ࡱ> gifdAbjbj>> DTT<P  >N^^^rrr8tLrljJ"":""### l l l l l l l$n`qP/l9^#####/l^^""hl111#r^"^" l1# l11A_~Cd";O+a4k~l0laPq.qhCd"edq^e##1#####/l/l1###l####q######### :DUALITY IN MONOPOLY Ilko Vranki, Mira Krpan University of Zagreb, Faculty of Economics and Business Trg J. F. Kennedy 6, 10000 Zagreb, Croatia {ivrankic, mkrpan}@efzg.hr Abstract: This paper follows our theoretical research on duality in microeconomic theory and applies the duality principles, which rest on the price taking behaviour of economic subjects, on the monopolistic behaviour. The standard approach of deriving the profit function for the monopolist from the production function and defined pseudo production function is accompanied by the alternative approach in which the starting point is the pseudo cost function. Finally, starting from the derived profit function the pseudo production function is recovered and a version of Hotellings lemma is given. All results are illustrated with a numerical example. Keywords: duality, pseudo production function, pseudo cost function, profit function, Hotellings lemma. INTRODUCTION Rationality of economic agents in microeconomic theory is described by the optimization problems in which convex sets play an important role. The possibility of characterizing them in two ways naturally gives rise to duality in microeconomic theory [3,4]. Duality in microeconomic theory includes derivation and recovering of the alternative representations of the consumer preferences and the production technology. Since duality in this case rests on the price taking behaviour, the question is how can the principles of duality be applied in the case of a monopolistic firm where the single producer has influence on the price which he charges for the product [1,2,5,7]? In this paper we show how the profit function for the monopolist for the given demand function can be obtained by introducing the pseudo cost function [6] and solving the profit maximization problem. The standard approach in which the production function and defined pseudo production function is the starting point is also derived. Finally, starting from the derived profit function we show how the pseudo production function can be recovered. Although Hotelling's lemma, a famous result from the microeconomic theory, defined below, cannot be directly applied in obtaining the monopolist's supply function for the given demand function, a version of Hotelling's lemma is illustrated. FROM THE PSEUDO PRODUCTION FUNCTION TO THE PROFIT FUNCTION The emphasis is on a monopolist whose objective is to maximize profit. The monopolist faces the following negatively sloped inverse demand function [5]  EMBED Equation.3 , (1) where  EMBED Equation.3  is the price of the monopolists product and  EMBED Equation.3  represents the influence on demand of other variables, for example income, and D is a function of  EMBED Equation.DSMT4  for which  EMBED Equation.3 . The monopolists technology is described by the production function  EMBED Equation.3 , where  EMBED Equation.3  is the vector of inputs used in the production of the monopolists product  EMBED Equation.3 . Since the total revenue function is described by  EMBED Equation.DSMT4 , the profit maximization model for the monopolist in which the quantities of inputs are the choice variables reduces to  EMBED Equation.3 , (2) where w is the vector of input prices. If we form the pseudo production function [5]  EMBED Equation.3  (3) as deflated revenue function and interpret the parameter w in (2) as the price of the pseudo product, then the known results from duality theory in microeconomics could be applied. It is assumed that sufficient regularity conditions are satisfied so that the maximum exists [1,5,7]. The first order necessary conditions for the profit maximization problem reduce to  EMBED Equation.DSMT4  (4) Therefore, the monopolist will hire the levels of inputs for which the marginal revenue of the corresponding input,  EMBED Equation.3 , is equal to its marginal cost,  EMBED Equation.3 . An interior solution is assumed. It is assumed also that the input market is perfectly competitive so that the monopolist takes input prices as given. Solving the first order necessary conditions leads us to the input demand functions. By substituting the derived input demand functions in the production function, for the given demand function, we obtain the monopolist's supply function. The profit function is obtained by substituting the derived functions in the goal function in (2). To illustrate this procedure, the production function  EMBED Equation.3  (5) is chosen [6]. Let us start from the linear inverse market demand function,  EMBED Equation.3 , (6) for which the pseudo production function is defined as follows  EMBED Equation.3 . (7) The profit maximization model reduces to  EMBED Equation.3 . (8) First order necessary conditions are expressed by the following system of equations  EMBED Equation.DSMT4 , (9) Dividing both expressions in (9) gets us to  EMBED Equation.3 . We can express  EMBED Equation.DSMT4  as a function of  EMBED Equation.DSMT4 ,  EMBED Equation.3 , whose substitution in one of (9) leads us to  EMBED Equation.3 . By solving it we get the monopolists demand function for the first input,  EMBED Equation.DSMT4 . By inserting it in  EMBED Equation.3 , we get the monopolists demand function for the second input,  EMBED Equation.DSMT4 . The supply function for the monopolist, for the given demand function, is then  EMBED Equation.3 . By inserting the derived supply and demand functions in the goal function of a producer, the profit function is obtained,  EMBED Equation.DSMT4  (10) FROM THE PSEUDO COST FUNCTION TO THE PROFIT FUNCTION Another method for obtaining the monopolist's profit function for the given demand function is to start from the monopolists cost function,  EMBED Equation.3 , which is an alternative way of describing technology [9]. The decision variable in this model of the monopolists profit maximization problem is the quantity of production,  EMBED Equation.3 . (11) Taking into account the definition of the pseudo production function,  EMBED Equation.3 , the quantity of production can be expressed as the function of pseudo production,  EMBED Equation.3 . In this case the profit maximization problem reduces to  EMBED Equation.3  (12) where  EMBED Equation.3  can be called the pseudo cost function. The first order necessary condition for the profit maximization problem gives the following equation which needs to be solved to get us to the pseudo production function,  EMBED Equation.3 . (13) Therefore, profit maximization for the given demand function in this model is characterized by the equality between the price of the pseudo product and the marginal pseudo cost. To illustrate how the profit function for the monopolist can be derived by starting from the pseudo cost function, we start from the chosen production function in (5). The cost function is derived from the model of cost minimization for the given level of production,  EMBED Equation.3  (14) From the theory of production it is known that the economic efficiency is characterized by the equality between the marginal rate of technical substitution  EMBED Equation.3  and the input price ratio  EMBED Equation.3  [8]. In our case it is  EMBED Equation.DSMT4 , from which the long-run expansion path is derived, which describes the optimal combinations of inputs at each output level as output expands, EMBED Equation.DSMT4 . Substituting it in the constraint, we get the conditional input demand functions, which give the cost minimizing input levels for the given output level,  EMBED Equation.3  and  EMBED Equation.3 , whose substitution in the goal function of (14) gives us the cost function  EMBED Equation.3 . (15) Our goal is to express it in terms of the pseudo production function and to obtain the pseudo cost function. For the given inverse demand function in (6) the pseudo production function is defined as  EMBED Equation.3 . Therefore, the quantity of production is related to the pseudo production function by the following quadratic equation  EMBED Equation.3  from which it follows  EMBED Equation.3  and the pseudocost function as a function of pseudoproduction function collapses to  EMBED Equation.3 . (16) Therefore, the following optimization model needs to be solved  EMBED Equation.DSMT4 . (17) The first order necessary condition is  EMBED Equation.3 . (18) Solving the equation for y gives the pseudoproduction function,  EMBED Equation.DSMT4  (19) whose substitution into the goal function leads to the same profit function as before,  EMBED Equation.DSMT4  (20) HOTELLINGS LEMMA According to Hotellings lemma [1,2,5,7,8], the derivative of the profit function for a perfectly competitive, price-taking firm with respect to the product price of a firm is equal to the firms quantity supplied. Since the monopolist is not the price taker, the question is how can be Hotellings lemma applied in this case. Starting from the monopolists profit function and looking at the parameter w as the price of the pseudo product, we can apply Hotellings lemma and get the pseudo production function [6],  EMBED Equation.3  (21) The supply function of a monopolist can be obtained from the pseudo production function for the given demand function,  EMBED Equation.3 or  EMBED Equation.3 . So we got the quadratic equation which needs to be solved to get the supply function,  EMBED Equation.3 , for the given demand function  EMBED Equation.DSMT4 . (22) FROM THE PROFIT FUNCTION TO THE PSEUDO PRODUCTION FUNCTION Starting from the profit function for the monopolist, how can we go back and recover the pseudo production function? Since the profit function gives the maximum profit for every combination of input prices,  EMBED Equation.3  and  EMBED Equation.3 , and the parameter w, its definition brings us to the following inequality  EMBED Equation.3 . (23) Definition of the pseudo production function enables us to rewrite the previous inequality as  EMBED Equation.3 , (24) from which it follows that the pseudo production function is the result of the following optimization problem  EMBED Equation.3 . (25) By normalizing the price of the pseudo product and dividing all the input prices by  EMBED Equation.3 ,  EMBED Equation.3 , the previous optimization problem reduces to  EMBED Equation.3  (26) and  EMBED Equation.3 . (27) Its solution is the monopolist's pseudo production function. Below we illustrate how to recover the pseudo production function from the profit function. Starting from our derived profit function in (10), the normalized profit function is  EMBED Equation.DSMT4  and the pseudo production function can be obtained as the solution to the following optimization problem  EMBED Equation.DSMT4 . (28) The system of equations that expresses the first order necessary conditions follows,  EMBED Equation.3  (29) Multiplying quantities of inputs gives  EMBED Equation.3  which implies  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . The product of the corresponding input with its price reduces to  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Inserting the given results in the goal function gives  EMBED Equation.DSMT4  (30) which is the pseudo production function we started from. CONCLUSION Since duality in microeconomics rests on price taking behaviour, the main idea was to apply the known duality principles to the monopolistic case. In deriving the profit function for the monopolist the standard approach is shown which includes starting from the production function and the defined pseudo production function. We give another approach by starting from the pseudo cost function. Finally, starting from the derived profit function the pseudo production function is recovered. A version of Hotellings lemma, important from the empirical standpoint, is given. Application of the duality theory in monopoly to the real data is left for the future research. References: [1] Appelbaum, E. (1975). Essays in the Theory and Applications of Duality in Economics. PhD Thesis. Vancouver:University of British Columbia. [2] Appelbaum, E. (1979). Testing Price Taking Behaviour. Journal of Econometrics, 9: 283294. [3] Blume, L. E. (2008). Convex Programming. In Durlauf, S. N., Blume, L. E. (Eds.). The New Palgrave Dictionary of Economics Online. doi:10.1057/9780230226203.0314. [4] Blume, L. E. (2008). Duality. In Durlauf, S. N., Blume, L. E. (Eds.). The New Palgrave Dictionary of Economics Online. doi:10.1057/9780230226203.0411. [5] Diewert, W.E. (1982). Duality Approaches to Microeconomic Theory. In Intriligator, M.D., Arrow, K..J. (Eds.). Handbook of Mathematical Economics (pp. 535599). Amsterdam: North Holland. [6] Krpan, M. (2011). Dualnost u mikroekonomskoj teoriji. PhD Thesis. Zagreb: Sveu iliate u Zagrebu, Ekonomski fakultet-Zagreb. [7] Lau, L. J. (1978). Applications of Profit Function. In Fuss, M., McFadden, D. (Eds.). Production Economics: A Dual Approach to Theory and Applications (pp. 133 216). Vol II, Part IV. 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W 1 =w 1 w,W 2 =w 2 wCompObj;=1fObjInfo>3Equation Native 4_1501240347:DAFʌʌOle 7CompObj@B8fObjInfoC:Equation Native ;E FMicrosoft Equation 3.0 DS Equation Equation.39q),) F * (x)=maxy:yd"W 1 x 1 +W 2 x 2 +(1,W 1 ,W 2 ){},"W 1 ,W 2 FMicrosoft Equation 3.0 DS Equation Equation.39q0,.) F * (x)='min W 1 ,W 2  W 1 x 1 +W 2 _1501240403FFʌʌOle ACompObjEGBfObjInfoHDEquation Native E_1501405789KFʌʌOle JCompObjJLKix 2 +(1,W 1 ,W 2 ) FMathType 6.0 Equation MathType EFEquation.DSMT49qS\*T * DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APObjInfoMMEquation Native N_1501405642^SPFʌʌOle VG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  p(1,W 1 ,W 2 )== a 2 4(2W 1 12 W 2 12 ++b)  FMathType 6.0 Equation MathTyCompObjOQWiObjInfoRYEquation Native Zu_1501405841IqUFʌʌpe EFEquation.DSMT49qSY$hT h DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  F * (x)==min W 1 ,W 2  W 1 x 1 ++W 2 x 2 ++ a 2 4(2W 1 12 W 2 12 ++b)  FMathType 6.0 Equation MathType EFEquation.DSMT49qOle dCompObjTVeiObjInfoWgEquation Native hS\TT T DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   "f"W 1 ==x 1 "- a 2 4(2W 1 12 W 2 12 ++b) "-2 W 1"- 12 W 2 12 ==0 "f"w 2 ==x 2 "- a 2 4(2W 1 12 W 2 12 ++b) "-2 W 1 12 W 2"- 12 ==0_1501240885?{ZFʌʌOle xCompObjY[yfObjInfo\{ FMicrosoft Equation 3.0 DS Equation Equation.39q(]_[ x 1 x 2 =a 2 16(2W 112 W 212 +b) "4Equation Native |_1501476708gw_FʌʌOle CompObj^`i FMathType 6.0 Equation MathType EFEquation.DSMT49qSó$7T 7 DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  2W 1 ObjInfoaEquation Native _1501476707dFʌʌOle 12 W 2 12 ++b== a2x 1"- 14 x 1"- 14 FMathType 6.0 Equation MathType EFEquation.DSMT49qCompObjceiObjInfofEquation Native _1501476706lbiFʌʌS$'T ' DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  W 1 12 W 2 12 == a4x 1"- 14 x 1"- 14 "- b2Ole CompObjhjiObjInfokEquation Native  FMathType 6.0 Equation MathType EFEquation.DSMT49qS$6T 6 DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  W 1 x 1 == a4x 1 14 x 2 14 "- b2x 1 12 x 2 12 FMathType 6.0 Equation MathType EFEquation.DSMT49q_1501476705nFʌOle CompObjmoiObjInfopEquation Native _1501405942sFOle CompObjrtiS$aT a DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  W 2 x 2 == a4x 1 14 x 2 14 "- b2x 1 12 x 2 12 FMathType 6.0 Equation MathType EFEquation.DSMT49qS/$5T 5 DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  F * (xObjInfouEquation Native K_1501243711xFOle )==2 a4x 1 14 x 2 14 "-2 b2x 1 12 x 2 12 ++ a2x 1 14 x 2 14 ==ax 1 14 x 2 14 "-bx 1 12 x 2 12 , FMicrosoft Equation 3.0 DS Equation Equation.39q_: (1,w 1 w,w 2 w)CompObjwyfObjInfozEquation Native {_1501243792v,}FOle CompObj|~fObjInfoEquation Native C FMicrosoft Equation 3.0 DS Equation Equation.39q' F * (x) FMicrosoft Equation 3.0 DS Eq_1501243817FOle CompObjfObjInfouation Equation.39q' F * (x)Oh+'0H    (08@SOR '15 Samo Drobne Normal.dotmKEquation Native C1TablerSummaryInformation(xDocumentSummaryInformation8DXI^vjkxSKQ ˵^[ "(؂[XuЎ*T 骠8"t:8."TAIzchK^(htVRZwDu:ƢV6bdo1 ( %\&VBNLfvVV`d5LMTvJm/Rژ'I!]\/mEBL+;I=[HxxpI cT ^~O23~i~e_$֏g:ˮcЩcŗ%1m><]7G-ы1)l`~[u~z~~DU i8C\܋׏t2ilOB+';omdd+1gּhHID6Dd 0D , 3 A+?"?+25J%!zhg X`! 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