Basic Mathematics for Economists - Rosser - Chapter 7 pps

... 0 .79 3832 0 .77 21834 0.854804 0.82 270 2 0 .79 2094 0 .76 2895 0 .73 503 0 .70 8425 0.854804 0.82 270 2 0 .79 2094 0 .76 2895 0 .73 503 0 .70 84255 0.8219 27 0 .78 3526 0 .74 7258 0 .71 2986 0.680583 0.649931 0.8219 27 ... 0.8219 27 0 .78 3526 0 .74 7258 0 .71 2986 0.680583 0.6499316 0 .79 0315 0 .74 6215 0 .70 4961 0.666342 0.630 17 0.5962 67 0 .79 0315 0 .74 6215 0 .70 4961 0.666342 0.630 17 0.5962 67 7 0 .75 9918 0 .71 0681 0.6650 57 0.62 275 ... 0. 876 972 0 .76 9081 0. 674 463 0.51 871 7 0. 072 3 97 0.0 375 531.15 0. 871 784 0 .76 0008 0.662564 0.503554 0.064296 0.032 376 1.20 0.866630 0 .75 1048 0.650880 0.488842 0.0 571 05 0.0 279 151.25 0.861508 0 .74 2197...

... 3.1Student no. 12345 678 9101112Height (cm) 178 175 170 166 168 185 169 189 175 181 177 180Weight (kg) 72 68 58 52 55 82 55 86 70 71 65 68© 1993, 2003 Mike Rosser Example 3.18Simplify (6 − 5x)(10 ... Rosser 8.Yousell900sharesviayourbrokerwhochargesaflatrateofcommissionof£20onalltransactionsoflessthan£1,000.Yourbankaccountiscreditedwith£340fromthesharesale.Whatpricewereyoursharessoldat?9.Yournetmonthlysalaryis£1,950.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).© 1993, 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 ... Yourself, Exercise 3 .7 1. Solve for x when 16x = 2x + 56.2. Solve for x when14 =6 + 4x5x3. Solve for x when 45 = 24 +3x.4. Solve for x if 5x2+ 20 = 1,000.5. If q = 560 − 3p solve for p when q...

... Rosser (YoucancheckthesesolutionsarethesameasthoseinExample5.14whichhadthesamesupplyanddemandfunctions.)Inamodelwithtwodependentvariables,likethissupplyanddemandmodel,oncethereducedformequationforonedependentvariablehasbeenderivedthenthereducedformequationfortheotherdependentvariablecanbefound.Thisisdonebysubstitutingthereducedformforthefirstvariableintooneofthefunctionsthatmakeupthemodel.Thus,inthisexample,ifthereducedformequationforequilibriumquantity(3)issubstitutedintothe demand function p = 15 −0 .75 qit becomes p = 15 −0 .75 (12 − t)giving p = 6 + 0 .75 t (4)which is the reduced form equation for equilibrium ... Rosser (YoucancheckthesesolutionsarethesameasthoseinExample5.14whichhadthesamesupplyanddemandfunctions.)Inamodelwithtwodependentvariables,likethissupplyanddemandmodel,oncethereducedformequationforonedependentvariablehasbeenderivedthenthereducedformequationfortheotherdependentvariablecanbefound.Thisisdonebysubstitutingthereducedformforthefirstvariableintooneofthefunctionsthatmakeupthemodel.Thus,inthisexample,ifthereducedformequationforequilibriumquantity(3)issubstitutedintothe ... break-even output?5. If y = 16 + 22x and y =−2.5 + 30.8x, solve for x and y.© 1993, 2003 Mike Rosser From this new reduced form equation we can see that (when I is 180 and t is 0. 375 ) for every...

... 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 27 ... -6 48 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 -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 1 17 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 ...

... 22.5( 275 ,5 67. 6)K−2.5= 0 (3)40 =22.5( 275 ,5 67. 6)K2.5K2.5=22.5( 275 ,5 67. 6)40= 155,006 .78 K = 119.16268Substituting this value into (1) givesL = 275 ,5 67. 6(119.1628)1.5= 211.84 478 This ... 20K0.5L0.25R0.4= 20(23. 47) 0.5(23. 47) 0.25 (75 .1)0.4= 1,200The cheapest cost level for producing this output will therefore be20K + 10L + 5R = 20(23.4) + 10(23. 47) + 5 (75 .1) = £1, 079 .60Example 11.15A ... − 20(3.2)0.4L1.15= 060 = 1.5924287L1.15 37. 678 296 = L1.1523. 47 = LSubstituting this value for L into (5) and (6) givesK = 23. 47 R = 3.2(23. 47) = 75 .1Checking that these values do give...

... =−0.1644 < 0Therefore, the second-order condition for a maximum is satisfied when t = 66. 67. Maximumtax revenue is raised when the per-unit tax is £66. 67. © 1993, 2003 Mike Rosser y=20+4xEFCD0y322420Bx ... rule. Assume, for example, that you wish to find anexpression for the slope of the non-linear demand functionp = (150 − 0.2q)0.5(1)ThebasicrulesfordifferentiationexplainedinChapter8cannotcopewiththissortoffunction. ... 0.2q3]100= 18,000 − 200 = £ 17, 800andTR = pq = 1,800q − 0.6q3= 18,000 − 600 = £ 17, 400Therefore consumer surplus = £ 17, 800 − £ 17, 400 = £400© 1993, 2003 Mike Rosser 12.6 IntegrationIntegrating...

... example: t = 1 .75 years, r = 2.5%= 0.025, A = 56 (e billion). Therefore, thefinal value of GNP will bey = Aert= 56e0.025(1 .75 )= 56e0.04 375 = 58.504384Thus the forecast for GNP is e58,504,384,000.So ... solution is thusyt= 34e1.5tWhen t = 7 then using this definite solution we can predicty 7 = 34e1.5 (7) = 34e10.5= 34(36,315.5) = 1,234, 72 7© 1993, 2003 Mike Rosser Test Yourself, Exercise 14.21. ... 1.5683122 = 7. 0 574 048 millionThus the predicted final population is 7, 0 57, 405.© 1993, 2003 Mike Rosser Example 14.16Given the differential equation dy/dt =−1.5y + 12 derive a function for y in...

... 7? 51? 39? It all depends on which order you perform thecalculations and the type of calculator you use.There are set rules for the order in which basic arithmetic operations should be performed,whichareexplainedinChapter2.Nowadays,theseareprogrammedintomostcalculatorsbut ... quantificationof economic predictions requires the use of mathematics. Although non-mathematical economic analysis may sometimes be useful for making qual-itative predictions (i.e. predicting the direction ... on from each other, each chapter assumes that students arefamiliar with material covered in previous chapters. It is therefore very important that you© 1993, 2003 Mike Rosser AlgebraMuch economic...

... 4 .74 973 63log 3,484 = 3.542 078 1© 1993, 2003 Mike Rosser To divide, we subtract the log of the denominator since56,200 ÷ 3,484 = 104 .74 973 63÷ 103.542 078 1= 104 .74 973 63−3.542 078 1= 101.2 076 582= ... 2.50Evaluate 4,632 .71 × 251. 07 using logs.SolutionUsing the [LOG] function key on a calculatorlog 4,632 .71 = 3.6658351log 251. 07 = 2.39 979 48Thus4,632 .71 × 251. 07 = 103.6658351× 102.39 979 48= 106.0656299= ... 1993, 2003 Mike Rosser To divide by a fraction one simply multiplies by its inverse.Example 2.163 ÷16= 3 ×61= 18Example 2. 17 44 7 ÷849=44 7 ×498=111× 7 2= 77 2= 3812Test...

... Rosser TestYourself,Exercise4.10Fortheproductionfunctionsbelow,assumefractionsofaunitofKandLcanbeused,and(a)deriveafunctionfortheisoquantrepresentingthespecifiedoutputlevelintheformK=f(L)(b)findthelevelofKrequiredtoachievethegivenoutputlevelifL=100,and(c)saywhattypeofreturnstoscalearepresent.1.Q=9K0.5L0.5,Q=362.Q=0.3K0.4L0.6,Q=243.Q=25K0.6L0.6,Q=8004.Q=42K0.6L0 .75 ,Q=5,2505.Q=0.4K0.3L0.5,Q=656.Q=2.83K0.35L0.62,Q=52 7. UselogstoputtheproductionfunctionQ=AKαLβRγintoalinearformat.4.11SummingfunctionshorizontallyIneconomics,thereareseveraloccasionswhentheoryrequiresonetosumcertainfunctions‘horizontally’.Studentsaremostlikelytoencounterthisconceptwhenstudyingthetheoryofthird-degreepricediscriminationandthetheoryofmultiplantmonopolyand/orcartels.By‘horizontally’summingafunctionwemeansummingitalongthehorizontalaxis.Thisideaisbestexplainedwithanexample.Example4.21Aprice-discriminatingmonopolistsellsintwoseparatemarketsatpricesP1andP2(measuredin£).Therelevantdemandandmarginalrevenueschedulesare(forpositivevaluesofQ)P1=12−0.15Q1P2=9−0. 075 Q2MR1=12−0.3Q1MR2=9−0.15Q2Itisassumedthatoutputisallocatedbetweenthetwomarketsaccordingtotheprice-discriminationrevenue-maximizingcriterionthatMR1=MR2.DeriveaformulafortheaggregatemarginalrevenueschedulewhichisthehorizontalsumofMR1andMR2.(Note:InChapter5,weshallreturntothisexampletofindouthowthissummedMRschedulecanhelpdeterminetheprofit-maximizingpricesP1andP2whenmarginalcostisknown.)SolutionThetwoschedulesMR1andMR2areillustratedinFigure4.22.Whatwearerequiredtodo ... 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© 1993, 2003 Mike Rosser ... rule. For any linear function in the format y = a +bx, then b will always representits slope.Example 4 .7 Find the slope of the function y =−2 + 3x.SolutionThe value of y increases by 3 for...