming a Lagrange function, deriving the relevant KKT conditions and using some information to simplify the Lagrange function that will become the objective function of the dual LS problem. Lagrange function: choose λ as a Lagrange multiplier vector L = 1 2 u u + λ (y Xβ u) with relevant KKT conditions L = u λ = 0 (1) u L = X λ = 0 (2) β From KKT (1), λ = u, that is, the dual variable (Lagrange multiplier) is identically equal to the primal variable. This is a case of self-duality. It follows that u λ = u u. In view of λ = u, the KKT (2) can be reformulated as X u = 0 that represents the orthogonality condition of the Least-Squares estimates, as illustrated in figure 3. Figure 3. The geometry of the Least-Squares method 9
Using u λ = u u and β X u = 0 it is possible to simplify the Lagrange function to read L = 1 u u + λ (y Xβ u) = 1 u u + u y u u u Xβ = u y 1 u u 2 2 2 Therefore, the dual of the Least-Squares method is the following model Dual max??? = y u 1 2 u u X u = 0. The meaning of the dual LS model must refer to the terminology of Information Theory where a message is decomposed into a signal and noise. Therefore, the regression function (defined by the sample information) must be interpreted as message = signal + noise y = Xβ + u But, in the LS model, the (self-dual) vector u exhibits a double role (and meaning): in the primal model it measures the quantity of noise; in the dual model it represents the shadow price of the sample information, y. We can conclude that y u the total gross value of sample information 1 1 2 u u = 2 u λ the total cost function of noise Hence the dual objective function maximizes the Net Value of Sample Information (NVSI) max NVSI = y u 1 2 u u Final question. Where is the vector of parameter β in the dual LS model? Answer: it corresponds to the Lagrange multiplier vector of the dual constraints. That is, by solving the dual LS model one obtains the LS estimates of the unknown parameter vector β as the values of the Lagrange multipliers of X u = 0. It is also to remember that the LS variance of the noise is obtained as σö 2 = y uö/ T. Example. The following numerical example provides the Least-Squares estimates of the parameters of a linear regression by the dual specification and shows that the estimates are consistent in the sense that the estimates approach the true value of the parameters as the sample size increases. 10
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