A Geometric Analysis of Global Profit Maximization for a Two-Product Firm

Transcription

1 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 6 Number Summer A Geometrc Analyss of Global Proft Maxmzaton for a Two-Product Frm Stephen K. Layson ABSTRACT Ths paper analyzes several fundamental ssues that arse n verfyng a global maxmum for a seemngly smple economc problem, proft maxmzaton for a two-product frm. A new gradent path geometrcal method of verfyng a global maxmum s presented that s analogous to the use of phase dagrams to solve for the eulbrum of a frst-order dfferental euaton system n two varables. An mportant advantage of the geometrcal approach s that t always verfes a global proft maxmum when the proft functon s concave and t also verfes a global maxmum n many cases where the proft functon s nether concave nor uasconcave. Introducton Often n economcs the dstncton between a local and global maxmum s gnored. The reason for ths s the mplct assumpton that economc objectve functons typcally have a sngle local maxmum whch s also the global maxmum. 3 Whether ths assumpton s generally vald or not, the theory of a global maxmum s one of the cornerstones on whch economcs rests so we should be clear about ths dstncton. In Economcs two basc assumptons are that frms are attemptng to fnd the global maxmum of proft and consumers are attemptng to fnd the global maxmum of utlty. Smon and Blume (994, p. 58), the authors of the leadng text n mathematcal economcs, explan well the mportance of global maxma and the closely lnked concept of concave functons n economcs: The property that crtcal ponts of concave functons are global maxmzers s an mportant one n economc theory. For example, many economc prncples, such as margnal rates of substtuton euals the prce rato, or margnal revenue euals margnal cost are smply the frst order necessary condtons of the correspondng maxmzaton problem. Ideally an economst would lke such a rule to also be a suffcent condton guaranteeng that utlty or proft s maxmzed so that t can provde a gudelne for economc behavor. Ths stuaton does occur when the objectve functon s concave. Furthermore, an economst, who wants to analyze how the maxmzer n a parameterzed problem depends on the parameters nvolved, wll usually apply the mplct functon theorem to the euatons of the frst order necessary condtons for maxmzaton. The only stuaton n whch t can be guaranteed that the soluton to these perturbed euatons s ndeed a maxmum for all values of the parameters occurs when the objectve functon s concave. Stephen K. Layson, Department of Economcs, 457 Bryan Buldng, UNCG, Greensboro, NC USA (336) layson@uncg.edu. I wsh to thank M. Taylor Rhodes for hs expert assstance wth the fgures n ths paper as well as edtoral help. For example, the possblty that there could be a dfference between the global and local maxmum n an economc applcaton s never mentoned n Samuelson (97) or Slberberg and Suen (00). 3 For a smple yet mportant example of an economc maxmzaton problem wth two local maxma see Battalo and Ekelund (97).

2 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 6 Number Summer As ponted out n the precedng uotaton 4, n economcs the most common procedure for justfyng an unconstraned global maxmum s to ether establsh or assume that the objectve functon s dfferentable and concave, so that the satsfacton of the frst order condtons guarantees a global maxmum. Note that the rather strong assumpton of concavty can be relaxed some. For example, Ponsten (967) 5 proved that a local maxmum of a strctly uasconcave functon s also a global maxmum. Also, for objectve functons wth closed and bounded domans, we can often use Weerstrass s theorem to prove the exstence of both a global maxmum and a global mnmum. 6 Sydsaeter and Hammond (995, pp ) gve a nce treatment of how to fnd the global maxmum and the global mnmum for ths case. Ths paper analyzes several fundamental ssues that arse n verfyng a global maxmum for a seemngly smple economc problem, proft maxmzaton for a two-product frm. Secton II of the paper gves several plausble examples of proft functons that are nether concave nor uasconcave. Secton III of the paper presents a new gradent path geometrcal method of verfyng a global maxmum that s analogous to the use of phase dagrams to solve for the eulbrum of a frst-order dfferental euaton system n two varables. 7 An mportant advantage of the geometrcal approach to optmzaton problems s that t always verfes a global maxmum when the objectve functon s concave and t also verfes a global maxmum n many cases where the objectve functon s nether concave nor uasconcave. Concavty, Quasconcavty and the Frm Proft Functon Consder a two-product frm s proft functon gven by the C functon, π (, ) where the output of product s, s the output of product, R (, ) s the frm s revenue functon and C (, ) s the frm s cost functon. Euaton () expresses the proft functon as revenue mnus cost. () π, ) = R(, ) C(, ) ( The doman of the proft functon s all nonnegatve values of and wth no upper bounds placed on ether or. Because proft s revenue mnus cost, the proft functon s necessarly concave f the revenue functon s concave and the cost functon s convex. Thus, there are two ways n whch the proft functon fals to be concave, ether the revenue functon s not concave or the cost functon s not convex. Ths secton wll demonstrate that the assumpton that the proft functon s concave s really very strngent. Let = π j π denote the frst order partal dervatves of the proft functon and let π j = π denote the second order partal dervatves of the proft functon. The well known necessary condtons for concavty of the proft functon s that at all ponts n the doman π 0, π 0 and ππ π. The most common type of volaton of the concavty assumpton occurs when ether π or π are postve at some pont. Even when π and π are everywhere negatve, however, volatons of the other condton for concavty, ππ π are also ute possble. 4 The last sentence n the precedng uotaton needs to be ualfed. A functon does not necessarly have to be concave for the satsfacton of the frst order condtons to guarantee a maxmum for all values of a parameter. Consder the x + ax functon f ( x; a) = e. Ths functon s not concave yet the maxmzng value of x s a/ for all values of the parameter a. Whle f s not concave n ths example t s a monotonc ncreasng transformaton of a concave functon. 5 Also see Mangasaran (965) and Martos (965). 6 See for example Smon and Blume (994, Theorem 30., p.83). 7 See Layson (00) for an earler dscusson of the gradent path concept and Chang and Wanwrght (005, pp ) for a good dscusson of two-varable phase dagrams.

3 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 6 Number Summer Euaton () expresses the second dervatves of the proft functon n terms of the second dervatves of the revenue and cost functons () π, ) = R (, ) C (, ). ( A suffcent condton for π to be postve s that margnal cost fall more sharply than margnal revenue. For example, for a two-product frm that sells both products n perfectly compettve markets so that R = R = 0, fallng margnal cost of producng ether product ( C < 0) at any pont n the doman wll volate the concavty assumpton. For a two-product monopoly the possblty of π >0 can arse ether by rsng margnal revenue ( R > 0) and/or fallng margnal cost ( C < 0). 8 Postve values for and/or π can also easly volate the weaker assumpton of uasconcavty. The π necessary condton for uasconcavty s that π π + π π π π π 0 at all ponts n the doman. In the case where π = 0 at all ponts n the doman the necessary condton for uasconcavty smplfes to π π π π 0 whch s clearly volated (at non-crtcal ponts) when π and π are both postve. The followng 3 examples explctly demonstrate dfferent ways n whch the proft functon can fal to be ether concave or uasconcave. Example One Ths example llustrates a case where the proft functon fals to be ether concave or uasconcave because of fallng margnal cost. A prce-takng, two-product frm has a cubc cost functon gven by (3) 3 3 C(, ) = F + α + α β β + γ + γ. 3 3 It s assumed that the parameters F, α, α, β, β, γ and γ are all postve. The margnal cost of producng each product s gven by (4) C = α β + γ =,. To nsure that the margnal cost of producng each commodty s always postve t s assumed that α γ > β and α γ > β. The second dervatves of ths cost functon are (5) C = β + γ =, 8 See Formby, Layson, and Smth (98) for a dscusson of the possblty of upward slopng margnal revenue.

4 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 6 Number Summer (6) C = 0 j j. For = 0, C β < 0. Because ths s a prce-takng frm t follows from euatons () and (5) that at = = 0, π = = > 0 C β. Thus the proft functon s not concave. Because π 0 n ths example, t follows that at = = 0 the necessary condton for uasconcavty, π π + π π π π π 0 s also volated. = Example Two In ths example the proft functon s not concave nor uasconcave even though π 0 and π 0 at all ponts n the doman. A prce-takng, two-product frm s cost functon s n n (7) C(, ) = F + a + a + b + b + h. In the cost functon above a, b, n and h are parameters wth restrctons F 0, a 0, b > 0 and n >. As wll be demonstrated below, for the cost functon gven by euaton (7), for h 0 a necessary condton for convexty of the cost functon (concavty of the proft functon) s that n = n =. The second dervatves of the cost functon are: n (8) C = ( n ) n b 0 (9) C = ( n ) n b (0) C = h n 0 Because n ths example R = R the proft functon. The neualtes = h 0 0, the convexty of the cost functon s euvalent to concavty of π = C and π = C 0 are satsfed at all ponts n the 0 doman n ths example, but f the neualty, π π π can only be satsfed at all ponts n the doman when = n. If ether n or s not exactly eual to then the proft functon s not concave. n = n To see ths use euatons (8)-(0) to rewrte the necessary condton π π π as () n n ( n )( n ) nnb b h Frst note that for n = n neualty () smplfes to 4b b h and as long as the parameters satsfy = ths latter neualty the proft functon s concave. 9 For n >, note that, whch s part of the lefthand-sde of neualty (), goes to zero as goes to zero. Hence for n n >, neualty () fals to hold 9 Chang and Wanwrght (005, 33-3) dscuss an optmzaton problem of ths type n ther classc text.

5 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 6 Number Summer 007 n n at all ponts n the doman. For <, goes to zero as goes to nfnty. Hence for n <, neualty () also fals to hold at all ponts n the doman. It follows that the proft functon s not concave for ether n > or n <. Smlar reasonng wll verfy that the proft functon s not concave for ether n > or n <. It s easy to demonstrate that ths proft functon can also fal to be uasconcave. For one example, (many more can be found) consder the case where: n >, n > and = = 0. In ths case π = π = 0 and the necessary condton for uasconcavty, π π + π π π π π 0 smplfes to ( p a)( p a ) h 0. Obvously ths latter neualty fals to hold f, for example, p > a, p > a and h>0. Example Three In examples and the proft functon faled to be concave because the cost functons were not convex. The next example llustrates a case where the proft functon generally fals to be concave because the revenue functon s only concave n a specal case. Consder a two-product monopoly wth a lnear, convex, cost functon C(, ) = F + a + a that faces nverse demand functons gven by: p = α β γ n () (3) p = γ. α n β Note that for the specal case where n = n = the nverse demand functons above are lnear. It s assumed n euatons () and (3) that α, α, β, β, n, n > 0. The sgns of γ and γ may be ether postve or negatve dependng on whether the two goods are substtutes or complements. The revenue functon n ths example s n (4) (, ) + α ( ) + n + R = α γ + γ β β. Because C = C = 0 n ths example, the second dervatves of the proft functon are: n (5) π = R = ( n +) n β 0 (6) π = R = ( n +) n β (7) π = R = ( γ + γ ) n 0 The neualtes π 0 and π 0 are satsfed at all ponts n the doman n ths example, but f ππ π γ + γ 0, the neualty, can only be satsfed at all ponts n the doman when n = n = β (γ + γ ). If γ + and 4β γ 0 and ether n or n s not exactly eual to then the proft functon s not concave. To see ths use euatons (5)-(7) to rewrte the necessary condton π π π as

6 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 6 Number Summer 007 n n (8) ( n + )( n + ) n n β β ( γ + γ. ) Frst note that for n = n neualty (8) smplfes to 4β β ( γ + γ and as long as the parameters = ) satsfy ths latter neualty the proft functon s concave. For n >, note that, whch s part of the left-hand-sde of neualty (8), goes to zero as goes to zero. Hence for n >, neualty (8) fals to n hold at all ponts n the doman. For n <, goes to zero as goes to nfnty. Hence for n <, neualty (8) also fals to hold at all ponts n the doman. It follows that the proft functon s not concave for ether n > or n <. Smlar reasonng wll verfy that the proft functon s not concave for ether n > or n <. n It s easy to demonstrate that ths proft functon can also fal to be uasconcave. For one example, consder the case where n >, n > and = = 0. In ths case π = π = 0 and the necessary condton for uasconcavty, π π + π π π π π 0 smplfes to a a ( γ + γ ) 0. Obvously ths latter neualty fals to hold f γ + γ > 0. The Geometrcal Approach for Verfyng a Global Proft Maxmum As the last secton has demonstrated, t s easy to construct examples of plausble proft functons that are nether concave nor uasconcave. Ths secton presents a geometrcal method of verfyng a global maxmum that does not reure that the objectve functon be ether concave or uasconcave. To llustrate the geometrcal approach consder the smple concave two-product proft functon dscussed n Chang and Wanwrght s (005, 33-3) classc text (9) π (, ) = p + p. The frst order condtons for maxmzng proft are π = p 4 = 0 and π = p 4 = 0. Fgure shows the graphs of these two frst order condtons n the (, ) plane. The sngle ntersecton of the lnear frst order condtons s the unue crtcal pont of the proft functon and s labeled (, ) n fgure. The ntersectng frst order condtons n fgure dvde the doman nto 4 regons labeled I through IV. The rghtward pontng arrows denote regons n whch π > 0 and the leftward pontng arrows denote regons where π < 0. The upward pontng arrows denote regons n whch π > 0 and the downward pontng arrows denote regons n whch π < 0.

7 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 6 Number Summer Fgure b P π 0 =. 5P II I (, ) π 0 = 0 III. 5P IV P a In regon I of fgure, the gradents all pont n a southwest drecton, n regon II the gradents all pont n a southeast drecton, n regon III the gradents all pont n a northeast drecton and n regon IV the gradents all pont n a northwest drecton. Startng at any arbtrary pont n the doman other than the crtcal pont, (, ), the gradents wll pont towards hgher values of π. Fgure llustrates a few gradent paths whch are analogous to phase paths n two-varable phase dagrams. As one moves along a path n the drecton ndcated by the gradents the value of π steadly rses. If all non-crtcal ponts n the doman have gradent paths leadng to the same pont, ths end pont s the global maxmum. It s mportant to note that when a gradent path crosses the π = 0 locus, the gradent path s vertcal at ths crossng pont and when a gradent path crosses the π = 0 locus, the gradent path s horzontal at ths crossng pont. Therefore, f one starts at any pont n ether regon II or IV of fgure, the gradent path must lead to (, ) wthout passng nto regons I or II. If one starts at a pont n regon I however, there are 3 possbltes: () the gradent path may approach (, ) wthout crossng nto ether regon II or IV, () the gradent path may cross from regon I nto regon II where t stays untl t reaches (, ) or (3) the gradent path may cross from regon I to regon IV where t stays untl t reaches (, ). If one starts at a pont n regon I on the axs such as pont a n fgure the unrestrcted gradent wll pont n the southwest drecton. Because of the non-negatvty restrcton on n ths problem the gradent path travels leftward along the axs untl t crosses nto regon IV where t then approaches (, ). Smlarly, f one starts at a pont n regon I on the axs such as pont b n fgure the unrestrcted gradent wll pont n the southwest drecton. Because of the non-negatvty restrcton on when = 0 n fgure the gradent path travels downward along the axs untl t crosses nto regon II where t then approaches (, ).

8 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 6 Number Summer If one starts at any pont n regon III, there are 3 possbltes: () the gradent path may approach (, ) wthout crossng nto regons II or IV, () the gradent path may cross from regon III nto regon II where t stays untl t reaches (, ) or (3) the gradent path crosses from regon III nto regon IV where t stays untl t reaches (, ). Because all possble non-crtcal ponts n the doman have gradent paths 0 that lead to the pont, ) ths pont s the global maxmzer of π. ( Note n fgure that the π = 0 locus s steeper than the π = 0 locus at the crtcal pont (, ).Ths condton, along wth the condton that π, π < 0 are euvalent to the second order suffcent condtons for a local maxmum. Applyng the mplct functon rule of dfferentaton to the frst order condtons, π, ) 0 and π, ) = 0 the pont slopes of the two loc are, respectvely: ( = ( (0) d d π = 0 π = π ( π 0 ) () d d π = π = 0 π ( π 0 ). Recall that the second order suffcent condtons for a local maxmum are that the second dervatves evaluated at a crtcal pont satsfy π, π < 0 and π π > π. If at a crtcal pont π, π < 0 s satsfed then the crtcal pont wll be a local maxmzer f the π = 0 locus s steeper than the π = 0 locus. To demonstrate ths, consder the 3 possble sub-cases where π, π < 0 : () π > 0, () π < 0 and (3) π = 0. From euatons (0) and () observe that f π > 0 then both loc have postve slopes and f the π = 0 locus s steeper than the π = 0 locus ths s euvalent to π π > whch mples π π π π π >. If π 0 then both loc have negatve slopes and f the π 0 locus s steeper than the < = π π π = 0 locus ths s euvalent to > whch mples that ππ > π. Fnally, f π = 0 then the π π π = 0 locus s vertcal and the π 0 locus s horzontal so the π 0 locus s clearly steeper than the π = 0 locus, and the condton ππ > π s satsfed. The proft functon gven by euaton (9) s strctly concave and the ntersecton of the frst order condtons n fgure gave us the global maxmzer. An advantage of the geometrcal technue s that t can also verfy a global maxmum, when one exsts, for non-concave functons. Consder example dscussed n the prevous secton, where a prce-takng two-product frm has a cubc proft functon gven by = = () π (, ) ( p α ) + ( p α = ) 3 3 F + β + β γ γ One can also llustrate comparatve statc effects wth the geometrcal approach. For example, an ncrease n p wll shft the π = 0 locus n fgure upward causng the value of to rse and the value of to fall. An ncrease n p wll shft the π = 0 locus n fgure rghtward causng the value of to rse and the value of to fall.

9 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 6 Number Summer As shown n the prevous secton the proft functon gven by euaton () s nether concave nor uasconcave. The frst order condtons for maxmzng proft n ths case are: (3) π (, ) p α + β γ 0 ; =,. = = Assumng that p > α for =,, the sngle soluton to the frst order condtons β + β + 4γ ( p α ) s = for =,. The graphs of the two frst order condtons for ths example are γ shown n fgure. The π = 0 locus s a vertcal lne at and the π = 0 locus s a horzontal lne at. Fgure π 0 = II I (, ) π 0 = III IV 0 It s easy to verfy from fgure that the crtcal pont (, ) s a global proft maxmzer. From euaton () note that π > 0 for 0 < and π < 0 for >. Thus, to the left of the π = 0 locus all the horzontal arrows pont n a rghtward drecton and to the rght of the π = 0 locus all the horzontal arrows pont n a leftward drecton. Also note that π > 0 for 0 < and π < 0 for >. Thus, below the π = 0 locus all the vertcal arrows pont n an upward drecton and above the π = 0 locus all the vertcal arrows pont n a downward drecton. In regon I of fgure, the gradents all pont n a southwest drecton, n regon II the gradents all pont n a southeast drecton, n regon III the gradents all pont n a northeast drecton and n regon IV the gradents all pont n a northwest drecton. At any pont on the π = 0 locus other than the crtcal pont, the

10 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 6 Number Summer gradents pont vertcally towards the crtcal pont and at any pont on the π = 0 locus other than the crtcal pont, the gradents pont horzontally towards the crtcal pont. It s mportant to note n fgure that f a gradent path touches the vertcal π = 0 locus or the horzontal π = 0 locus before t reaches the crtcal pont, the gradent path wll reman on the π = 0 or π = 0 locus untl the path reaches the crtcal pont. Therefore, f one starts at any pont n regon I of fgure, the gradent path must lead to (, ) wthout passng nto regons II or III. Smlarly f one starts at any pont n regons II, III or IV, the gradent path must lead to (, ) wthout passng nto another regon. Because all ponts n the doman other than (, ) have gradent paths endng at (, ), (, ) s the global maxmum. An alternatve way of fndng the global maxmum for ths problem would be to recognze that because π < 0 for > and π < 0 for >, one can make the doman closed and bounded by the restrctons that 0 and 0. Then apply the extreme value theorem methods to show that (, ) s the global maxmzer. However, as the next example demonstrates, ths method won t work n cases where there are no fnte values for and, say and, where π < 0 for > ˆ and π < 0 for > ˆ. For the fnal example of the geometrcal method, consder a specal case of example 3 n dscussed n + faces nverse demand functons gven by: the prevous secton. A two-product monopoly wth a cost functon C(, ) = F a + a (4) p = α β +. 5 (5) p = α +. β. 5 From the nverse demand functons above note that the two products produced by the monopoly are complements. The proft functon n ths example s 3 3 (6) π (, ) = ( α a) + ( α a) + β β. The frst order condtons for proft maxmzaton are (7) π = α a + 3β = 0 (8) π = α a + 3β = 0. It s easy to verfy from the second dervatves that the proft functon n ths example s not concave. At = 0, π = π 0 and π, hence the necessary condton for concavty π π s = = = π

11 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 6 Number Summer volated. Nevertheless, t s easy to use the geometrcal method to verfy that the ntersecton of the two frst order condtons gves the global proft maxmzer. Fgure 3 π 0 = II I π 0 = (, ) a α 0 III IV a α Fgure 3 shows the graphs of these two frst order condtons n the (, ) plane. It s assumed n fgure 3 that α > a and α > a.the sngle ntersecton of the frst order condtons s the unue crtcal pont of the proft functon and s labeled (, ) n fgure 3. The ntersectng frst order condtons n fgure 3 dvde the doman nto 4 regons labeled I through IV. Startng at any pont n regons I or III (ncludng the borders), the gradent paths wll lead to the crtcal pont (, ) wthout passng nto ether regon II or IV. Startng at any pont n ether regon II or IV, the gradent paths wll ether drectly approach the crtcal pont (, ) or the gradent paths wll move nto ether regon I or III where they wll reman untl the reach the crtcal pont. Because all ponts n the doman other than the crtcal pont have gradents paths whch end at the crtcal pont, the crtcal pont ( ) n fgure 3 s the global maxmzer., Concluson Ths paper analyzes and dscusses some of the serous dffcultes that arse n verfyng the global maxmum for a seemngly smple economc problem, proft maxmzaton for a two-product frm. The most common procedure for justfyng a global proft maxmum n economcs s to ether establsh or assume that the proft functon s dfferentable and concave, so that the satsfacton of the frst order condtons guarantees a global proft maxmum. Ths paper demonstrates, however, that many plausble proft functons are not concave nor even uasconcave. A new gradent path geometrcal method of verfyng a Tryng to prove that ths functon s uasconcave by showng that π π + π π π π π 0 at all ponts n the doman appears to be an ntractable problem.

12 JOURNAL OF ECONOMICS AND FINANCE EDUCATION Volume 6 Number Summer global maxmum s presented that s analogous to the use of phase dagrams to solve for the eulbrum of a frst-order dfferental euaton system n two varables. An mportant advantage of the geometrcal approach to optmzaton problems s that t always verfes a global maxmum when the objectve functon s concave and t also verfes a global maxmum n many cases where the objectve functon s nether concave nor uasconcave. References Alpha C. Chang and Kevn Wanwrght, Fundamental Methods of Mathematcal Economcs, fourth edton, New York, 005. Raymond C. Battalo and Robert B. Ekelund,Jr., Output Change under Thrd Degree Dscrmnaton, The Southern Economc Journal, 97, 39, John P. Formby, Stephen K Layson, and W. James Smth, The Law of Demand, Postve Slopng Margnal Revenue and Multple Proft Eulbra, Economc Inury, Aprl 98, Stephen K. Layson, Verfyng A Global Maxmum, Workng Papers n Economcs #ECO00, Center for Appled Research, UNCG, 00. Olv L. Mangasaran, Pseudo-Convex Functons, Journal of SIAM Control, Ser. A, 965, 3, 8-9. B. Martos, The Drect Power of Adjacent Vertex Programmng Methods, Management Scence, 965,, 4-5. J. Ponsten, Seven Knds of Convexty, SIAM Revew, 967, 9, 5-9. Paul A. Samuelson, Foundatons of Economc Analyss, Atheneum, New York, 97. Eugene Slberberg and Wng Suen, The Structure of Economcs: A Mathematcal Analyss, Irwn McGraw- Hll, New York, 00. Carl P. Smon and Lawrence Blume, Mathematcs for Economsts, Norton, New York 994. Knut Sydsaeter and Peter Hammond, Mathematcs for Economc Analyss, Prentce Hall, Englewood Clffs, 995. Knut Sydsaeter, Peter Hammond, Atle Seerstad, and Arne Strom, Further Mathematcs for Economc Analyss, Prentce Hall,Ednburgh Gate, 005.