Basic Mathematics for Economists - Rosser - Chapter 9 pptx

... 0% and 100%. Therefore c willtake a value between 0 and 1 and so the second-order condition for a maximum will be met.© 199 3, 2003 Mike Rosser The second-order condition for a minimum is met ... variable.Example 9. 2Show that TR is a maximum when q is 18 for the non-linear demand schedule.p = 194 .4 − 0.2q2SolutionTR = pq = ( 194 .4 − 0.2q2)q = 194 .4q − 0.2q3 For a stationary point ... positive© 199 3, 2003 Mike Rosser NM–y–xxyFigure 9. 3a constant and so the second-order derivative, which represents the rate of change of theslope, will always be zero.Example 9. 4InChapter5weconsideredanexampleofabreak-evenchartwhereafirmwasassumedtohave...

... manually. To illustrate this point, if you have an old basic calculator, use it to answer the problem103× 3 = ?You may get the answer 9. 999 999 9. However, if you use a modern mathematical calculatoryou ... on from each other, each chapter assumes that students arefamiliar with material covered in previous chapters. It is therefore very important that you© 199 3, 2003 Mike Rosser AlgebraMuch economic ... theirfirst-year mathematics for economics course successfully and go on to achieve an honoursdegree. There is no reason why you should not do likewise if you are prepared to put in theeffort.© 199 3,...

... 12 2 115 14.25 -2 .25 13 3 92 .5 15.38 1.13 14 4 103.75 14.81 -0 .56 15 5 98 .125 15. 09 0.28 16 6 100 .93 75 14 .95 -0 .14 17 7 99 .53125 15.02 0.07 18 8 100.23438 14 .99 -0 .04 19 9 99 .882813 15.01 ... 2. 19 12 2 183.75 22.03 -3 .28 13 3 144.375 26 .95 4 .92 14 4 203.4375 19. 57 -7 .38 15 5 114.84375 30.64 11.07 16 6 247.73438 14.03 -1 6.61 17 7 48. 398 438 38 .95 24 .92 18 8 347.40234 1.57 -3 7.38 ... 1.57 -3 7.38 19 9 -1 01.10352 57.64 56.06 20 10 571.65527 -2 6.46 -8 4. 09 © 199 3, 2003 Mike Rosser 13 Dynamics and difference equationsLearning objectivesAfter completing this chapter students...

... 460(1.08)n= 92 5, what is n?Solution460(1.08)n= 92 5(1.08)n= 2.0108 696 Putting in log formn log 1.08 = log 2.0108 696 n =log 2.0108 696 log 1.08=0.30338 39 0.0334238= 9. 076 894 5WeshallreturntothistypeofprobleminChapter7whenweconsiderforhowlongasum ... 3.6658351log 251.07 = 2. 399 794 8Thus4,632.71 × 251.07 = 103.6658351× 102. 399 794 8= 106.0656 299 = 1,163,134.5using the [10x] function key.The principle is therefore to put all numbers ... eighth root. Thuslog 226.34 = 2.354761318log 226.34 = 0. 294 3452Therefore8√(226.34) = 100. 294 3452= 1 .96 945 09. © 199 3, 2003 Mike Rosser SolutionBetween points A and B price falls by 5 from...

... Rosser 8.Yousell900sharesviayourbrokerwhochargesaflatrateofcommissionof£20onalltransactionsoflessthan£1,000.Yourbankaccountiscreditedwith£340fromthesharesale.Whatpricewereyoursharessoldat? 9. Yournetmonthlysalaryis£1 ,95 0.YouknowthatNationalInsuranceandpensioncontributionstake15%ofyourgrosssalaryandthatincometaxisleviedatarateof25%ongrossannualearningsabovea£5,400exemptionlimit.Whatisyourgrossmonthlysalary?10.Youhave64squarepavingstonesandwishtolaythemtoformasquarepatioinyourgarden.Ifeachpavingstoneis0.5metressquare,whatwillthelengthofasideofyourpatiobe?11.AfirmfacesthemarginalrevenuescheduleMR=80−2qandthemarginalcostscheduleMC=15+0.5qwhereqisquantityproduced.YouknowthatafirmmaximizesprofitwhenMC=MR.Whatwilltheprofit-maximizingoutputbe?3.8ThesummationsignThesummationsigncanbeusedincertaincircumstancesasashorthandmeansofexpress-ingthesumofanumberofdifferenttermsaddedtogether.(isaGreekletter,pronounced‘sigma’.)Therearetwowaysinwhichitcanbeused.Thefirstiswhenonevariableincreasesitsvalueby1ineachsuccessiveterm,astheexamplebelowillustrates.Example3.43Anewfirmsells30unitsinthefirstweekofbusiness.Salesthenincreaseattherateof30unitsperweek.Ifitcontinuesinbusinessfor5weeks,itstotalcumulativesaleswillthereforebe(30×1)+(30×2)+(30×3)+(30×4)+(30×5)Youcanseethatthenumberrepresentingtheweekisincreasedby1ineachsuccessiveterm.Thisisratheracumbersomeexpressiontoworkwith.Wecaninsteadwritesalesrevenue=5i=130iThismeansthatoneissummingalltheterms30iforvaluesofifrom1to5.Ifthenumberofweeksofbusinessnwasnotknownwecouldinsteadwritesalesrevenue=ni=130iToevaluateanexpressioncontainingasummationsign,onemaystillhavetocalculatethevalueofeachtermseparatelyandthenaddup.However,spreadsheetscanbeusedtodotediouscalculationsandinsomecasesshort-cutformulaemaybeused(seeChapter7).© 199 3, 2003 Mike Rosser 3. Simplify8xy + 2x2+ 24x2x4. A firm has to pay fixed costs of £200 and then £16 labour plus £5 raw materi-als for each ... 3.38Solve for x if 16x − 4 = 68 + 7x.SolutionSubtracting 7x from both sides9x − 4 = 68Adding 4 to both sides9x = 72Dividing both sides by 9 gives the solutionx = 8© 199 3, 2003 Mike Rosser SolutionInthisexamplea=1,b=0andc=−1.Thetwonumbers−1and1give−1whenmultipliedtogetherand0whenadded.Thusx2−y2=(x−y)(x+y)Tocheckthis,multiplyout:(x−y)(x−y)=x2−xy+y2−yx=x2−y2Example3.26Attempttofactorizetheexpression3x2+8x+23.SolutionAs23isapositiveprimenumber,theonlypairsofpositiveintegersthatcouldpossiblybemultipliedtogethertogive23are1and23.Thus,whateverpermutationsofcombinationswithtermsinxthatwetry,theterminxwhenbracketsaremultipliedoutwillbeatleast24x,e.g.(3x+23)(x+1)=3x2+26x+23,whereasthegivenexpressioncontainstheterm8x.Itisthereforenotpossibletofactorizethisexpression.Unfortunately,itisnotalwayssoobviouswhetherornotanexpressioncanbefactorized.Example3.27Attempttofactorizetheexpression3x2+24+16.SolutionAlthoughthenumberslookpromising,ifyoutryvariouspermutationsyouwillfindthatthisexpressiondoesnotfactorize.Thereisnoeasywayoffactorizingexpressionsanditisjustamatteroftrialanderror.Donotdespairthough!Asyouwillseelateron,factorizingmayhelpyoutouseshort-cutmethodsofsolvingcertainproblems.Ifyouspendagestryingtofactorizeanexpres-sionthenthiswilldefeattheobjectofusingtheshort-cutmethod.Ifitisnotobvioushowanexpressioncanbefactorizedafterafewminutesofthoughtandexperimentationwithsomepotentialpossiblesolutionsthenitisusuallymoreefficienttoforgetfactoriza-tionandusesomeothermethodofsolvingtheproblem.Weshallreturntothistopicin Chapter6 .© ... expression for the number of euros that can be bought for £x (any given quantityof sterling), and(ii) evaluate it for x = 250.© 199 3, 2003 Mike Rosser the inflation rate. Adjusted for inflation,...

... mathematical form of the relationship is not actually known then a functionmay be written in what is called a general form. For example, a general form demand© 199 3, 2003 Mike Rosser ... graph?© 199 3, 2003 Mike Rosser has been entered only the calculated value appears in the cell where the formula is. However,when you put the cursor on a cell containing a formula, the full formula ... 10%.© 199 3, 2003 Mike Rosser 0 –x–yy=x3y=x3y=x2y=x2xyFigure 4.14Table 4.3x 01234567 89 y = x0.50 1 1.414 1.732 2 2.236 2.4 49 2.646 2.828 30y312y=x0.514 9 xFigure...

... so that MC1= MC2= MCTherefore q = q1+ q2= (2MC −85) + (0.5MC − 65)q = 2.5MC − 15060 + 0.4q = MC (1)© 199 3, 2003 Mike Rosser ADemandB0 q£ 9. 99 2.20 94 120 166 250MRЈ6016.0012.16MCMR8.32Figure ... break-even output?5. If y = 16 + 22x and y =−2.5 + 30.8x, solve for x and y.© 199 3, 2003 Mike Rosser From this new reduced form equation we can see that (when I is 180 and t is 0.375) for every ... produced.)Should the firm change its production mix if per-unit profits alter to(a) £25 each for both A and B, or (b) £30 for A and £20 for B?© 199 3, 2003 Mike Rosser 0pp = 420 –0.2q p =60+0.4q qX42030060600Figure...

... 27 -6 37 38 -1 42 15 5.5 -2 07 16.5 -6 58 27.5 -6 25 38.5 -1 08 16 6 -2 38 17 -6 67 28 -6 12 39 -7 3 17 6.5 -2 68 17.5 -6 75 28.5 -5 98 39. 5 -3 7 18 7 -2 97 18 -6 82 29 -5 83 40 0 19 7.5 -3 25 18.5 -6 88 29. 5 ... 29. 5 -5 67 40.5 38 20 8 -3 52 19 -6 93 30 -5 50 41 77 21 8.5 -3 78 19. 5 -6 97 30.5 -5 32 41.5 117 22 9 -4 03 20 -7 00 31 -5 13 42 158 23 9. 5 -4 27 20.5 -7 02 31.5 -4 93 42.5 200 24 10 -4 50 21 -7 03 32 -4 72 ... 24.5 -6 82 35.5 -2 97 10 3 -3 7 14 -5 98 25 -6 75 36 -2 68 11 3.5 -7 3 14.5 -6 12 25.5 -6 67 36.5 -2 38 12 4 -1 08 15 -6 25 26 -6 58 37 -2 07 13 4.5 -1 42 15.5 -6 37 26.5 -6 48 37.5 -1 75 14 5 -1 75 16 -6 48...

... 0 .95 2381 0 .94 3 396 0 .93 45 79 0 .92 592 6 0 .91 7431 0 .96 1538 0 .95 2381 0 .94 3 396 0 .93 45 79 0 .92 592 6 0 .91 74312 0 .92 4556 0 .90 70 29 0.8 899 96 0.8734 39 0.8573 39 0.84168 0 .92 4556 0 .90 70 29 0.8 899 96 0.8734 39 0.8573 39 ... 0.820815 0.7 195 74 0.268103 0. 192 9200.60 0 .93 0731 0.866260 0.806255 0. 698 427 0.23 794 9 0.166 190 0.65 0 .92 5 197 0.85 599 1 0. 791 961 0.67 791 1 0.211 199 0.1431740.70 0 .91 9700 0.845848 0.77 792 7 0.658008 ... 0.627412 0.5820 09 0.5402 69 0.501866 0.730 69 0.6768 39 0.627412 0.5820 09 0.5402 69 0.501866 9 0.702587 0.6446 09 0. 591 898 0.54 393 4 0.5002 49 0.460428 0.702587 0.6446 09 0. 591 898 0.54 393 4 0.5002 49 0.46042810...

... 90 10 28800 1600 1608 81 1 27378 178 178 9 80.1 0.1 27217 .98 17 .98 1 79. 8 10 80.01 0.01 27201. 799 8 1. 799 8 1 79. 9811 80.001 0.001 27200.18 0.1 799 98 1 79. 998 12 80.0001 0.0001 27200.018 0.01 799 998 ... 27200.018 0.01 799 998 1 79. 999 813 80.00001 0.00001 27200.0018 0.001800000 1 79. 999 9802Test Yourself, Exercise 8.31. Given the demand schedule p = 120 − 3q derive a function for MR and find theoutput ... b<0).With a per-unit tax of t, the supply schedule becomesp = c + dq +t© 199 3, 2003 Mike Rosser Therefore,dpdq=−30 + 0.5qWhen q = 8,dpdq=−30 + 0.5(8) =−30 +4 =−26Also, when q = 8,p = 90 0 −...