Monopolist s mark-up and the elasticity of substitution

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1 Croatian Oerational Research Review 377 CRORR 8(7), Monoolist s mark-u and the elasticity of substitution Ilko Vrankić, Mira Kran, and Tomislav Herceg Deartment of Economic Theory, Faculty of Economics and Business, University of Zagreb, J. F. Kennedy 6, Zagreb, Croatia {ivrankic, mkran, therceg}@efzg.hr Abstract. This aer analyzes the mark-u of the rice of a roduct over marginal costs for a monoolist using Aelbaum s theoretical model. The rofit maximization model of an industry that uses the monoolist s roduct as its inut is formulated. Our goal is to exress the monoolist s mark-u as a function of the elasticity of substitution for the resective industry and to analyze how changes in the elasticity of substitution affect the mark-u ratio. Consequently, the CES roduction function along with its substitution arameter is chosen. An analytical descrition of changes in the elasticity of substitution and its influence on the monoolist's mark-u is given. All scenarios are sulemented by geometrical illustrations, economic interretations and numerical examles. Keywords: rofit maximization, monoolist, mark-u, elasticity of substitution Received: Setember 3, 6; acceted: June 9, 7; available online: November 3, 7 DOI:.7535/crorr.7.4. Introduction Identifying and measuring market ower in roduct markets has drawn the attention of numerous economists, due to the influence of market ower on overall economic erformance [4,9]. The most direct indicator of monooly ower is the mark-u ratio, which is defined as the ratio of difference between the rice and a firm s marginal cost to rice, known in literature as the Lerner index [7]. Since marginal costs are not directly observable, emirical measuring them emirically is relatively difficult. A new methodology in estimating the mark-u of rices over marginal costs at an aggregate level was devised in the late 98-s [6] and uses the Solow residual. Due to some criticism of Hall s aroach to its emirical alication, Roeger modified Hall s original method [8]. Aelbaum [] in his analysis, unlike other studies in which the degree of monooly is measured, offers a framework for testing the rice using a behavioral hyothesis for the industry and alying it to the U.S. crude etroleum and natural Corresonding author htt:// 7 Croatian Oerational Research Society

2 378 Ilko Vrankić, Mira Kran and Tomislav Herceg gas industry. This framework for analyzing a non-cometitive industry or firm is the basis for our aer. The mark-u of roduct rice over marginal costs as a direct indicator of imerfect cometition is analyzed using a theoretical model to formulate the rofit maximization model of an industry that uses the monoolist s roduct as its inut. The industry s technology is described by the CES roduction function and is chosen given the existence of the substitution arameter, which in turn is used in calculating the elasticity of substitution and measures the ease of substitution between the monoolist s inut and other inuts of the resective industry. The solution of the necessary first-order conditions for the industry s rofit maximization model yields the demand function for the monoolist s roduct. The next ste involves using a derived demand function and solving a monoolist s rofit maximization model. The goal of this aer is to exress the monoolist s mark-u as a function of the industry s elasticity of substitution and to analyze how changes in the elasticity of substitution affect the mark-u ratio. In ractice, this model is alicable to many industries. For instance, one such industry is the grahite lead industry, where grahite encils initially used ure grahite for the lead which was very exensive, resulting in low volumes manufactured. This will be later exlained relying on the lower boundary of the elasticity of substitution between inuts. Once ure grahite was relaced with cheaer unclean grahite and mixed with clay, both of which are abundantly available, demand for ure grahite fell and in time the grahite monooly collased as did its roducer. This again will be exlained later using the uer boundary of the elasticity of substitution.. The monoolist's mark-u Let s analyze first an industry and suose that its technology is reresented by a constant elasticity of substitution (CES) roduction function, y G( y, x ) ( ) x,,,, () where y and x are quantities of inuts used in the roduction rocess by an industry, is the distribution arameter, is the scale arameter and is the substitution arameter. Let s assume that one of the inuts the industry uses is roduced by a monoolist and the quantity is reresented by y and x is the vector of other inuts used in the roduction rocess of an industry.

3 Monoolist s mark-u and the elasticity of substitution 379 An industry maximizes rofit by choosing otimal quantities of inuts subject to given rices of inuts and the rofit maximization model of an industry is formulated as []: max w G( y, x) y wx, () y, x where w is the rice of an industry roduct, is the monoolist s roduct rice and w is the vector of other inut rices. By differentiating the goal function with resect to the decision variables, the first-order necessary conditions are obtained, w G( y, x) y (3) w w x G y (, x ), and the solution to the system of equations gives the demand functions for inuts, y ( ), x ( ). We suose that the second-order sufficient conditions are satisfied. Given that we have chosen the CES roduction function, the rofit maximization model of an industry is defined as y, x max w y ( ) x y wx, (4) where we assume, without the loss of generality, that the industry, in addition to the monoolist s roduct as inut emloys one more inut where the quantity is reresented by x, and rice is reresented by w. By differentiating the goal function in (4) with resect to the quantities of inuts, the following system of equations is obtained. The solutions are the demand functions for the monoolist s roduct and for the other inut, G( y, x) w w y x y ( ) y (5) G( y, x) w w w y x x x ( ) ( ).

4 38 Ilko Vrankić, Mira Kran and Tomislav Herceg Due to functional form of the chosen CES roduction function, the comutation is somewhat tedious, hence it will be resented further on in the aer. Let s exress the quantity of inut, x, as a function of the monoolist s roduct quantity, x w y x y ( ) ( ) w y w ( ) w w ( ) ( ) y x x x y (6) Our interest in this aer is to obtain the demand function for the monoolist s roduct and incororate it into the monoolist s rofit maximization model in order to exress the mark-u of monoolist s roduct rice over marginal cost as a function of the elasticity of substitution. The next ste in deriving the demand function for the monoolist s roduct involves inserting (6) into the first equation of (5), and hence obtain the demand function for the monoolist s roduct, w ( ) ( ) w y y y w y w ( ) w ( ) y w (7)

5 Monoolist s mark-u and the elasticity of substitution 38 ( ) w y w y w y w y w w w ( ) w (8) (9) Let s now focus on the monoolist s rofit maximization model. We chose to take the monoolist s roduct rice as his decision variable instead of the outut quantity, hence the monoolist s rofit maximization roblem is formulated as follows: max y ( ) c,, y ( ), () where is the monoolist s roduct rice, y ( ) is the industry s demand function for the monoolist s roduct and c,, y( ) is the cost function which describes monoolist s technology, and deends on the rice of inuts that the monoolist uses in his roduction rocess,,, and the given level of roduction, here exressed as a function of rice, y ). ( By differentiating the goal function with resect to the decision variable, the first order necessary condition is obtained: c,, y( ) y( ) y ( ), () y where we observe a deviation of the monoolist s roduct rice from the marginal cost, which is the mark-u and is given by the exression within the brackets.

6 38 Ilko Vrankić, Mira Kran and Tomislav Herceg Let s now assume that the monoolist s technology is reresented by the Cobb- Douglas roduction function, y F( y, y ) y y,,, () where y is the monoolist s outut quantity, F ( y, y ) is the roduction function describing the monoolist s technology, and y, y are inut quantities that the monoolist emloys in the roduction rocess. Given that our chosen decision variable in the monoolist s rofit maximization model is the roduct rice, from which the roduct quantity can be easily obtained just by inserting the otimal roduct rice into the demand function for the monoolist s roduct, the starting function reresenting the monoolist s technology has to be the cost function instead of the roduction function. From the duality theory in microeconomics, we know that the cost function reresents technology as the roduction function equally well [,3,5]. The cost function is derived from the model of cost minimization subject to the given outut level [7,]: min y, y y y s. t. F( y, y ) y y y. (3) Base on microeconomic theory, first-order necessary conditions describe economic efficiency in which the marginal rate of technical substitution, F / y MRTS, F / y is equal to the inut rice ratio,. In our case, economic efficiency is reresentted by the following equality, y y, (4) from which the equation of the long run exansion ath in roduction can be derived, y By inserting it in the constraint in (3), the conditional inut demand functions are obtained, y. (5)

7 Monoolist s mark-u and the elasticity of substitution 383 y (,, y ) y and (6) y (,, y ) y Finally, by inserting the derived inut demand functions in the goal function of (3), the monoolist s cost function is obtained. c(,, y ) y (,, y ) y (,, y ) c(,, y ) y y c(,, y ) ( ) y. (7) Now, let s go back to the equation () and exress the first-order necessary condition for the monoolist s rofit maximization model in our case. The exression for the marginal cost exressed as a function of the roduct rice, c,, y( ), is obtained by inserting y ( ) in y what gives us y y c(,, y ), (8) ( ) ( )( ) ( ),, ( ) ( ) w w y c y Now we are in the osition to discuss how the monoolist s mark-u deends on the elasticity of substitution. (9).

8 384 Ilko Vrankić, Mira Kran and Tomislav Herceg 3. Monoolist s mark-u and the elasticity of substitution The elasticity of substitution measures the ease of substitution between the roduction factors due to their rice changes. It is the elasticity of the inut quantities ratio with resect to the marginal rate of technical substitution, x d ln y G( y, x) / y d ln G( y, x) / x. () The industry s CES roduction function is characterized by the constant elasticity of substitution which is shown below. The marginal rate of technical substitution is exressed as a function of the inut quantities ratio, G( y, x) / y x y MRTS G( y, x) / x ( ) y ( ) x, () from which it can be concluded that the elasticity of substitution is determined by the substitution arameter, y ln MRTS ln ( ) ( ) x ln ( )ln y d MRTS x y d ln x. () d ln MRTS Below we will show how the monoolist s mark-u ratio changes when the substitutability of an industry s inuts y and x, measured by the elasticity of substitution, changes in a few scenarios. For this urose, let s assume that the rice of an outut in a erfectly cometitive industry is, w, the other industry s inut rice is, w, and both rices of the monoolist s inuts are equal to,. Let the scale arameter in the industry's roduction function be.5,.5, thus assuming decreasing returns to scale. Let the distribution arameter be equal to.5. To reresent the monoolist's technology,

9 Monoolist s mark-u and the elasticity of substitution 385 we assume that the arameters in the monoolist's Cobb-Douglas roduction function, and, reresenting the elasticities of outut with resect to each inut, are each equal to.5. Since the elasticity of scale is equal to the sum of these elasticities, we thus also assume that there are decreasing returns to scale in the monoolist's technology. 3.. Numerical illustrations Let the elasticity of substitution be.5,,5. The substitution arameter is then equal to. By inserting all given data in the already derived results for the 3 monoolist s demand function in (9), we get the following exression for the monoolist s demand function y 3 ( ) y 5 ( ). Combining it with the first order necessary condition for the monoolist s rofit maximization roblem given in () and with the exression for the marginal cost given in (9), which for the chosen arameters reduces to (3) 5, c 3 3 y (4) we get the following results: the monoolist s roduced quantity is equal to.3, y.3, and the otimal roduct rice is equal to 9.38, The marginal cost is equal to 8.. Therefore, the mark-u of rice over marginal cost MC is equal to.96, and the mark-u ratio or the Lerner index, L, is equal to 58.5%.

10 386 Ilko Vrankić, Mira Kran and Tomislav Herceg The main interest in this aer was to show how the monoolist s mark-u ratio changes as the elasticity of substitution changes. Different values of the elasticity of substitution and the corresonding mark-u ratios are given in the Table. y MC (marginal cost) Mark-u Lerner index % % % % % % Table : The influence of the elasticity of substitution on the monoolist s mark-u ratio As can be seen from the table, the greater the elasticity of substitution, in other words, the greater the ease with which the industry can substitute inuts, one of which is the roduced by a monoolist, the lower the monoolist s mark-u ratio. As the elasticity of substitution becomes greater, the monoolist s rice is closer to the marginal cost of roduction. The deendence of the monoolist s roduct rice, the marginal cost of roduction and mark-u on the elasticity of substitution is illustrated in Figure. One can also notice that σ lies within a certain range, within which no solution was found. The minimum elasticity of substitution is affected by the industry s rice w, which does not allow the monoolist to undertake extortionate ricing. On the other hand, high substitutability of the monoolistic inut with an inut x causes monoolist s demand to fall below its average cost causing the monoolist to go bankrut. After that oint, this model no longer exists since the monoolist has disaeared.

11 Monoolist s mark-u and the elasticity of substitution 387 Figure : The monoolist s roduct rice, the marginal cost, and the monoolist s marku as a function of the elasticity of substitution Figure clearly shows that the greater the elasticity of substitution, the lower the monoolist s mark-u and the rice and marginal cost are closer to each other. Knowing that the elasticity of substitution is related to the curvature of the isoquant, in other words, the greater the elasticity of substitution, the less curved the isoquant is, allows us to illustrate below how the choice of the monoolist s roduct and other inut by an industry changes with a change to the elasticity of substitution for a cost minimization roblem. It is also obvious that the greater the elasticity of substitution, the flatter the exansion ath, leaning towards a erfectly cometitive inut.

12 388 Ilko Vrankić, Mira Kran and Tomislav Herceg Figure : Industry s choice as a function of the elasticity of substitution 4. Conclusion In this aer, the mark-u of roduct rice over marginal costs for the monoolist was analyzed using Aelbaum s theoretical model. The rofit maximization model of an industry which uses the monoolist s roduct as its inut was formulated. Our goal was to exress the monoolist s mark-u as a function of the industry s elasticity of substitution and to analyze how changes in the elasticity

13 Monoolist s mark-u and the elasticity of substitution 389 of substitution affect the mark-u ratio. Consequently, in reresenting the industry s technology, the CES roduction function was chosen with its substitution arameter used in the elasticity of substitution calculation. The first task was to solve the industry s rofit maximization model, where soluteon of the first-order necessary conditions yields the demand function for the monoolist s roduct. The next ste involved using the derived demand function for the monoolist and solving the rofit maximization model so as to exress the mark-u of roduct rice over marginal cost of a monoolist as a function of the elasticity of substitution. We showed for a number of numerical examles that the greater the elasticity of substitution, the lower the monoolist s mark-u ratio. This is so because the elasticity of substitution measures the ease with which the industry can substitute inuts, one of which is roduced by a monoolist. As the elasticity of substitution becomes greater, the monoolist s rice is closer to his marginal cost of roduction. There exists a range of the elasticity of substitution beyond which the model does not rovide a solution; the minimum being determined by the industry s selling rice which has to justify the monoolist s high mark-u and the maximum determined by the substitutability of the monoolistic inut which would render the monoolist unnecessary. In this aer, these boundaries were aroximated using numerical methods, but in subsequent research, a simler version of the model will be used in an endeavor to find functional relations and the domain of the Lerner index as a function of the elasticity of substitution between monoolistic and erfectly cometitive inuts. References [] Aelbaum, E. (979). Testing rice taking behavior. Journal of Econometrics, 9, [] Blume, L.E. (8). Convex Programming. In Durlauf, S. N. and Blume, L. E. (Eds.). The New Palgrave Dictionary of Economics. New York: Palgrave Macmillan. Available at: htt:// D96.html [Accessed 4/4/7]. [3] Blume, L.E. (8). Duality. In Durlauf, S. N. and Blume, L. E. (Eds.). The New Palgrave Dictionary of Economics. New York: Palgrave Macmillan. Available at: htt:// [Accessed 4/4/7]. [4] Bresnahan, T. (989). Emirical studies of industries with market ower. In Schmalensee, R. and Willig, R. (Eds.). Handbook of Industrial Organization (. 57). Amsterdam: North Holland.

14 39 Ilko Vrankić, Mira Kran and Tomislav Herceg [5] Diewert, W.E. (98). Duality aroaches to microeconomic theory. In Intriligator, M.D.; Arrow, K.J.(Eds.). Handbook of Mathematical Economics ( ). Amsterdam: North Holland. [6] Hall, R.E. (986). Market structure and macroeconomic fluctuations. Brookings Paers on Economic Activity,, [7] Mas Colell, A, Whinston, M. D, Green, J. (995). Microeconomic Theory. New York: Oxford University Press. [8] Roeger, W. (995). Can imerfect cometition exlain the difference between rimal and dual roductivity measures? Journal of Political Economy, 3 (), [9] Schmalensee, R. (989). Inter-industry studies of structure and erformance. In Schmalensee, R. and Willig, R. (Eds.). Handbook of Industrial Organization (. 95 9). Amsterdam: North Holland. [] Silberberg, E. and Suen, W. (). The Structure of Economics: A Mathematical Analysis. Boston, Mass.: McGraw-Hill.