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1 Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange multplers s under consderaton Applcatons n economcs are eamned Illustratve eamples are presented Introducton The Lagrange multplers method s readly used for solvng constraned etrema problems Let us concentrate on the ratonale for ths method Recall that for a functon f of n varables the necessary condton for local etrema s that at the pont of etrema all partal dervatves (supposng they est) must be zero There are therefore n equatons n n unknowns (the s ), that may be solved to fnd the potental etrema pont (called crtcal pont) When the s are constraned, there s (at least one) addtonal equaton (constrant) but no addtonal varables, so that the set of equatons s overdetermned Hence the method ntroduces an addtonal varable (the Lagrange multpler), that enables to solve the problem More specfcally (we may restrct to fndng a mama), suppose we wsh to fnd values,, n mamzng f (,, n ) subject to a constrant that permts only some values of the s That constrant s epressed n the form g (,, n ) = The Lagrange multplers method s based on settng up the new functon (the Lagrange functon) L(,, n, ) = f (,, n ) + g(,, n ), () where s an addtonal varable called the Lagrange multpler From () the condtons for a crtcal pont are L + g = = () L + g = = n n n L = g(,, n ) =, where the symbols L,g are to denote partal dervatves wth respect to the varables lsted n the ndces Of course, equatons () are only necessary condtons for a local mamum To confrm that the calculated result s ndeed a local mamum second order condtons must be verfed Practcall n all current economc problems there s on economc grounds only a sngle local mamum In a standard course of engneerng mathematcs the Lagrange multpler s usually presented as a clever mathematcal tool ( trck ) to reach the wanted soluton There s no large
2 spectrum of sensble eamples (mostly a lmted number of smple welltred school eamples) to show convncngly the power of the method The economc meanng of the Lagrange multpler provdes a strong stmulus to strengthen ts mportance Ths wll be central to our net consderatons Economc mleu To grasp the ssue we wll notce two useful meanngs Rearrange the frst n equatons n () as, of the Lagrange multplers g = = g = n n () Equatons () say that at mamum pont the rato of moreover t equals The numerators to g s the same for every and gve the margnal contrbuton (or beneft) of each to the functon f to be mamzed, n other words they gve the appromate change n f due to a one unt change n Smlarl the denomnators have a margnal cost nterpretaton, namel g gves the margnal cost of usng (or margnal takng from g ), n other words the appromate change n g due to a unt change n In the lght of ths we may summarze, that s the common beneftcost rato for all the s, e () Eample Let, be a producton functon, where l s a labour and k captal The cost to the frm of usng as nput l unts of labour and k unts of captal s, where and are the per unt costs of labour and captal respectvely If the frm has a fed amount, M, to spend on these nputs then the cost constrant s In order to mamze the functon, subject to ths constrant we set up the Lagrange functon (rewrtng constrant condton to Due to () t holds,,, = =, but s the margnal product of labour and s the margnal product of captal Then frst two equaton can be rearranged (accordng to ()) to gve
3 whch states that at the mamum pont the rato of margnal product to prce s the same for both nputs and t equals Eample A farmer has a gven length of fence F and wshes to enclose the largest possble rectangular area The queston s about the shape of ths area To solve t, let, y be lengths of sdes of the rectangle The problem s to fnd and y mamzng the area S (, = y of the feld, subject to the condton (constrant) that the permeter s fed at F = + y Ths s obvously a problem n constrant mamzaton We put f (, = S(,, g(, = F y = and set up the Lagrange functon () Condtons () are L(, ) = y+ ( F (5) L = y = = F = y These three equatons must be solved The frst two equatons gve = y =, e must be equal to y and due to (5) they should be chosen so that the rato of margnal benefts to margnal cost s the same for both varables The margnal contrbuton to the area of one more unt of s due to () gven by S = y whch means that the area s ncreased by y The margnal cost of usng s g = It means value from g; but snce g(, = F y, the value s taken from the avalable permeter F As mentoned above, the condtons () state that ths rato must be equal for each of the varables Completng the soluton (substtutng = y = n we get F F =, = Now let us dscuss the nterpretaton of If the farmer wants to know, 8 how much more feld could be enclosed by addng an etra unt of the length of fence, the Lagrange multpler provdes the answer 8 F (appromatel, e the present permeter should be dvded by 8 For nstance, let be a current permeter of the fence Wth a vew to our F soluton, the optmal feld wll be a square wth sdes of lengths = and the enclosed area wll be square unts Now f permeter were enlarged by one unt, the value = F = = 5 estmates the ncrease of the total area Calculatng the eact ncrease of 8 8 the total area, we get: the permeter s now, each sde of the square wll be, the total area of the feld s ( ) = 5, 6 square unts Hence, the predcton of 5 square unts gven by the Lagrange multpler proves to be suffcently close Eample Let an ndvdual s health (measured on a scale of to ) be represented by the functon f,, 5,
4 where and y are daly dosages of two drugs It may be verfed, that ths functon attans ts (local) mamum for, wth the correspondng value of, So, at that pont s the best health status possble Now we want to mamze f under the constrant that ths ndvdual could tolerate only one dose per da e We put, 5, g(, = and set up the Lagrange functon L(, ) = + y + y+ 5+ ( Condtons () are L = + = y+ = y = y Applyng Lagrange multplers method we get the soluton =,, = wth the correspondng value of, 8 The value may be nterpreted as the remander to the mamum value of health status Now we reduce the restrcton alterng the constrant equaton to + y = We epect f to ncrease Fndng the new soluton as before we have,5;,5; wth,5;,5 9,5 So, there s stll some remander (appromately =, precsely,5) to the optmal health status Further reducng constrant to leads to the soluton,, whch s the mamum of f (wthout constrant) For hgher sums of (overdose) we epect negatve values of Rewrte constrant condton g (, = as c (, = k,,,, where k s a parameter Then the Lagrange functon s of the form,,, For the partal dervatve of L wth respect to k we get L k= From the nterpretaton of a partal dervatve we conclude, that the value states the appromate change n L( and also f ) due to a unt change of k Hence the value of the multpler shows the appromate change that occurs n f n response to the change n k by one n the condton c (, = k, e, Snce usually c (, = k means economc restrctons mposed (budget, cost, producton lmtaton), the value of multpler ndcates so called the opportunty cost (of ths constrant) If we could reduce the restrcton, e to ncrease k by, then the etra cost s If we are able to realze an etra unt of output under the cost less than, then t represents the beneft due to the ncrease of the value at the pont of mama Clearly to the economc decson maker such nformaton on opportunty costs s of consderable mportance Eample The proft of some frm s gven by PR(, = + 8, + y, y, where, y represent the levels of output of two products produced by the frm Let us further assume that the frm knows ts mamum combned feasble producton to be 5 It represents the constrant + 5 Puttng g (, = 5 we set up the Lagrange functon L(, ) = + 8, + y,y + (5 Applyng the Lagrange multplers method we get the soluton =, y =, = wth the correspondng value of the proft PR (,) = 69 Now we reduce the restrcton alterng the constrant equaton to + y = 5 Fndng the new soluton as before we have
5 = =,666, =,, PR(, ) = 699,9 We see that the ncrease n proft brought about by ncreasng the constrant restrcton by unt has been 9,9  appromately the same as the value that we derved n the orgnal formulaton It ndcates that the addtonal ncrease of labour and captal n order to ncrease the producton has the opportunty cost of appromately Eample The utlty functon s gven by,,, where or y s the number of unts of a good X or Y respectvely Suppose the prce of X s,5 USD per unt, the prce of Y s USD per unt To calculate the optmal combnaton for an ncome of 5 USD we employ Lagrange multplers method The constrant s gven by,5 5 We put, 5,5 and form Lagrange functon,,, 5,5 Applyng ths method we get,,, wth the correspondng value of utlty,867 Now we moderate the constrant to 5 5 Applyng, the method agan we obtan the soluton,,5 wth the correspondng value of utlty Concluson ;,5,8 We see that the ncrease n utlty equals the value of In the nstructon of engneerng mathematcs the Lagrange multplers method s mostly appled n cases when the constrant condton g(, = cannot be epressed eplctly as the functon f () or = h( When solvng constraned etrema problems n economcs the bulk of the constrant condtons may be epressed eplctl so the reason to use the Lagrange multplers method would seem to be too sophstcated regardless of ts theoretcal aspects Wth a vew to the crucal mportance of the economc nterpretatons of Lagrange multplers s the use of the method prmarly preferred Concrete applcatons of the presented nterpretaton prncple may be developed n many economc processes A deeper study on the role of the Langrange multplers n optmzaton tasks may be found n Rockafellar (99) References I Jacques, Mathematcs for Economcs and Busness, AddsonWesle Readng, Massachusetts, 995 IMezník, On Economc Interpretaton of Lagrange Multplers, Proc of the th Internatonal Conference Turnng Dreams nto Realty: Transformatons and Paradgm Shfts n Mathematcs Educaton, Rhodes Unverst Grahamstown, South Afrca, September 7,, 9 W Ncholson, Mcroeconomc Theor The Dryden Press, New York, 998 R T Rockafellar, Lagrange Multplers and Optmalt SIAM Rev, 5(99), pp88 M Wsnewsk, Introductory Mathematcal Methods n Economcs, McGrawHll Comp, London, 99 5
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