Basic Mathematics for Economists phần 1 ppt

... is 15 0 and quantity rise is 15 0. Therefore using the concept of arc elasticitye = 15 0600 + 750 15 0350 + 200= 15 0 1, 350 15 0550= 15 0 1, 350×550 15 0= 1 27× 11 1 = 11 27£D0 40 10 0 12 060 ... fractions.0 .1 = 1/ 100. 01 = 1/ 1000.0 01 = 1/ 1,000 etc.Thus 0.234 is equivalent to 234 /1, 000.© 19 93, 2003 Mike Rosser Basic Mathematics for Economists Second EditionMike Rosser© 19 93, 2003 ... 2 .17 447÷849=447×498= 11 1 ×72=772= 38 1 2Test Yourself, Exercise 2.4 1. 1 6+ 1 7+ 1 8=2.37+29− 1 4=3.25×607× 21 15=4.45÷24 19 =5. 427− 1 23=6. 2 1 6+ 3 1 4−45=7. 3 1 4+...

... 20000 7% 17 706.578 4 25000 8% 14 864.679 5 25000 9% 12 152.86 10 6 12 000 10 % 9563.64 11 11 % 7090.04 12 IRR = 14 .14 % 12 % 4725.54 13 13 % 2464.06 14 14 % 299.94 15 15 % -17 72 .14 16 16 % -3757 .14 17 17 % ... 385 .15 6 15 .50% 1. 21% 385.98 7 15 .75% 1. 23% 386. 81 8 16 .00% 1. 24% 387.64 9 16 .25% 1. 26% 388.47 10 16 .50% 1. 28% 389.30 11 16 .75% 1. 30% 390 .13 12 17 .00% 1. 32% 390.95 13 17 .25% 1. 33% 3 91. 78 ... 3 91. 78 14 17 .50% 1. 35% 392. 61 15 17 .75% 1. 37% 393.43 16 18 .00% 1. 39% 394.26 17 18 .25% 1. 41% 395.08 18 18 .50% 1. 42% 395.90 19 18 .75% 1. 44% 396.73 20 19 .00% 1. 46% 397.55 21 19.25% 1. 48% 398.37...

... variables 10 .5 Total differentials and total derivatives 11 Constrained optimization 11 .1 Constrained optimization and resource allocation 11 .2 Constrained optimization by substitution 11 .3 The ... to Second EditionAcknowledgements 1 Introduction 1. 1 Why study mathematics? 1. 2 Calculators and computers 1. 3 Using the book2 Arithmetic2 .1 Revision of basic concepts2.2 Multiple operations2.3 ... CoventryUniversity.© 19 93, 2003 Mike Rosser 12 .5 Individual labour supply 12 .6 Integration 12 .7 Definite integrals 13 Dynamics and difference equations 13 .1 Dynamic economic analysis 13 .2 The cobweb:...

... increase will bep 1 ixip0ixi= 1, 944 +1, 666 +1, 012 + 910 + 2 ,16 0 + 7 81 + 1, 176 + 1, 500 1, 800 +1, 360 + 770 + 840 + 2 ,12 0 + 682 + 1, 128 + 1, 300= 11 ,14 9 10 ,000= 1. 115 This means that, ... measuring how much, onTable 3 .1 Student no. 12 345678 910 111 2Height (cm) 17 8 17 5 17 0 16 6 16 8 18 5 16 9 18 9 17 5 18 1 17 7 18 0Weight (kg) 72 68 58 52 55 82 55 86 70 71 65 68© 19 93, 2003 Mike Rosser8.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 =1 30iThismeansthatoneissummingalltheterms30iforvaluesofifrom1to5.Ifthenumberofweeksofbusinessnwasnotknownwecouldinsteadwritesalesrevenue=ni =1 30iToevaluateanexpressioncontainingasummationsign,onemaystillhavetocalculatethevalueofeachtermseparatelyandthenaddup.However,spreadsheetscanbeusedtodotediouscalculationsandinsomecasesshort-cutformulaemaybeused(seeChapter7).© ... of K© 19 93, 2003 Mike Rosserthe inflation rate. Adjusted for inflation, the real price increase for food is thus 10 0 1. 225 1. 115 − 1 = 10 0 (1. 099 1) = 9.9%Test Yourself, Exercise 3.8 1. RefertoTable3.1aboveandwriteanexpressionfortheaverageheightofthefirstn...

... D E 1 Ex 4 .17 TR = 80 - 0.2Q^2 2 3 Q TR 4 0 0 5 20 15 20 6 40 2880 7 60 4080 8 80 512 0 9 10 0 6000 10 12 0 6720 11 14 0 7280 12 16 0 7680 13 18 0 7920 14 200 8000 15 220 7920 16 240 ... applies. 1. MR 1 = 30 − 0.01Q 1 and MR2= 40 − 0.02Q22. MR 1 = 80 − 0.4Q 1 and MR2= 71 − 0.5Q23. MR 1 = 48.75 − 0 .12 5Q 1 and MR2= 75 − 0.3Q2and MR3= 12 0 − 0 .15 Q34. MC 1 = 20 + 0.25Q 1 and ... the demandschedules. Givenp 1 = 12 −0 .15 q 1 p2= 9 −0.075q2then MR 1 = 12 −0.3q 1 MR2= 9 −0 .15 q2and so q 1 = 40 −MR 1 0.3 (1) q2= 60 −MR20 .15 (2)© 19 93, 2003 Mike RosserExample...

... 11 10 86.05 16 6.05 16 12 12 92.40 206.35 17 13 15 43.85 2 51. 45 18 14 18 45.20 3 01. 35 19 15 22 01. 25 356.05 20 16 2 616 .80 415 .55 21 17 3096.65 479.85 22 18 3645.60 548.95 23 19 4268.45 622.85 ... 466.40 19 .35 c = -6.257 3 482.85 16 .45 d = 0.8 8 4 5 01. 20 18 .35 9 5 526.25 25.05 10 6 562.80 36.55 11 7 615 .65 52.85 12 8 689.60 73.95 13 9 789.45 99.85 14 10 920.00 13 0.55 15 11 10 86.05 16 6.05 ... -675 36 -268 11 3.5 -73 14 .5 - 612 25.5 -667 36.5 -238 12 4 -10 8 15 -625 26 -658 37 -207 13 4.5 -14 2 15 .5 -637 26.5 -648 37.5 -17 5 14 5 -17 5 16 -648 27 -637 38 -14 2 15 5.5 -207 16 .5 -658 27.5...

... 90 10 28800 16 00 16 08 81 1 27378 17 8 17 89 80 .1 0 .1 27 217 .98 17 .98 17 9.8 10 80. 01 0. 01 272 01. 7998 1. 7998 17 9.98 11 80.0 01 0.0 01 27200 .18 0 .17 9998 17 9.998 12 80.00 01 0.00 01 27200. 018 0. 017 99998 ... differentiation.Substituting (2) into (1) we getY = a + bY + IY (1 − b) = a + IY =a + I 1 −b=a 1 −b+I 1 −bThereforedYdI= 1 1 −bwhich is the formula for the multiplier.This multiplier ... 27200. 018 0. 017 99998 17 9.9998 13 80.000 01 0.000 01 27200.0 018 0.0 018 00000 17 9.9999802Test Yourself, Exercise 8.3 1. Given the demand schedule p = 12 0 − 3q derive a function for MR and find theoutput...

... 4.4p2+ 1. 4p 1 )= 517 p 1 − 3.5p2 1 + 0.8p 1 p2+ 770p2− 4.4p22+ 1. 4p 1 p2= 517 p 1 − 3.5p2 1 + 770p2− 4.4p22+ 2.2p 1 p2First-order conditions for a maximum are∂TR∂p 1 = 517 − 7p 1 + ... 0.2q22− (16 + 1. 2q 1 + 1. 5q2+ 0.2q 1 q2)= 600q 1 − 0.3q2 1 + 500q2− 0.2q22− 16 − 1. 2q 1 − 1. 5q2− 0.2q 1 q2= 16 + 598.8q 1 − 0.3q2 1 + 498.5q2− 0.2q22− 0.2q 1 q2© 19 93, 2003 Mike ... L0.75(2)Taking (1) to the power of 3L0.75=42 18 6K0.73=423 18 63K2 .1 (3)Setting (3) equal to (2)423 18 63K2 .1 = 31K0.3K 1. 8= 31( 186)3423= 2, 692.4 81 K = 80.4 711 79 (4)Substituting...

... = 10 0 9 t Qt Pt in Pt 10 0 16 0 12 .00 Stability => STABLE 11 1 70 16 .50 4.50 12 2 11 5 14 .25 -2.25 13 3 92.5 15 .38 1. 13 14 4 10 3.75 14 . 81 -0.56 15 5 98 .12 5 15 .09 0.28 16 6 10 0.9375 14 .95 ... Quantity = 16 8 9 t Qt Pt in Pt 10 0 17 5 23 .13 Stability => UNSTABLE 11 1 157.5 25. 31 2 .19 12 2 18 3.75 22.03 -3.28 13 3 14 4.375 26.95 4.92 14 4 203.4375 19 .57 -7.38 15 5 11 4.84375 30.64 11 .07 ... 11 .07 16 6 247.73438 14 .03 -16 . 61 17 7 48.398438 38.95 24.92 18 8 347.40234 1. 57 -37.38 19 9 -10 1 .10 352 57.64 56.06 20 10 5 71. 65527 -26.46 -84.09 © 19 93, 2003 Mike Rosser for price P 1 , corresponding...

... + 12 090 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 2295 10 0 10 5 11 0 11 5 12 0 12 5Figure 14 .1 © 19 93, 2003 Mike RosserY3= C3+ I3= (40 +0.6Y2) + 11 0= 40 +0.6(387.96) + 11 0= ... 17 7 266. 01 376. 01 18 8 265.60 375.60 19 9 265.36 375.36 20 10 265.22 375.22 21 11 265 .13 375 .13 22 12 265.08 375.08 23 13 265.05 375.05 24 14 265.03 375.03 25 15 265.02 375.02 26 16 ... 2 61 = 3 01 ThereforeY0= C0+ I0= 3 01 + 11 0 = 411 In the next time period, the lagged consumption function means that C 1 will be basedon Y0. ThusY 1 = C 1 + 11 0= (40 +0.6Y0) + 11 0=...