A Simple Economic Model about the Teamwork Pedagogy

Transcription

1 Appled Mathematcal Scences, Vol. 6, 01, no. 1, 13-0 A Smple Economc Model about the Teamwork Pedagog Gregor L. Lght Department of Management, Provdence College Provdence, Rhode Island 0918, USA glght@provdence.edu Abstract Ths note apples the construct of the elastct of substtuton n economcs to an analss of the teachng effectveness n usng the team approach n classrooms. We present examples as based on the well-known constant-elastct-of-substtuton CES producton functon, whch we further specalze nto the partcular cases of zero, one, and nfnte elastctes of substtuton between an two members of a team of three students. We construct a par-wse CES producton functon of three nputs, whch can easl be extended nto n nputs. We hghlght the mportant role of the elastct of substtuton n an producton and thus shed lght ncdentall back onto the state of the econom. Mathematcs Subject Classfcaton: 97D60, 91B38, 53A07, 97D40, 97C70 Kewords: CES producton functon, level set geometr, surface curvatures, elastct of substtuton, factor substtutablt 1 Introducton In man felds of studes the use of team projects n classroom assgnments s ether encouraged or consdered as a smple routne. Even n the feld of mathematcs known for ts abstractness, educators n recent ears have advocated such a pedagog. One tpcal argument n favor of ths approach of teachng and learnng s that of developng students abltes n teamwork so that n ther later work places the wll functon cooperatvel and productvel wth ther co-workers. To be sure, the setup of commttees to engage n group decsons s a ubqutous phenomenon; clearl there has to be some underlng premse for ths, and t appears to be that collectve wsdom and work s superor to that of an ndvdual. We, as classroom nstructors however, are alwas

2 14 G. L. Lght concerned about the problem of substtutablt among group members n an assgned task;.e., Student does less work because Student 1 does more. We wll appl the dea of the well-known CES constant elastct of substtuton producton functon n economcs to a modelng of students teamwork for a smlar treatment, cf. [6]. Students wll be regarded as the producton nputs and ther common score from a project wll be treated as the output. Secton below wll frst ntroduce our model as based on the CES producton functon and then present three llustratve examples showng the sgnfcant effect of nput substtutablt on the producton output. Secton 3 wll conclude wth a summar remark. The Model Consder the followng par-wse CES producton functon = f x 1,x,x 3 of three nputs cf. [10] for blateral elastctes of substtuton among multple nputs, and [4] for the prevalent use of the CES producton functon n economcs: : = , where 1 1 A {1,} a 1 x ρ a x ρ 1 s/ρ 1, 13 A {1,3} a 1 x ρ a 3 x ρ 13 3 s/ρ 13, 3 A {,3} a x ρ 3 + a 3 x ρ 3 3 s/ρ 3, A {,j} an nteracton coeffcent between and j n the producton of, a k a contrbuton factor b x k n the producton of, and ρ j, 1] {0} s a factor related to the substtutablt between and j n the producton of, to be explaned later, and s>0 s the elastct of scale measurng the percentage ncrease n due to a one-percent ncrease n all the nputs. We now dspla the followng standard textbook results: 1 the margnal product n j of x or x j, j x = s j x j = s j x j x j a x ρ j a x ρ j a j x ρ j j a x ρ j + a j x ρ j j + a j x ρ j j, and ; 3 the par-wse rate of techncal substtuton RT S j/ n0=d j =

3 Team pedagog 15 j j x dx + x j dx j, so that RT S j/ : = dx j dx = a xj a j a a j j x j x r 1 ρ j j/ 4 x j 1 ρj 5 notaton: r j/ x j x, 6 drt S j/ dr j/ = a a j 1 ρ ρj r j j/ ; 7 3 the par-wse elastct of substtuton between, j as defned b settng A k = A jk = 0 cf. e.g., [1, 9], for the sgnfcance of the construct of elastct of substtuton n economcs, ˆσ j/ := dr j/ r j/ = drt S j/ RT S j/ 1 1 ρ j [0, {1} ; 8 4 the output elastct of nput x so that 3 x =1 x 1 x x = s +s j k a x ρ j a x ρ j + a j x ρ j j a 1 x ρ 1 1 a 1 x ρ a 1 x ρ 1 a x ρ 1 a 1 x ρ a 1 x ρ 1 a x ρ k 13 a x ρ k + a k x ρ k k, 9 a 1 x ρ 13 1 a 1 x ρ a 1 x ρ 13 3 a x ρ 3 a x ρ 3 + a 3 x ρ 3 3 = s + s 1 3 +s + s 13 a 3 x ρ a 3 x ρ 3 +s a 1 x ρ a 1 x ρ 3 + s 13 3 a x ρ 3 + a 3 x ρ 3 3 = s the elastct of scale, as expected. 10

4 16 G. L. Lght Example 1 For the above par-wse CES producton functon Equaton 1, set A {1,} = A {1,3} = A {,3} =10, 11 a 1 = 0.6, a =0.3, a 3 =0.1, 1 ρ 1 = ρ 13 = ρ 3 =0.5, so that ˆσ j =,, j, and 13 s = Then we have [ = x x +0.6 x x x +0.1 x 3 ]. 15 Suppose that Then we have =1 x 1 =9, x =4, and x 3 =1. 16 = 98.6, 17 x 1 x 1 = 0.785, 18 x x = 0.189, and 19 x 3 x 3 = 0.06, so that 0 3 x x = 1 = s, as expected. 1 One ma nterpret the example here as Student 1,, and 3 respectvel spent 9, 4, and 1 hours n a team project and the obtaned a common score of 98.6 whch would be dstrbuted n accordance wth economc theor as however. Example Consder the same par-wse CES producton functon Equaton 1 and set A {1,} = A {1,3} = A {,3} =10, a 1 = 0.75, a =0.5, a 3 =0.1, 3 ρ 13 = ρ 3 = so that ˆσ 3/1 =ˆσ 3/ =0, 4 ρ 1 = 0 so that ˆσ /1 s et to be defned below, 5 s = 1, x 1 =81, x =16, and x 3 = ɛ 0. 6

5 Team pedagog 17 Then Smlarl, 13 = lm ρ 13 81ρ ɛ ρ 13 1/ρ 13 ρ 13 = 10 lm ρ ɛρ 13 1/ρ ɛ = 10ɛ 0. 7 It then follows that := , and ρ =10ɛ lm 10 ρ xρ x ρ 1 1/ρ 1,or 9 1 ρ1 0.75x ρ x ρ 1 0 = lm cf. [11], p ρ 1 = ln ln x ln x,.e., 31 1 =10x x = = Here we ma magne that the three students got 540/1000 = 54% out of a term project, for whch Student 1,, and 3 spent respectvel 81, 16, and nearl 0 hours; et snce ˆσ 3/1 =ˆσ 3/ =0, nether Student 1 nor Student could substtute for Student 3 s nput x 3 n the producton of, whch greatl lowered ther jont score. In passng, we note that the above well-known Cobb- Douglas producton functon Equaton 3 has ˆσ /1 = 1 1 ρ 1 =1. 34 Furthermore, consder the followng constraned optmzaton for the dfferental geometr nvolved n a constraned optmzaton and the topc of surface regulartes, cf. e.g., [, 3, 5, 7]: Mn x 1 + w x, s.t. 10x x 0.5 = 1, 35 x 1,x where,w, 1 are the gven parameters wth w the cost of x and 1 a pre-set output quantt. In our present context, Student 1 and have put respectvel ther tme costs of and w to be spent n dong the project, and the together seek to mnmze ther total tme cost ncurred for the work but

6 18 G. L. Lght subject to the attanment of ther desred score 1. Then we have the followng frst-order condtons: w = 1/ x 1 / x 1 = x x1 0.5 =, and 36 x x x x w =10 x w = 10 x ;.e., the crtcal nput 37 1 = 10x x 0.5 x 0.75 x = x 1 = w, and w w 0.5 = 1 3w Assumng that = w and 1 = 950, then ther optmal nput hours are x 1 = and x = 1 3 x 1 = Equaton 36, 40 where the unequal optmal nput quanttes are attrbuted to the fact that Student 1 has a hgher output elastct of a 1 =0.75 than that of Student of a =0.5. That s, a 1% ncrease n x 1 s to result n a 0.75% ncrease n 1. Example 3 Fnall we llustrate the case of ˆσ /1 = 1 = for the 1 ρ 1 sgnfcance of a zero or nfnte elastct of substtuton, see, e.g., [8] b settng ρ 1 =1but otherwse keepng all the same gven nformaton as n the precedng Example. Then we have x x =7.5x 1 +.5x 41 = = Here t s nterestng to note that due to the perfect substtutablt between x 1 and x, to obtan the same score of 1 = 950 Student 1 alone can acheve the outcome b spendng x 1 = 950 / hours n the project, compared wth the prevous x 1 15 n combnaton wth x 4 from the Cobb-Douglas technolog n the precedng Example.

7 Team pedagog 19 3 Summar Remark In ths paper, we have modeled teamwork or group decson-makng b a parwse CES producton functon from economcs. Although our llustraton here used onl three nputs students, one could easl extend our model nto n nputs b a sum of n terms of the same form of Aj a x ρ j + a j x ρ j s/ρj j. One ma also var an of the parameters to derve addtonal nsghts nto the matter. As mentoned n the begnnng, our work here contrbutes not onl to classroom pedagog but also to beond, snce the smple fact s that nput substtutablt n output s a common unversal concern. In partcular, f we dentf Student 1,, and 3 n our above examples wth captal, labor, and natural resources, then we have shown that a dmnshng non-renewable natural resource that s not substtutable s to lower the overall producton of the econom and that a large substtutablt between captal and labor ma result n large unemploment. Returnng to educatonal methodolog, we recommend nstructors desgn team projects that have hgh level of complementart. References [1] P. Bertolett, Elastctes of substtuton and complementart: a snthess, J. Productv. Anal., 4 No. 005, [] R.M. Freund, On the prmal-dual geometr of level sets n lnear and conc optmzaton, Sam J. Optm., 13 No , [3] J.S. Km, The structure of hpersurfaces wth some curvature condtons, Proc. Amer. Math. Soc. 15 No , [4] R. Klump and H. Pressler, CES producton functons and economc growth, Scand. J. Econ., 10 No , [5] W.V. Lovtt, A tpe of sngular ponts for a transformaton of three varables, Trans. Amer. Math. Soc., , [6] M. Medoff, The nput relatonshp between co-authors n economcs: a producton functon approach, Amer. J. Econ. Soc., 66 No. 007, [7] Y. Nevergelt, Wh Lagrange multplers wth extreme magntudes gve extrema of defnte Hermtan forms on quadrc surfaces, Sam J. Matrx Anal. Appl., 31 No ,

8 0 G. L. Lght [8] K. Nshmura and A. Vendtt, Captal deprecaton, factor substtutablt and ndetermnac, J. Dfference Eq. Appl., 10 No , [9] D.B. Renolds, Entrop and dmnshng elastct of substtuton, Resources Pol., 5 No , [10] H. Thompson, Substtuton elastctes wth man nputs, Appl. Math. Lett., 10 No , [11] H.R. Varan, Mcroeconomc Analss, Norton, New York, Receved: Jul, 011