III. INTEGRATION Eonomists seem muh more intereste in mrginl effets n ifferentition thn in integrtion. Integrtion is importnt for fining the epete vlue n vrine of rnom vriles, whih is use in eonometris s well s in iniviul eision mking uner unertinty. There s lot of mteril to over for integrtion, n nyhow, it s n essentil prt of lulus review. Integrtion involves tking the sum of lots of infinitely tiny retngles. Let s sy you hve some funtion f on some intervl from to, n you wnt to fin the re etween tht funtion n the -is. If f is liner funtion ( f = + ), this is no prolem. If f is nything else, my inlintion woul e to rek it up into something tht looks like stirse of sorts. Let s ivie up the intervl [, ] into n suintervls with the sme length, so we hve series like = 0 < < < K < n < n =, with i i =. We use the vlue tht f tkes t the right en of eh intervl to pproimte its vlue on tht whole intervl. Then the re A uner the urve is pproimtely the sum of ll these little retngles; eh of whih hs with n height f( i ): n ( i ) i= A f As efore with erivtives, finer prtitioning proues etter pproimtion. An infinitesiml prtitioning gives perfet pproimtion. When is infinitesiml, we write, n the summtion sign is reple with n integrl: A = f As note, whtever omes etween the n is sometimes lle the integrn. Epete vlue prolems sometimes use isrete proility istriution, n sometimes ontinuous istriutions. Lifetime utility mimiztion prolems up the vlue of the utility funtion over spn of time; sometimes these re moele using isrete time, n sometimes with ontinuous time. Eh time, the isrete se involves summtion, n the ontinuous se n integrl. For me, thinking of it this wy mkes these prolems less intimiting. The ie of integrtion is tht lmost every funtion is the erivtive of some other funtion. If f is the erivtive of F, then F is lle the ntierivtive of f. The funtion F is lso wht you get when you integrte f: f = F+ is equivlent to: f F = This is lle n inefinite integrl euse no limits re speifie, so we re not evluting it over ny integrl or spe. We re left with n unientifie, lle Summer 00 mth lss notes, pge
onstnt of integrtion. When evluting the integrl over n intervl, this onstnt goes wy. Beuse it resolves the prolem of this onstnt, n integrl over preetermine intervl is lle efinite integrl. This result is more or less the first funmentl theorem of lulus: Theorem: Let f e ontinuous on [, ]. If F is n ntierivtive for f on [, ], then f( ) = F F. Here re some of the sis rules of integrtion. These will help you solve integrls, when they re solvle. Integrls re liner. Tht is, onstnts n pss through the integrl n there s no prolem with reking the integrl into itive prts: [ + ] = + f g f g The power rule in reverse works most of the time. Tht is: n+ n = n + +, provie tht n. (If n =, rell the nturl logrithm.) Emple: 4 + 7 9 Chnge of vrile when you hve funtion net to its erivtive. If you re integrting some omplite funtion of with respet to, ut you notie tht it looks s if the integrn ontins something multiplie y its erivtive, this formul might e helpful: [ ] = ( ) f u u f g g This is the opposite of the hin rule. Emple: 3 + 6 Integrtion y prts is your frien. You might hve to integrte some omplite funtion, ut you notie tht one prt looks like the erivtive of n esy funtion, wheres the other prt looks like it hs pretty strightforwr erivtive. [ ] = [ ] f g f g f g The is essentilly oule hnge of vrile. Emple: e Summer 00 mth lss notes, pge
Aitionlly, you shoul keep in min ll the speil erivtives, like those of f= ln n f= e, when integrting. On top of these, there re mny funtions whih hve speil ntierivtives tht you hve to memorize, e le to erive, or t lest reognize. Trigonometri funtions re some of these. Funtions with lots of squres n squre roots might e s well. Chpter 9 of Sysæter, Strøm, n Berk hs lists mny of the funtions n their ntierivtives. Most si lulus tetooks lso hve tle of integrls. Sine I ve never enountere these in ouple of yers of gr shool, I m not going to over them. There re few things to rememer out efinite integrls. First of ll, swithing the orer of the limits hnges the sign of the integrl. Seon, the integrl over single point hs vlue of zero (this omes up in proility). Thir, it s fine to split up the limits of integrtion: = f f = 0 f = + f f f Some other useful properties reltive to tking erivtives of integrls. The first two sy wht hppens when you ifferentite with respet to the limits of integrtion. The net is for ifferentiting with respet to prmeter insie the integrl. The lst is Leinitz rule, generliztion of ll these, tht lso tells you wht to o when ifferentiting with respet to the inepenent vrile of f: t t vt ut f = f f = f f, t f(, t) = (when,, n o not epen on t) t vt f, t f(, t) = f( v() t, t) v ( t) f( u() t, t) u ( t)+ ut t The integrls we hve tlke out hve een for funtions efine on lose intervls, whih re ompt sets. An integrl over n intervl from to in whih either or (or oth!) equls ± or in whih the integrn f is unefine t some point in [,] is n improper integrl. Emple: euse the upper limit is infinity Emple: euse the funtion isn t efine t = 0 Summer 00 mth lss notes, pge 3
In these ses, the trik is to stik in some onstnt like for the offening vlue, n then tke evlute the limit of the integrl s pprohes tht vlue. If the vlue is in the mile of the integrl, you hve to split the integrl into multiple prts. Emple: = = lim lim = lim = Emple: = lim + lim 0 0 Integrtion will frequently e use to fin the epete vlue of funtion, often epete utility. Though proility is going to e overe on the lst y of this ourse, we ll o epettions riefly now, so we n work some eonomi emples. My frien Emun n I hve et. He flips oin n if it omes up hes, he wins two ollrs from me. If it omes up tils, I win one ollr from him. Wht is my epete vlue of this et? Clulting tht I get $, 50% of the time, plus negtive $3, 50% of the time, I guess the epete ernings re negtive one ollr. An I e orret. There is some set of ll possile events S. In this emple, there were only two events. There is given proility tht ny prtiulr event s in this set will hppen, enote y P( outome = s). Provie my frien n I hve pproimtely the sme psyhi ility to influene the outome of the oin, P( outome = hes)= P( outome = tils)=. The epete vlue of funtion h is efine s: E[ hs ]= hs P( outome = s) s S Tht is, the sum over ll possile sttes of the proility of rriving in tht stte, times the vlue of the funtion in tht stte. In the emple, ll I re out ws the monetry vlue of the pyoff I ws setting hs = s. Another possiility is tht this funtion is my utility funtion. Sy I hve this utility funtion: U= ln= ln( w+ s) where w is my welth efore the et. Utility is the nturl logrithm of my onsumption, whih is equl to my initil welth plus my winnings in the et. Then my epete utility is: w E[ U ]= ( w+ s) P( s)= ( w )+ ( w + )= ln ln 3 ln 3 ln s { hes,tils} w + When we hve ontinuous set of outomes (sy, we flip the oin n trillion times n look t the perentge of hes n tils), then there is proility ensity funtion f with the property tht for ny suset T of S, 0 Summer 00 mth lss notes, pge 4
= P outome T T f s s Notie tht with ontinuous istriutions, the proility tht you get ny one element s of set S is zero, sine the integrl from s to s is zero. On the other hn, the proility tht you get something in S must e etly one; tht is, f s s S (this is property tht permissile PDFs must hve). Anlogous to the isrete se, the epet vlue of funtion hs is efine s: E hs s S [ ]= hsf ss Emple: The numer of yers it tkes stuents to omplete the grute progrm here is t istriute f()= t 0. 87e 87., where t 0. How long n n inoming gr stuent epet to enure this torture? Emple: My utility funtion for the numer of yers spent in gr shool is: Ut ()= C t I might grute in four yers with little isutility; it might tke me few ees, in whih se it is very pinful. I wnt to know from the strt: wht is the epete utility of stying in gr shool? Emple: The strting slries of new PhDs in eonomis is istriute: y f( ( µ ) y)= ep σ π σ (An yes, in this emple it is possile for them to e negtive ples mke you py for the privilege of working there.) If µ =56,4 n σ =8,73, then wht is my epete strting slry? Emple: All my slry in my first yer of work will e onsume. My utility funtion for onsumption in this yer is: U = α β+ γ Wht s my epete utility? Referenes: Simon n Blume: Appeni A4. Sysæter, Strøm, n Berk: Chpter 9. Sls n Hille: Chpters 5, 8, n 7. Summer 00 mth lss notes, pge 5