Multivariable Calculus MATH 205
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Date Created: 10/29/15
Fall Semester 07 08 Akila Weempana LECTURE 5 INTEGRAL CALCULUS I INTRODUCTION H In the last lecture we talked about differentials and used differentials to talk about some important economic concepts like elasticities economic growth and simple comparative static analysis Since Econ 205 is a pre requisite that will be all we will cover on differential calculus theory You should be very comfortable with any other topic related to differential calculus at the level of Math 205 Today s lecture does a similar review of concepts regarding integral calculus from Math 205 Once again we will do a quick review and spend most of the time discussing economic applications The most important applications will be calculating consumer and producer surplus bond pricing and some econometric applications with density functions If you need to brush up on your integrals once again I urge you to do so immediately Chapter 12 of Klein will serve both as a review of what you studied in your calculus classes and as supplementary reading for this lecture THEORY The inde nite integral ffmdz is the family of anti derivatives of a function dFm 1m Since constant terms have a derivative of zero we can only de ne up to an arbitrary constant absent any further information hence the family of anti derivatives In other words 0 In essence you can think of integration as the reverse of di erentiation Suppose that we have a function m that is the derivative of some function Integration can be used with a little bit of additional information to uncover the constant terms to recover the original function using the information contained in The de nite integral f fzdz is the area under the curve fx over the range m a to m b The value of f fmdz Fb 7 Fa where f fmdz o In Economics this means that we can derive a total cost function using information about marginal costs derive the underlying utility function given information about marginal util ity and derive a cumulative density function given information about a probability density function etc Basic Rules of Inde nite Integrals 0 Since almost everyone uses integration less than differentiation in academic settings7 it may be wise to do a quick survey of some basic rules of integration Some useful rules to remember where a is an arbitrary constant term are n1 zndm 1c fadxamc n emdxemc fidmlnlmlc fx gltzgtidz f mm new afxdx a 0 Note that c is an arbitrary constant Since constants disappear during differentiation without additional information we can not identify the constant terms in the original function by integration7 ie given f m 2x we can not state whether x 2 3 or x 2 5 c There are also two important rules of substitution that you should be familiar with 1 If ugx then f fug zdm f fudu 2 fudvuv7vfdu Examples 1 Find the value of the following integral f frgdm Let u gm x2 3 Then du g zdm 2zdm We can then rewrite the integral as f idu the solution to which is In ul 0 So 22dxlnm23 0 N Find the value of the following integral flnmdz Let u lnz and v z Then du 1mdz and dv dm Since we have an integral of the form fudv we have a solution of the form M 7 fvdu zlnm 7 fm zlnz 7 fdz so lnmdm zlnm 7 m 0 Basic Rules of De nite Integrals o Other important rules to keep in mind are f mm 0 Ab mz 7bafzdz c b c fzdm fmdzb fzdzwherealtbltc III APPLICATIONS Recovering Total Cost from Marginal Cost 0 Given a marginal cost function of the form MCQ we can recover the underlying total cost function as TCQ fMCQdQ For example7 if we have a marginal cost function of the form MCQ 3 2Q7 the underlying total cost function is TCQ 3Q Q2 c 0 Note that the total cost function can only be pinned down upto a constant What is the constant in a total cost function It is xed costs7 so using a marginal cost function we are unable to recover the level of xed costs that a rm faces Consumer and Producer Surplus o In microeconomics7 integrals play a Vital role in calculating consumer and producer surplus This is especially true when the demand curve and supply curve are not linear Let s think about how to determine consumer and producer surplus using the supplydemand diagram below 0 Let the demand function be denoted by Q DP and the supply function be denoted by Q SP The equilibrium price and quantity7 ie where demand and supply intersect are denoted as P and Q respectively P 5M2 P P cs PS D1ltQgt B Q I 0 Consumer surplus is the area under the demand curve above Pi ie the difference between willingness to pay and price paid for each unit sold up to Q This area is labeled CS in the above diagram 0 Producer surplus is the area under the supply curve below Pi ie the difference between the marginal cost of production represented by the supply curve and the market price7 which represents the producer s pro ts This area is labeled PS in the above diagram 0 We can write down algebraic expressions for producer and consumer surplus using integrals For example consumer surplus is the area under the demand curve and above P between Q 0 and Q Q We can write this in one of two ways Qt CS D 1QPldQ P or equivalently CS DPdP Q 125 A Ptsmmdg PM or equivalently PS SPdP Example 0 Suppose the demand curve for a product is given by DP 10 7 2P and the supply curve is given by SP 4 P Equilibrium price and quantity can be calculated as P 2 and Q 6 The inverse demand and supply functions can be calculated as D 1Q 5 7 g and 5 1 Q c2 7 4 The y intercepts are at 15 5 for demand and E 74 for supply Using integrals7 we can calculate consumer surplus to be 6 CS D 1Q 7 2 day 0 6 6 57Q 72 162 3 7ng 0 2 0 2 Q2 6 3 7 7 lt Q 4 9 0 Alternatively we could have found consumer surplus as 0 CS 5 DPdP 2 5 10 7 2P dP 2 7 10P7P2H50725720749 0 Using integrals we can calculate producer surplus to be PS 62 75 1Ql W 6 6 l27Q74ldQ 67mm 0 0 2 6 6627 0 Alternatively we could have found producer surplus as 18 0 PS 2 SP dP 74 2 4PldP 74 P2 2 4P 7 0 Of course since the curves are linear we can avoid all of this and calculate the magnitudes of consumer and producer surplus geometrically Consumer surplus is the area covered by a right angled triangle with base 6 and height 3 so CS 12 6 3 9 Producer surplus is the area under a right angled triangle with base 6 and height 6 so PS 12 6 6 18 827716818 4 Welfare Effects of Price Changes 0 We can also use integrals to think about the welfare effects of changes in equilibrium price and quantity For example what happens to consumer and producer surplus when price goes up Graphically let s think about what happens when there is an increase in demand that raises equilibrium price to P and equilibrium quantity to Q P P11 Os PS 0 New consumer surplus is the area under the new demand curve above P and the new producer surplus is the area under the supply curve below P Example Gini Suppose the new demand curve D for the product is Q 13 7 2P and the supply curve is still given by Q 4 P New equilibrium price and quantity can be calculated as P and Q 7 Consumer surplus is now CS f07 D 1Q 7 3 dQ The new demand curve can be inverted as P 13 7 QW so 05 f i 3 dc f5 39 162 which simpli es to 7Q Q2 0547 Producer surplus is now PS f07 3 7 S 1Q dQ f07 3 7 Q 7 4 dQ which simpli es to 49 7 0 4 Q2 7 2 Consumer surplus increases from 9 to 1225 and producer surplus increases from 18 to 245 PS A177 QdQ mi Coef cients Another useful application of integrals described in Klein is the calculation of a Gini Coef cient7 a widely used measure of income inequality The Gini coef cient is derived using integral calculus from a graph known as a Lorenz curve A Lorenz curve simply plots the cumulative share of overall income earned by each percentile of the population against each percentile of the population7 ordered from poorest to richest A 45 degree line is drawn on the diagram to represent complete equality the case where every percentile of the population earns the same share of overall income The more equal the income distribution is7 the closer the Lorenz curve is to matching the 45 degree line The Gini coef cient is de ned as the area between the 45 degree line and the Lorenz curve over the area under the 45 degree line In terms of the graph below we can de ne Gim m M u r 11 the Gini 11 can be calculated as I3 is 7 MPH dp folpdp 7 fol Lpl dp 7 G i 2 i Proportion of population 100 45 degree line B p Lorenz Curve Cumulative share of income 100 Random Variables We can also use integrals to think about basic concepts related to econometrics and statistics In statistics we often use random variables variables whose value re ect the outcome of some probabilistic event Random variables have a probability distribution which describes the values that the random variable can take on and the probability of achieving each of those outcomes For example suppose X is a random variable whose value X is the outcome from a single roll of a 6 sided fair die The probability distribution of X fx is f116 f216 f316 f416 f516 f616 and fi0 for any other value i For any discrete random variable X with the probability distribution function fx the fol lowing rules must hold 0 S Hz 1 and EM 1 ie the probability of observing any outcome has to be non negative and the sum of the probability of observing the independent outcomes can t exceed 1 Random numbers can be continuous as well as discrete Suppose that Z is a continuous random variable that can take on a continuum of values 2 Since 2 can take on an in nite number of values we can t describe the probability of taking any given value we can only talk about the probability that 2 taking on a range of values as Pra 2 g b f fzdz 2 0 As in the discrete case the second condition states that all the probabilities have to add up to 1 so ff fzd2 1 Note that 0 if Z does not take on the value 2 Possible distribution functions for a discrete random variable X and a continuous random variable Z are given below 2 Other key concepts associated with random variables include the cumulative distribution function de ned as ProbX z j The expected value of a random variables de ned as The variance of a random variable de ned as E W EM 2W 2 EM 2ME M2 EM M2 This can be calculated as Examples Let s do some examples with a couple of commonly used probability distributions highlighted by Klein the uniform distribution and the exponential distribution The uniform distribution has an upper limit 1 and a lower limit a and has a distribution function ME ME for z in 11 7 a 0 otherwise o The GDP for the uniform distribution is then Fm Ofmdxbiadx m bia a 70 bia o The expected value of a random variable that is uniformly distributed is Ea 1mfmdmabbfadm bziaz 2b7a ba M T o The variance of a random variable that is uniformly distributed is we m e m2 EM 7 2 00 b 2 b b 2 m2fmdm 7 a m2fmdm 7 m 00 4 1 4 1 b ba2 1 3531 ba2 Vadm biaaxzdmi 4 ga77 4 bgias ba2 b2aba2 b22aba2 m7 4 fif b272aba2 122 b a Vadm 12 o A random variable that is exponentially distributed has a distribution function x A57 for0ltzltoo o The GDP for the exponential distribution is then I I AETAEJ dm 700 0 feiAz m 0 17 57M o The expected value of a random variable that is exponentially distributed is 1 mfmdm 000 new dz De ne u z dv Ae Mdz then 7m 7 OOO 757M dm dudmv 75 M 7 Udu 7m 7 co eAm co 0 0 co co o The variance of a random variable that is exponentially distributed is vane 7 new WM mam n2 We can calculate the value of the rst term in the above expression as Therefore 000 mzkei w dm A mzkei w dm De ne u 2011 Ae Mdz then du dem U 757M 00 0 7 0 757M 2xdm M 7 Udu 7m257 w 2 00 7A1 XO Ame dm 0 co 7267Am 0