III. INTEGRATION Economists seem much more intereste in mrginl effects n ifferentition thn in integrtion. Integrtion is importnt for fining the expecte vlue n vrince of rnom vriles, which is use in econometrics s well s in iniviul ecision mking uner uncertinty. There s lot of mteril to cover for integrtion, n nyhow, it s n essentil prt of clculus review. Integrtion involves tking the sum of lots of infinitely tiny rectngles. Let s sy you hve some function f ( x) on some intervl from to, n you wnt to fin the re etween tht function n the x-xis. If f is liner function ( f ( x) = x + ), this is no prolem. If f is nything else, my inclintion woul e to rek it up into something tht looks like stircse of sorts. Let s ivie up the intervl [,] into n suintervls with the sme length, so we hve series like = x 0 < x 1 < x 2 < < x n1 < x n =, with x i x i1 = "x. We use the vlue tht f x tkes t the right en of ech intervl to pproximte its vlue on tht whole intervl. Then the re A uner the curve is pproximtely the sum of ll these little rectngles; ech of which hs with x n height f ( x i ) : A # n i=1 f ( x i )"x As efore with erivtives, finer prtitioning prouces etter pproximtion. An infinitesiml prtitioning gives perfect pproximtion. When x is infinitesiml, we write x, n the summtion sign is replce with n integrl: A = f ( x)x As note, whtever comes etween the n x is sometimes clle the integrn. Expecte vlue prolems sometimes use iscrete proility istriution, n sometimes continuous istriution. Lifetime utility mximiztion prolems up the vlue of the utility function over spn of time; sometimes these re moele using iscrete time, n sometimes with continuous time. Ech time, the iscrete cse involves summtion, n the continuous cse n integrl. For me, thinking of it this wy mkes these prolems less intimiting. The ie of integrtion is tht lmost every function is the erivtive of some other function. If f is the erivtive of F, then F is clle the ntierivtive of f. The function F is lso wht you get when you integrte f: = F( x) + c is equivlent to: f ( x) = F This is clle n inefinite integrl ecuse no limits re specifie, so we re not evluting it over ny integrl or spce. We re left with n unientifie c, clle constnt of integrtion. When evluting the integrl over n intervl, this constnt goes wy. Becuse it resolves the prolem of this constnt, n integrl ( x) Fll 2007 mth clss notes, pge 21
over preetermine intervl is clle efinite integrl. This result is more or less the first funmentl theorem of clculus: [ ], Theorem: Let f e continuous on[,]. If F is n ntierivtive for f on, then = F( ) " F( ). Tht s the theory. For prcticl mtters, here re some of the sics rules of integrtion, to help you evlute integrls (when they re solvle). Integrls re liner. Tht is, constnts cn pss through the integrl n there s no prolem with reking the integrl into itive prts: + g( x) & "# f x $% x = & f x x + & g x x The power rule in reverse works most of the time. Tht is: x n x = xn+1 + c, n + 1 provie tht n "1. (If n = 1, recll the nturl logrithm.) Exmple: " 14x 2 + 7 x 9 x When you hve function next to its own erivtive, try chnge of vriles. If you re integrting some complicte function of x with respect to x, ut you notice tht it looks s if the integrn contins something multiplie y its erivtive, this formul might e helpful: " f ( u)u = # $ f g( x) g ( x) % & x This is the opposite of the chin rule. Exmple: x 3x 2 + 6 x Integrtion y prts is your frien. You might hve to integrte some complicte function, ut you notice tht one prt looks like the erivtive of n esy function, wheres the other prt looks like it hs pretty strightforwr erivtive. " #$ f x g x %& x = f ( x) g x ( f "( x) g( x) This is essentilly oule chnge of vrile. Exmple: xe x x #$ %& x Aitionlly, you shoul keep in min ll the specil erivtives, like those of f x = ln x n f ( x) = e x, when integrting. On top of these, there re mny Fll 2007 mth clss notes, pge 22
functions with specil ntierivtives tht you hve to memorize, e le to erive, or t lest recognize. Trigonometric functions re some of these. Functions with lots of squres n squre roots might e s well. Chpter 9 of Sysæter, Strøm, n Berck hs lists mny of the functions n their ntierivtives. Most sic clculus textooks lso hve tle of integrls. Since I ve never encountere these in couple of yers of gr school, I m not going to cover them. There re few things to rememer out efinite integrls. First of ll, switching the orer of the limits chnges the sign of the integrl. Secon, the integrl over single point hs vlue of zero (this comes up in proility). Thir, it s fine to split up the limits of integrtion: = " f ( x)x = 0 c = + c Some other useful properties relte to tking erivtives of integrls. The first two sy wht hppens when you ifferentite with respect to the limits of integrtion. The next is for ifferentiting with respect to prmeter insie the integrl. The lst is Leinitz rule, generliztion of ll these, tht lso tells you wht to o when ifferentiting with respect to the inepenent vrile of f: t t f x x = f = " f " f x,t f ( x,t)x = x (when,, n x o not epen on t) "t v( t) u t " f ( x,t)x = f v( t),t v t # f u t (,t) u" t + v t u t $t $ f x,t The integrls we hve tlke out hve een for functions efine on close intervls, which re compct sets. An integrl over n intervl from to in which either or (or oth) equls ± or in which the integrn f is unefine t some point in [,] is n improper integrl. x Exmple: Exmple: x " ecuse the upper limit is infinity 1 x 2 1 xx " ecuse the function x isn t efine t x = 0 1 x x Fll 2007 mth clss notes, pge 23
In these cses, the trick is to stick in some constnt like c for the offening vlue, n then tke evlute the limit of the integrl s c pproches tht vlue. If the vlue is in the mile of the integrl, you hve to split the integrl into multiple prts. Exmple: Exmple: x $ c x $ *1 " = lim 1 x 2 c# % & " 1 x 2 ( ) = lim c# % & x ( ) 1 xx c xx 0 xx " = lim 1 x c#0 " + lim 1 x c$0 " c x c 1 $ = lim c# 1* 1 % & c( ) = 1 Integrtion will frequently e use to fin the expecte vlue of function, often expecte utility. Though proility is going to e covere on the lst y of this course, we ll o expecttions riefly now, so we cn work some economic exmples. Exmple: My frien Oliver n I hve et. He flips coin n if it comes up hes, he wins two ollrs from me. If it comes up tils, I win one ollr from him. Wht is my expecte vlue of this et? Clculting tht I get $1 50% of the time, plus negtive $3 50% of the time, I guess tht this is losing prospect in generl. An I e correct, ut I like to know how to stte this formlly. Let s let X e some rnom vrile, something whose vlue is etermine y chnce. The set S will enote ll possile vlues of the outcome of this rnom vrile. (In this exmple, X cn e the mount tht I win, n the set of ll possile outcomes is S = {3,+1}.) There is some proility istriution P[X] tht etermines the likelihoo of ech outcome ( P[X = 3] = 0.5, n P[X = +1] = 0.5 ). The expecte vlue of X is efine s: E[X] = " sp[x = s] ss Tht is, the sum over ll possile vlues of the outcome, times the chnce of tht outcome occurring. This will give you the verge vlue of X over mny, mny inepenent repetitions of the experiment. We cn lso tke the expecte vlue of some function of X. For exmple, I might get some utility from my monetry winnings, represente y U(X) ; I wnt to know my expecte utility. This woul e clculte s: E[U(X)] = " U(s)P[X = s] ss There is nothing specil out the function U; ny g(x) or h(x) will o. Exmple: My utility function is U(c), where c is my consumption, equl to my initil welth plus or minus ny winnings. I strt with $10, n with equl proility, I either lose $3 or win $1. My expecte winnings re: Fll 2007 mth clss notes, pge 24
E[X] = (0.5)(3) + (0.5)(1) = 1 My expecte consumption is: E[10 + X] = (0.5)(10 3) + (0.5)(10 + 1) = 9 n my expecte utility is: E[ln(10 + X)] = (0.5)ln(10 3) + (0.5)ln(10 + 1) = 0.5( ln 7 + ln9) When we hve continuous set of outcomes (sy, we flip the coin n trillion times n look t the percentge of hes n tils), then there is proility ensity function f with the property tht for ny suset T of S, = f s P X T " T s Notice tht with continuous istriutions, the proility tht you get ny one element s of set S is zero, since the integrl from s to s is zero. On the other hn, the proility tht you get something in S must e exctly one; tht is, f ( s S )s " 1(this is property tht permissile PDFs must hve). Anlogous to the iscrete cse, the expect vlue of rnom vrile is: E[X] = " ss sf (s)s n the expecte vlue of some function h(x) is efine s: E[h(X)] = " ss h(s) f (s)s Exmple: The numer of yers it tkes gr stuents to complete PhD is istriute f ( t) = 0.187e 1.87t, wheret 0. How long cn n incoming gr stuent expect to e in gr school? Exmple: Your utility function for the numer of yers spent in gr school is: U ( t) = C t 2 (You might grute in four yers with little isutility; it might tke you few eces, in which cse it is very pinful.) Before you strt, you wnt to know: wht is the expecte utility of gr school? Exmple: The strting slries of new PhDs in economics is istriute: f ( y) = 2 1 2" exp $ & # y # µ % 2 2 ) ( Fll 2007 mth clss notes, pge 25
If µ = 56,412 n = 8,273, then wht is their expecte strting slry? Exmple: The utility person gets from this slry is etermine y the function: U ( c) = c " #c 2 + $ Wht s his expecte utility? References: Simon n Blume: Appenix A4. Sysæter, Strøm, n Berck: Chpter 9. Sls n Hille: Chpters 5, 8, n 17. Fll 2007 mth clss notes, pge 26