Pragmathematics 2014 lecture 3 (2)
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This document discusses linear relationships and break-even analysis. It introduces linear relationships using the example of a hat company's total revenue (TR) and total costs (TC). TR is defined as the price per hat multiplied by the number of hats sold (N), forming the relationship TR = 50N. TC is the sum of fixed costs ($1000) and variable costs per hat ($30) multiplied by N, giving TC = $1000 + $30N. The break-even point is where TR = TC, which occurs when N = 50 hats. The document then provides a more general explanation of linear relationships as lines on a graph defined by the equation y = a + bx.
Pragmathematics 2014 lecture 3 (2)
- 1. 2. Relationships Relationshipshave alreadybeenintroducedabove, for example the relationshipbetweenx and2x showninthe picture The graph showsthe relationshipbetween2x andx, where the valuesof 2x are measuredonthe North-Southaxis (calledthe ordinate) and the valuesof x are measuredonthe East-Westaxis (calledthe abscis). One typicallyusesthe symbol ‘y’torepresentthe valuesof the function, inthiscase: y = 2x and one alsousuallycallsthe ordinate ‘the y-axis’. Likewise,the abscisismore oftenreferredtoas‘the x-axis’. The relationshipy=2x betweenyandx isa complex relationshipalready. The picture showsthat itis curved,ratherthan a simple straightline. Let ustherefore startwiththe more simple straight-line relationships 4.1 Simple linearrelationship An importantphenomenonforus,humanbeings,isthatof ‘relationship’; we all understandthe word‘relationship’: some systematicassociationbetweentwoormore separate entities; for example the relationshipbetweentwoormore people,
- 2. or the relationshipbetweenpopulationdensityandairpollution, or the relationshipbetweendrivingspeedandgasolineconsumption,etc. If we understandarelationship, we can use that knowledge topredictormanage that phenomenon, e.g.the future incidence orrespiratorydiseasesfromanincrease inurbanization, the gasoline consumptionof aregionif a maximumspeedisimposed,etc. The study of relationships,therefore,isanimportantpartof mathematics for those whoare interestedindoingandapplyingscientificresearch. Let usfirstexplainthe concept of a relationshipbetweentwovariables(‘bivariate’). We use an example frombusinesseconomics, namelya‘breakevenanalysis’. Remember:thisis just an example,todemonstrate mathematics in a concrete case; this is not about the economics! The goal of a breakevenanalysisistodetermine the minimumsalesvolumethatacompanyneeds before itstartsmakinga profit. Take a companythat makesPanamaHats (Ecuadorians,Iknow: thisis nota hat fromPanama, but a ‘sombrerode pajatoquilla’fromyourcountry!). Let ussay that the companysellsitshatsfor 50$ apiece. The revenue (‘TR’) of thatcompanythenequalsthe numberof hatsit sells, representedbythe symbol N, multipliedbythe price itreceivesforeachhat(50$): Total Revenue =TR = 50$*N Thisis a ‘relationship’;we caneasilyrepresentitasinthe graphbelow.
- 3. A more general expressionforarelationshipof the kind TR = 50*N is y = ax here y standsfor Total Revenue,x fornumberof hats, ‘a’ forthe revenue perhat(50$); y and x are variables. Since Total Revenue dependsonthe numberof hatssold, TR (or, more generallyy) isthe dependentvariable, and N (ormore generallyx) isthe independentvariable. a is a ‘parameter’ The parametera is alsocalledthe ‘slope’of the relationship; it expressesbyhow muchy increasesif x increasesbyone unit (here,TRincreasesby50$ for each increase of salesbyone hat) if the price changesto 60$ perhat, then‘a’ isnot 50 anymore,but60. 0 1000 2000 3000 4000 5000 6000 7000 0 25 50 75 100 125 150 TR N
- 4. We can easilycompute the total revenueforagivennumbersof hatssold, irrespectiveof the price perhat; we justneedto change 50 into60 (or any othernumber) inthe formula. Excel is handy forsuch computations,asdemonstrated already in sheet3. Nowletus lookat the costs of thisfirm. Its costsare of twokinds: on the one hand fixedcost(F) foroperatingabusiness, these costsremainthe same,irrespective of the numberof hatssold, evenif youdo notsell a single hat (e.g.the salaryof the manager,the salaryof the salesclerk,the rental of the premises); on the otherhand variable costsperhat (v) these are costs fore.g.raw materialsandlaborthat increase withthe numberof hats. If the numberof hats doubles,the variable costsdouble; youneeddouble the materialsanddoublethe numberof hoursworked. Let ussay that the fixedcostof the firmare 1000$ and the cost perhat (‘variable’cost) 30$. The total cost (‘TC’) forproducingN hats thenis,as showninthe graph Total Costs= TC = 1000$ + 30$*N or more generally Total Costs= F + v*N or still more generally z = c + d*x,
- 5. where c isthe parameterthat standsfor the fixedcosts, d isthe parameterthatstandsfor the variable costperunit(perhat), z is the variable representingthe total costs and x is the variable representingthe numberof hatssold; z is the dependentvariable,x isthe independentvariable: We can nowrepresentbothrelationships, Total Revenue andTotal Costinone graph,as below: In thisgraph (called‘break-evenanalysis’), we see that our hat makerneedstosell acertainnumberof hats before histotal revenuesare highenoughtocoverhistotal costs, i.e.before he startsmakinga profit. That number,N°,of hats (we write N°to designate one specificvalue of N) where the total revenuesexactlyequalthe total costs iscalledthe ‘break-evenpoint’. As we can see inthe graph, that isthe pointwhere bothlines intersect,the pointwhere 50N = 1000 + 30N whichcan be rearrangedinto(usingthe rulesof algebramentionedabove) 50N - 30N = (50-30)N = 20N = 1000
- 6. fromwhichwe can compute N°= 1000/20 = 50 hats N° isthe solutiontothe ‘breakevenproblem’. Withthisexample,we have introducedthe conceptof aline or a ‘linearrelationship’: boththe relationshipbetweennumberof hatsand total revenue and that betweennumberof hatsand total costsare linearrelationships. Both are picturedas straightlines. We are not aware of it, but we are usinglinearrelationshipsall the time to explain orpicture thingstoourselvesorto others. Like forexample the answertothe question “Whenwill yoube here”? “Well,let’ssee,Icome bybicycle; howfar is itto your place?” “It isabout 50 kilometers” “Well,Ican leave at three o’clock,andI ride some 20 kmper hour, so I shouldarrive around5:30.” The reasoningor relationshiphere is: time of arrival = 3 + (1/20)*distance,i.e.3+ (1/20)*50 = 5:30 or if you want,y = 3 +(1/20)x where yis arrival time,x isthe distance. That is actuallyalinearrelationship, if you make a picture of it,it wouldbe a line, showingtime of arrival ‘infunction’of the distance tobe traveled; if the distance is80 km, thentime = 3 + (1/20)*80 = 7. You will thinkthatthisisa stupidlysimpleexample, but that isexactlythe point: the mathematicsyoureadbeloware in fact verysimple. The general expressionforalinearrelationshipis y = a + bx, where yis the dependentvariable,x isthe independentvariable or‘running’variable; and a and b are parameters (tobe givenspecificvalues), ‘a’ iscalledthe ‘intercept’; that isthe value of ywhenx = 0, as showngraphicallybelow. ‘b’is calledthe slope orinclinationof the line;
- 7. it expressesbyhowmuchy increaseswhenx increasesbyone unit. If,for example,b=0.5, theny increasesby0.5 each time x increasesbyone unit(showninthe graph). Dependingonthe value of the parametersaand b, youget differentrelationshipsbetweenx andy, but alwayslinearones. To familiarizeyourself withlinearrelationships, we showexamplesy= a + bx for positive (+2),zero(0) ornegative (-3) valuesof the parametera and forpositive (+4),zeroor negative (-0,5) valuesof the parameterb a =-3 and b = -0,5 a =-3 and b = 0
- 8. a =-3 and b = 4 a =0 and b = -0,5 a=0 and b=0
- 9. a=0 and b=4 a=2 and b=-0,5 a=2 and b=0
- 10. a=2 and b=4 If y = a + bx,theny dependsonx: if we knowthe value of the parametersaand b for a specificrelationship, we can compute exactlythe value of ythat correspondswitheachvalue of x; likewise,since x canbe expressedas(y-a)/borx = -a/b + y/b, x can alsobe computedexactlyforanygivenvalue of y. In the case of a linearrelationshipy= a + bx x andy are saidto be ‘linearlydependent’: if you knowthe value of one,youknowthe value of the other. In a sense,youmightsaythat if youknow one of them(x or y) the knowledge of the otherissuperfluous; youcan compute one outof the other. Linear dependence will be mentionedagain below, whenwe doresearchwithseveral variables and itmay be that the informationcontainedinone variable issuperfluous, because itis‘contained’inone ormore other variables. For example:
- 11. if we thinkthat the incidence of cancerinpeople isdrivenbytheirextentof smoking as well asby theirintake of saturatedfats, but people whosmoke alsotendtoeata lotof saturatedfats, it will be difficulttofindouthowmuchof the incidence of cancer isdue to smokingandhow muchto unhealthyeating Smokingandunhealthyeatingthentendtogo‘handinhand’, tendto be linearlyrelatedto each other. It thenbecomesdifficulttoseparate outthe effectoncancer of smokingonthe one handand unhealthyeatingonthe other. But that isfor laterdiscussion… Now,back to linearrelationships. As we all know,if youtake a page of paperand plottwo pointsonit, youcan drawexactlyone,andonlyone,straightline throughthese twopoints. Anytwo points(x,y) suffice todefine aline orlinearrelationship. If we knowthata line runsthroughthe points(x = 2, y = 0) and (x =5, y = 6), since the line (the linearrelationship) isof the general formy= a +bx, thenthe followingmusthold 0 = a + 2b (y = 0 whenx = 2) 6 = a + 5b (y = 6 whenx = 5) Subtractingthe firstfromthe secondrelationship, we obtain 6-0 = a-a + 5b-2b or 6 = 3b, and hence b= 2 substitutingb= 2 in the firstexpression,(0= a + 2b) we findthat 0 = a +2*2 and hence a = -4. The expressionof the relationshipy= a +bx through(x = 2, y = 0) and(x =5, y = 6) therefore isy= -4 +2x We can verifythat(x = 2, y = 0) and(x =5, y = 6) meet the condition: indeed: 0 = a + 2b or 0 = -4 + 2*2 (forthe point2,0) and 6 = a + 5b or 6 = -4 + 5*2 (forthe point5,6) Once you knowthe parametersaand b for a relationshipof the formy= a + bx, youcan use itto compute any value of ywithinthe original range of valuesof x, i.e.between2and 5, for example x =3; that iscalled‘interpolation’; e.g. if x = 3, theny = -4 +2*3 = 2
- 12. You can alsouse it to compute a value of y for a value of x outside of itsoriginal interval, i.e.outside the range from2 to 5, e.g.if x = 1000, theny = -4 +2*1000 or 1996. The latteris called ‘extrapolation’. Wheninterpolating,youare ‘inknownterritory’; whenextrapolating,youare ‘innewterritory’. Scientistswilloftendevelopamodel basedonobservationsof pastconditions (i.e.interpolation) and use that model topredictwhatwill happenundernew conditions (i.e.extrapolation) If two linesy= a+bx and z = c+dx intersect (i.e.yand z have the same value), like thatwasthe case above inthe break-evenanalysis withthe Total Revenuesline andthe Total Costs, that will,bydefinition,be foraspecificvalue of x,call itx°,such that y = z, i.e. a+bx°= c+dx° or, usingthe rulesof algebrabx° - dx°= c – a or (b-d)x°=(c-a) and hence x°= (c-a)/(b-d) For example,inthe examplewithhats,where y=50x and z = 1000 + 30x i.e.witha = 0, b = 50, c = 1000 and d = 30 that iswhenx°= 1000/(50 – 30) = 50 hats, as alreadyshownabove. The expressionx°=(c-a)/(b-d) illustratesthe usesandadvantagesof mathematics: it isa general solutiontoaproblem(‘where dotwolinesintersect?’), whichcan thenbe tailoredtoany specificsituation (‘whendothe Total Revenuesof ahat manufacturerequal the Total Costs?) i.e.,to specificvaluesof the parametersa,b,c, d; thisavoidshavingtocompute a specificsolution everytime youhave a problemof the same structure but withdifferentparameters (e.g.the intersectionof twolines) If the price perhat dropsto 40*, forexample, the break-evenvolume becomes1000/(40-30) = 100.
- 13. If a =10, b = 100, c = 2000 and d = 20, thenx° can be computedwiththe same rule as (c-a)/(b-d) =(2000-10)/(100-20) x° = (c-a)/(b-d),again,isa rule to remember. You do not have to prove it again, the proof has beengiven; you can now use it for any problemwhere two linearrelationshipsintersect. As we sawin the graph above,the relationships TR = 50*N andTC = 1000 + 30*N are lines,orlinearrelationships. Thismeansthat if we sell one more hat, the total revenue increasesbyafixed50$ (irrespective of whetherwe alreadysell few ormanyhats), or if we manufacture one more hat, the costsincrease by30$ (again,irrespective of howmanyhatswe are alreadymanufacturingatthatpoint). These relationshipscanbe visualizedonasheetof paper, i.e.ona flatmap,a plane, withan axisforTR or TC (the up-downaxis,orNorth-Southaxis,usuallyidentifiedasy) and an axisforN (the left-to-rightaxis,orEast-Westaxis,usuallyidentifiedasx). Each pointof that plane hasa coordinate forN and forTR (orTC) or, more generally,forx andfory. A line,then,isthe collectionof points(x,y) inamapor plane that satisfythe conditiony= a + bx for some specificvaluesof aand b. For example,the line TC= 1000 +30N isthe collectionof pointswithcoordinates(N,TC) such that TC = 1000 + 30N Evidently,1030 liesonthat line (itisthe value of TC forN =1) and so does1300 (the value of TC for n = 10); but the point(N,TC) = (10, 2000) obviouslydoesnotlie onthe line. Similarly,the line y= 2 + 3x isthe collectionof all points(x°,y°) inthe plane withaxesx andy whichmeetthe requirementthat y° = 2 + 3x°
- 14. The point(2,8) issuch a point,since 8 = 2+3*2; so isthe point(5,17); and so are also the points(0,2) and (-1,-1); on the otherhand,the point(3, 5) doesnot lie onthisline, since itis nottrue that 5 = 2 + 3*3. Thisis illustratedgraphicallybelow. Why didwe needtomentionthis? Below,we will see thatinreality, the observations(points) thatwe canmake of a relationship do notalwayslie neatlyona straightline, butalmost. We will thensee howwe cannevertheless‘guess’ whatthe linearrelationshipisbelow inthe sectiononestimatingthe parametersof alinearrelationship.
- 15. 4.2 Multivariate linearrelationships. In our example,total revenue dependsonlyon the numberof hatssold; the relationshipisbetweentwovariables,TRandN (ory and x); it isbivariate. Linearrelationshipscaninvolvemore thantwophenomenaorvariables. Let usillustrate thatbyexpandingthe Panamahatexample. If the firmalsomakesties,andsellsthemat20$ apiece, thenitstotal revenue dependsnotonlyonthe numberof hats sold(N), but alsoon the numbertiessold(T): Total Revenue = 50*N + 20*T or in general terms: y = a + bx + cz (withparametersa,b andc; inthe example a= 0, b = 50 and c = 20). If we wantto representthisgraphically, we are not lookingatlinesona map(i.e.ina flat‘plane’) anymore, but ratherat a plane ina space withaxesTR, N andT; (we are ‘movingup’one dimensioninourreasoning: the unidimensional lineof the previousexample nowbecomesa two-dimensionalplane and the 2-dimensional planeinwhichwe drew te line nowbecomesa 3-dimensional space,i.e.‘space’aswe are usedtothinkof it, witha right-left,front-backandup-downdimensionoraxis). Each pointof that plane meetsthe conditionthatTR = 50*N + 20*T. That plane goesthroughthe origin (TR = 0 whenN = 0 and T = 0; i.e. throughthe point0,0, 0), thisexpressesthatthe revenuesare zero whenthe firmdoesnotsell anyhat or tie. The plane slopesupwardswithN andwithT, more steeply(50per unit) withN thanwithT (30 per unit). The relationshipbetweenTotal revenue y),hatssold(x) andtiessold(z) isillustratedin the picture below(agraphical model…).
- 16. While TR = 50$*N or y = a + bx isa bivariate linearrelationship, TR + 50*N + 20*T or y = a + bx + cz is a multivariate linearrelationship (inthiscase it is trivariate,betweenthree variables:y,x andz). Such a trivariate relationshipsmaybe useful todescribe,andbuild, a slopingflatsurface,e.g.inarchitecture orindesign. But it isalsouseful torepresentormodel phenomena where one dependentvariable iscaused by more than one independentvariable, such as whenthe revenuesof afirmdependonN and T, or whenthe incidence of cancerdepends on the extentof smokingandonthe intake of saturatedfats. If the firmalsosells Sshirts,at 80$ per shirt, thenthe linearrelationshipwouldbecomeTR= 50N + 20T + 80S, and so on… A final note on linearrelationships:lineartransformation. To illustrate whatwe meanbya lineartransformation, letus returnto the example of measuringtemperature. Temperature remainsthe same, irrespectiveof whetherwe measure itin°Cor in °F. It musttherefore be possible to‘translate’or‘transform’ °Cinto°Fand vice versa. Since we knowthat waterfreezesat32°F, or at 0°C and since a 1° difference in°Fisa 5/9° difference in°C, the transformationisgivenby°C= (°F-32)*(5/9) 0 1 2 3 4 5 6 7 8910 0 100 200 300 400 500 600 700 0 1 2 3 4 5 6 7 8 9 10 Z Y X
- 17. or °C = – 32*(5/9) + (5/9)*°F, thisisa linearrelationshipof the form°C= a * b*°F for example,65°Fcorrespondsto(65-32)*(5/9) °C = 18,33°C Usingthe rulesof algebra,we can alsosay that °C*(9/5) = °F – 32, or °F = 32 + °C*(9/5), againa linearrelationship,of the form°F= c + d*°C Anotherexampleof alineartransformationisgiven by the operationof standardizingameasurement The standardizedvalue zof a measurementx isobtainedas z = (x-m)/s with‘m’the meanor average of the measurementsand‘s’theirstandarddeviation. Usingthe rulesof algebra,thiscan be writtenasz = -(m/s) +x/s if we substitute the symbol ‘a’for–(m/s) and‘b’for (1/s) thenwe can write z = a + bx,a relationshipwithlinearform.
- 18. 4.3 Estimating the parametersof a linearrelationship Typically,inscientificresearch,we donotknow exactlythe formor model takenon bya specificphenomenon,e.g.the relationshipbetweentwovariables, such as the incidence of lungcancerandair pollution; we may assume thatit islinear,e.g.of the formy = a + bx, but we usuallydonotknowthe value of the parametersa and b. We mayhave to guessthe value of these parametersourselves, or try to infertheirvalue fromthe pasthistoryof the phenomenon. To use a simple andfamiliarexample, letus reverttothe hat manufacturer. Whenthe managerof the hat companysays that hisTotal Revenue is50$*N. that may actuallyonlybe hisassumptionorguess. Is thisreallytrue?Dothe salesfiguresconfirmthis? Sometimes,hatsmaybe soldata lowerprice, e.g.whenthe customerbuysmore thanone, or sometimesrichcustomersmaybe chargeda higherprice… Then,whatreallyisthe relationshipbetweenrevenue andnumberof hats? Can we inferthe parameterof the relationshipTR= a*N frompast observationsonnumberof hatsandrevenues? Andwhat isthe relationshipbetweenTotal Costsandnumberof hats; isit really1000$ + 30$*N as the managersays? Can we inferthe parametersof the cost functionTC= F + v*N frominspectingpastresultsof hatmanufacturers, fromthe numberof hatsproducedlastyear andtotal costsin that year? Assume thatthe Hat makers’associationprovidesuswiththe Total Costs and the numberof manufacturedhatsfor13 firms,asin the table below.
- 19. Firm HATS(N) Total Costs(TC) 1 1100 45200 2 400 15400 3 980 29800 4 1100 44300 5 600 18040 6 1200 55000 7 1050 40300 8 500 16500 9 400 13500 10 1250 42900 11 800 31250 12 900 29800 13 300 15060 If we make a graph,as below,thenwe see thatthe observations,the data do notall lie neatly onone straightline TC=c + dN; it isnot the case thatTC isexactlyequal to1000 + 30*N for each firm. Thenwhat isthe (linear) relationshipbetweentotal costandnumberof hats TC = c + d*N whichbestreflectsthe reality(the data)?; whatis the ‘best’ line TC = c + d*N. That is to say:what isthe bestvalue forthe parameters,c°and d°, so that the relationshipTC = c° + d°*N closelyapproximatesthe observationsinthe graph?
- 20. If I knewthese ‘best’valuesof ‘c’and‘d’(call these c° andd°), thenfor Firm1, the relationshipwouldpredictTotal Costsof c° + d°*1100; since the real costs forthat firmare 45200; the difference betweenrealityandthe predictionis(45200- c° – d°*1100); for Firm2 the difference wouldbe (15400 -c° - d°*400), etc. We lookforvaluesforc° and d° that make these differencessmall. But we do notknowc° and d°; the bestwe can dois estimate themonthe basisof the available data. To find,or ‘estimate’c°andd°,we apply the ‘leastsum of squared differences’or‘leastsquares’principle that wasintroducedabove: lookforthe value of ‘c’and ‘d’ whichmake the sumof the 13 squareddifferences (45200- c – d*1100)² + (15400 -c - d*400)² + … + (15600 – c – d*300)² as small as possible. For these valuesof ‘c’and‘d’ the resulting‘fitted’orpredictedTotal Costs will lie ‘asclose aspossible’tothe real Total Cost figures; the squareddifference betweenrealityandthe predictionwill thenbe the smallest Since there are 13 squareddifferences (oneforeachmonth), we sum them,addthemup andlook(we ask the computerto lookforthis…) for the value of ‘c’ and‘d’ that make that ‘sumof squares’minimal.
- 21. That is an operationeasilyperformedinExcel; e.g.by the functionLINESTinExcel underFormulas>more functions>statistics>linest or evenas a facilityofferedtogetherwiththe scatterplotof the data.. The estimatedrelationshipy= a + bx,inthiscase TC = c + d*N isseento be y = 2634 + 31,45*N It iscalledthe ‘simple linearregressionofy on x’, here itis the regressionof TCon N. We say‘linear’because the relationshipisof the form‘y = a + bx’ and we say ‘simple’because thisinvolvesonlytwovariables,Xandy. Simple linearregressionsare usedveryofteninempirical researchtomodel relationshipsbetweenone dependentvariable (y) andone independentvariable (x) and where there isnoreasonto thinkthatthe relationshipisnotlinear(seebelow). If y is assumedtodependonmore than one variable,forexample y = a + bx + cz thenthe bestfittinglinearrelationshipwill be a‘multiple linearregression’.