ECONOMICS 00A MATHEMATICAL HANDOUT Fall 07 A. CALCULUS REVIEW Mark Machia Derivatives, Partial Derivatives ad the Chai Rule You should already kow what a derivative is. We ll use the epressios ƒ() or dƒ()/ for the derivative of the fuctio ƒ(). To idicate the derivative of ƒ() evaluated at the poit = *, we ll use the epressios ƒ(*) or dƒ(*)/. Whe we have a fuctio of more tha oe variable, we ca cosider its derivatives with respect to each of the variables, that is, each of its partial derivatives. We use the epressios: ƒ(, ) ad ƒ(,) iterchageably to deote the partial derivative of ƒ(,) with respect to its first argumet (that is, with respect to ). To calculate this, just hold fied (treat it as a costat) so that ƒ(,) may be thought of as a fuctio of aloe, ad differetiate it with respect to. The otatio for partial derivatives with respect to (or i the geeral case, with respect to i) is aalogous. For eample, if ƒ(,) = + 3, we have: ƒ(, ) ƒ(, ) = ƒ(,) = + 3 ad = ƒ(,) = The ormal vector of a fuctio ƒ(,...,) at the poit (,...,) is just the vector (i.e., ordered list) of its partial derivatives at that poit, that is, the vector: ƒ(,..., ) ƒ(,..., ) ƒ(,..., ),,..., ƒ (,..., ), ƒ (,..., ),..., ƒ (,..., ) Normal vectors play a key role i the coditios for ucostraied ad costraied optimizatio. The chai rule gives the derivative for a fuctio of a fuctio. Thus if ƒ() g(h()) we have ƒ() = g(h()) h() The chai rule also applies to takig partial derivatives. For eample, if ƒ(,) g(h(,)) the ƒ(, ) h(, ) g( h(, )) Similarly, if ƒ(,) g(h(,),k(,)) the: ƒ(, ) h(, ) k(, ) g ( h(, ), k(, )) g ( h(, ), k(, )) The secod derivative of the fuctio ƒ() is writte: d ƒ( ) ƒ() or If the material i this sectio is ot already familiar to you, you probably wo t be able to pass this course.
ad it is obtaied by differetiatig the fuctio ƒ() twice with respect to (if you wat to calculate the value of the secod derivative at a particular poit *, do t substitute i * util after you ve differetiated twice). A secod partial derivative of a fuctio of several variables is aalogous, i.e., we write: ƒii(,...,) or to deote differetiatig twice with respect to i. ƒ(,..., ) i We get a cross partial derivative whe we differetiate first with respect to i ad the with respect to some other variable j. We will deote this with the epressios: ƒ(,..., ) ƒij(,...,) or Here s a strage ad woderful result: if we had differetiated i the opposite order, that is, first with respect to j ad the with respect to i, we would have gotte the same result. I other words, ƒji(,...,) ƒij(,...,) or equivaletly ƒ(,...,)/ij ƒ(,...,)/ji. i j Approimatio Formulas for Small Chages i Fuctios (Total Differetials) If ƒ() is differetiable, we ca approimate the effect of a small chage i by: ƒ = ƒ(+) ƒ() ƒ() where is the chage i. From calculus, we kow that as becomes smaller ad smaller, this approimatio becomes etremely good. We sometimes write this geeral idea more formally by epressig the total differetial of ƒ(), amely: dƒ = ƒ() but it is still just shorthad for sayig We ca approimate the chage i ƒ() by the formula ƒ ƒ(), ad this approimatio becomes etremely good for very small values of. Whe ƒ() is a fuctio of a fuctio, i.e., it takes the form ƒ() g(h()), the chai rule lets us write the above approimatio formula ad above total differetial formula as dg( h( ))) g( h( )) g( h( )) h( ) thus dg( h( )) g( h( )) h( ) For a fuctio ƒ(,...,) that depeds upo several variables, the approimatio formula is: ƒ(,..., ) ƒ(,..., ) ƒ = ƒ(+,...,+) ƒ(,...,)... Oce agai, this approimatio formula becomes etremely good for very small values of,,. As before, we sometimes write this idea more formally (ad succictly) by epressig the total differetial of ƒ(), amely:
ƒ(,..., ) ƒ(,..., ) dƒ... or i equivalet otatio: dƒ = ƒ(,...,) + + ƒ(,...,) B. ELASTICITY Let the variable y deped upo the variable accordig to some fuctio, i.e.: y = ƒ() How resposive is y to chages i? Oe measure of resposiveess would be to plot the fuctio ƒ() ad look at its slope. If we did this, our measure of resposiveess would be: absolute chage i y y dy slope of f ( ) = f ( ) absolute chage i Elasticity is a differet measure of resposiveess tha slope. Rather tha lookig at the ratio of the absolute chage i y to the absolute chage i, elasticity is a measure of the proportioate (or percetage) chage i y to the proportioate (or percetage) chage i. Formally, if y = ƒ(), the the elasticity of y with respect to, writte Ey,, is give by: E y, = y proportioate chage i y y y proportioate chage i y If we cosider very small chages i (ad hece i y), y/ becomes dy/ = ƒ(), so we get that the elasticity of y with respect to is give by: y y = y dy E, ( ) y f y y y Note that if ƒ() is a icreasig fuctio the elasticity will be positive, ad if ƒ() is a decreasig fuctio, it will be egative. It is crucial to ote that while elasticity ad slope are both measures of how resposive y is to chages i, they are differet measures. I other words, elasticity is ot the same as slope. For eample, if y is eactly proportioal to, i.e., if we have y = c, the slope of this curve would be c, but its elasticity is give by: dy Ey, = c y c I other words, wheever y is eactly proportioal to, the elasticity of y with respect to will be oe, regardless of the value of the coefficiet c. Here s aother eample to show that elasticity is ot the same as slope. The fuctio y = 3 + 4 obviously has a costat slope (amely 4). But it does ot have a costat elasticity. To see this, use the formula agai: 3
E y, which is obviously ot costat as varies. dy 4 = 4 y 34 34 Fially, we ote that if a fuctio has a costat elasticity, it must take the form ƒ() c for some costats c > 0 ad. We prove this by the calculatio: E ƒ( ), = ( ) d ƒ( ) c ƒ( ) c Note that will be positive if the fuctio is icreasig, ad egative if it is decreasig. C. LEVEL CURVES OF FUNCTIONS If ƒ(,) is a fuctio of the two variables ad, a level curve of ƒ(,) is just a locus of poits i the (,) plae alog which ƒ(,) takes o some costat value, say the value k. The equatio of this level curve is therefore give by ƒ(,) = k. For eample, the level curves of a cosumer s utility fuctio are just his or her idifferece curves (defied by the equatio U(,) = u0), ad the level curves of a firm s productio fuctio are just the isoquats (defied by the equatio ƒ(l,k) = Q0). The slope of a level curve is idicated by the otatio: ƒ(, ) k ƒ0 or where the subscripted equatios are used to remid us that ad must vary i a maer which keeps us o the ƒ(,) = k level curve (i.e., so that ƒ = 0). To calculate this slope, recall the vector of chages (,) will keep us o this level curve if ad oly if it satisfies the equatio: 0 = f ƒ(,) + ƒ(,) which implies that ad will accordigly satisfy: so that i the limit we have: ƒ(, ) k ƒ (, ) ƒ (, ) ƒ(, ) k ƒ (, ) ƒ (, ) This slope gives the rate at which we ca trade off or substitute agaist so as to leave the value of the fuctio ƒ(,) uchaged. This cocept will be of frequet use i this course. 4
slope = b ƒ, gk ƒ(, ) ƒ(, ) ƒ(, ) = k A applicatio of this result is that the slope of the idifferece curve at a give cosumptio budle is give by the ratio of the margial utilities of the two commodities at that budle. Aother applicatio is that the slope of a isoquat at a give iput budle is the ratio of the margial products of the two factors at that iput budle. I the case of a fuctio ƒ(,...,) of several variables, we will have a level surface i dimesioal space alog which the fuctio is costat, that is, defied by the equatio ƒ(,...,) = k. I this case the level surface does ot have a uique taget lie. However, we ca still determie the rate at which we ca trade off ay pair of variables i ad j so as to keep the value of the fuctio costat. By eact aalogy with the above derivatio, this rate is give by: ƒ (,..., ) i j i ƒ (,..., ) j ƒ(,..., ) k j ƒ0 i Fially, give ay level curve (or level surface) correspodig to the value k, the better-tha set of that level curve or level surface is the set of all poits at which the fuctio yields a higher value tha k, ad the worse-tha set is the set of all poits at which the fuctio yields a lower value tha k. D. POSSIBLE PROPERTIES OF FUNCTIONS Cardial vs. Ordial Properties of a Fuctio Cosider some domai X. For coveiece, we ca epress each poit i X by the boldface symbol = (,...,), * = ( *,..., * ), etc. Every real-valued fuctio ƒ() = ƒ(,...,) over X implies (amog other thigs) the followig two types of iformatio: ordial iformatio: how the fuctio raks differet poits i its domai, i.e., the rakig ord over poits i X, as defied by A ord B ƒ(a) ƒ(b) cardial iformatio: how the fuctio raks differeces betwee pairs of poits i the domai, the rakig crd over all pairs of poits i X, as defied by {A, B} crd {C, D} ƒ(a) ƒ(b) ƒ(c) ƒ(d) 5
It is easy to prove that the rakigs ord ad crd will always be trasitive (ca you do this?) ad we defie their associated relatios ord, ~ord, ord ad crd, ~crd, crd i the obvious maer. Note that a pair of fuctios ƒ() ad g() over X will have the same ordial iformatio as each other if ad oly if they are liked by some icreasig trasformatio of the form g() (ƒ()) for some strictly icreasig fuctio () Similarly, a pair of fuctios ƒ() ad g() will have the same cardial iformatio as each other if ad oly if they are liked by some icreasig affie trasformatio of the form g() a ƒ() + b for some costats a, b with a > 0 Although it is importat to recall that every fuctio has both cardial ad ordial iformatio, we typically refer to a fuctio as ordial if oly its ordial iformatio is relevat. A eample is a stadard utility fuctio U(,...,) over ostochastic commodity budles (,...,), whose ordial iformatio is precisely the cosumer s preferece relatio over such budles. We refer to a fuctio as cardial if its cardial iformatio is relevat. A eample of a cardial fuctio is a cosumer s vo Neuma-Morgester utility fuctio which (uder the epected utility hypothesis) represets their attitudes toward risk (if you have ot yet see this cocept, do t worry.) The cardial iformatio of a fuctio icludes its ordial iformatio, sice * ord ƒ(*) ƒ() ƒ(*) ƒ() ƒ() ƒ() {*, } crd {, } But do t thik that a fuctio s cardial iformatio (the rakig crd) ehausts the iformatio of the fuctio. For eample, kowig all the cardial iformatio of a productio fuctio still leaves you with o idea of how much output it yields from ay iput budle! (I.e., productio fuctios are more tha just cardial fuctios.) Scale Properties of a Fuctio The fuctio ƒ(,...,) ehibits costat returs to scale or is homogeeous of degree oe, if ƒ(,...,) ƒ(,...,) for all,..., ad all > 0 i.e. if multiplyig all argumets by implies the value of the fuctio is also multiplied by. A fuctio ƒ(,...,) is said to ehibit scale ivariace or is homogeeous of degree zero if ƒ(,...,) ƒ(,...,) for all,..., ad all > 0 i.e., if multiplyig all argumets by leads to o chage i the value of the fuctio. Say that ƒ(,...,) is homogeeous of degree oe, so that we have ƒ(,...,) ƒ(,...,). Differetiatig this idetity with respect to yields: settig = the gives: ƒ ( i,..., ) i ƒ(,..., ) for all,..., ad > 0 i ƒ ( i,..., ) i ƒ(,..., ) for all,..., i which is called Euler s theorem, ad which will tur out to have very importat implicatios for the distributio of icome amog factors of productio. Here s aother useful result: if a fuctio is homogeeous of degree, the its partial derivatives are all homogeeous of degree 0. To see this, take the idetity ƒ(,...,) ƒ(,...,) ad this time differetiate with respect to i, to get: 6
ƒi(,...,) ƒi(,...,) for all,..., ad > 0 or equivaletly: ƒi(,...,) ƒi(,...,) for all,..., ad > 0 which establishes our result. I other words, if a productio fuctio ehibits costat returs to scale, the margial products of all the factors will be scale ivariat. More geerally, a fuctio ƒ(,...,) is homogeeous of degree if ƒ(,...,) ƒ(,...,) for all,..., ad all > 0 i.e., if multiplyig all argumets by implies that the value of the fuctio is multiplied by. Aother way to thik of this property is that ƒ(,...,) has a costat scale elasticity of. To check your uderstadig of argumets ivolvig homogeeous fuctios, see if you ca prove the followig geeralizatio of the previous paragraph s result: If ƒ(,...,) is homogeeous of degree k, the its partials are all homogeeous of degree k. Cocave ad Cove Fuctios A real-valued fuctio ƒ() = ƒ(,...,) over a cove domai X is said to be cocave if ad cove if ƒ( + ( ) *) ƒ() + ( ) ƒ(*) for all, *, ad all (0,) ƒ( + ( ) *) ƒ() + ( ) ƒ(*) for all, *, ad all (0,) ƒ() is strictly cocave/cove if the relevat iequality holds strictly for all, * ad. If ƒ() is a cocave/cove fuctio of a sigle variable, the above coditios imply that the chord likig ay two poits o its graph will lie everywhere o or below/above it. If ƒ() is twice differetiable, the it will be cocave/cove if ad oly if its secod derivative ƒ() is everywhere opositive/oegative. However, the lik with secod derivatives is ot so eact for strict cocavity/coveity: the fuctio ƒ() = 4 is twice differetiable ad strictly cove over the etire real lie, evet though its secod derivative ƒ() is ot positive at = 0. Fially, ƒ() = ƒ(,...,) is affie if it takes the form ƒ(,...,) i= aii + b for some costats a,...,a, b. Affie fuctios are simultaeously weakly cocave ad weakly cove. Quasicocave ad Quasicove Fuctios A real-valued fuctio ƒ() = ƒ(,...,) over a cove domai X is quasicocave if ad quasicove if ƒ( + ( ) *)ƒ() for all, * such that ƒ(*) = ƒ(), ad all (0,) ƒ( + ( ) *)ƒ() for all, * such that ƒ(*) = ƒ(), ad all (0,) We use the terms strictly quasicocave/quasicove if the relevat iequality always holds strictly. There are a couple other equivalet defiitios of these cocepts. Perhaps the most ituitive is ƒ() = ƒ(,...,) is quasicocave for every *, the set { ƒ() ƒ(*) is a cove set } ƒ() = ƒ(,...,) is quasicove for every *, the set { ƒ() ƒ(*) is a cove set } Fuctios that are cocavecove, but ot strictly so, are sometimes termed weakly cocaveweakly cove. 7
Alteratively, ƒ() = ƒ(,...,) is quasicocave for all, *, ƒ( + ( )*) mi [ƒ(*),ƒ()] for all (0,) quasicove for all, *, ƒ( + ( )*) ma [ƒ(*),ƒ()] for all (0,) The fudametal differece betwee cocavity/coveity ad quasicocavity/quasicoveity is: Quasicocavity/quasicoveity are ordial cocepts, i that they solely pertai to properties of a fuctio s implied rakig ord : quasicocave for all *, the set { ord * is a cove set } quasicove for all *, the set { ord * is a cove set } Thus if a fuctio is quasicocave/quasicove, so is every icreasig trasformatio of it. Cocavity/coveity are cardial cocepts, i that they solely pertai to properties of the implied rakig crd. O the assumptio that the fuctio is cotiuous, they ca be show to be equivalet to the coditio that wheever ƒ(*) ƒ(), the ƒ(*) ƒ(½*+½) / ƒ(½*+½) ƒ(). This property ca be epressed i terms of the rakig crd as: cocave if {*, } crd {, } the {*, ½*+½} crd {½*+½, } cove if {*, } crd {, } the {*, ½*+½} crd {½*+½, } Thus, if a fuctio is cocave/cove, so is every icreasig affie trasformatio of it. Note that every cocave/cove fuctio is ecessarily quasicocave/quasicove, but ot vice versa: thik about the fuctio ƒ(,) over X = {(,) 0, 0}. E. DETERMINANTS, SYSTEMS OF LINEAR EQUATIONS & CRAMER S RULE The Determiat of a Matri I order to solve systems of liear equatios we eed to defie the determiat A of a square matri A. If A is a matri, that is, if A = [a], we defie A = a. I the case: if A = a a a a we defie A = aa aa that is, the product alog the dowward slopig diagoal (aa), mius the product alog the upward slopig diagoal (aa). I the 3 3 case: if A = a a a a a a a a a 3 3 3 3 33 the first form (i.e., recopy the first two colums). The we defie: a a a a a a a a a a a a a a a 3 3 3 3 33 3 3 A = aaa33 + aa3a3 + a3aa3 a3aa3 aa3a3 aaa33 i other words, add the products of all three dowward slopig diagoals ad subtract the products of all three upward slopig diagoals. Ufortuately, this techique does t work for 4 4 or bigger matrices, so to hell with them. 8
Systems of Liear Equatios ad Cramer s Rule The geeral form of a system of liear equatios i the ukow variables,..., is: a + a + + a = c a + a + + a = c a + a + + a = c for some matri of coefficiets A = a a a a a a a a a c c ad vector of costats C =. c Note that the first subscript i the coefficiet aij refers to its row ad the secod subscript refers to its colum (thus, aij is the coefficiet of j i the i th equatio). We ow preset Cramer s Rule for solvig systems of liear equatios. The solutios * ad * to the liear system: a + a = c are simply: a + a = c c a a c c a a c ad * * a a a a a a a a The solutios *, * ad * 3 to the 3 3 system: a + a + a33 = c a + a + a33 = c a3 + a3 + a333 = c3 are: c a a a c a a a c 3 3 c a a a c a a a c 3 3 c a a a c a a a c 3 3 33 3 3 33 3 3 3 3 a a a3 a a a3 a a a3 a a a a a a a a a 3 3 3 a a a a a a a a a 3 3 33 3 3 33 3 3 33 9
Note that i both the ad the 3 3 case we have that * i is obtaied as the ratio of two determiats. The deomiator is always the determiat of the coefficiet matri A. The umerator is the determiat of a matri which is just like the coefficiet matri, ecept that the j th colum has bee replaced by the vector of right had side costats. F. SOLVING OPTIMIZATION PROBLEMS The Geeral Structure of Optimizatio Problems Ecoomics is full of optimizatio (maimizatio or miimizatio) problems: the maimizatio of utility, the miimizatio of epediture, the miimizatio of cost, the maimizatio of profits, etc. Uderstadig these will be a lot easier if we cosider what is systematic about such problems. Each optimizatio problem has a objective fuctio ƒ(,...,;,...,m) which we are tryig to either maimize or miimize (i our eamples, we ll always be maimizig). This fuctio depeds upo both the cotrol variables,..., which we (or the ecoomic aget) are able to set, as well as some parameters,...,m, which are give as part of the problem. Thus a geeral ucostraied maimizatio problem takes the form: ma ƒ(,...,;,...,m).,..., Although we will ofte have may parameters,...,m i a give problem, for simplicity we shall assume from ow o that there is oly a sigle parameter. All of our results will apply to the may-parameter case, however. We represet the solutios to this problem, which obviously deped upo the values of the parameter(s), by the solutio fuctios: ( ) ( ) It is ofte useful to ask how well have we doe? or i other words, how high ca we get ƒ(,...,;), give the value of the parameter? This is obviously determied by substitutig i the optimal solutios back ito the objective fuctio, to obtai: ( ) ( ) ƒ(,..., ; ) ƒ( ( ),..., ( ); ) ad () is called the optimal value fuctio. Sometimes we will be optimizig subject to a costrait o the cotrol variables (such as the budget costrait of the cosumer). Sice this costrait may also deped upo oe or more parameters, our problem becomes: ma ƒ(,...,;),..., subject to g(,...,;) = 0 I this case we still defie the solutio fuctios ad optimal value fuctio i the same way we just have to remember to take ito accout the costrait. Although it is possible that there could 0
be more tha oe costrait i a give problem, we will oly cosider problems with a sigle costrait. For eample, if we were lookig at the profit maimizatio problem, the cotrol variables would be the quatities of iputs ad outputs chose by the firm, the parameters would be the curret iput ad output prices, the costrait would be the productio fuctio, ad the optimal value fuctio would be the firm s profit fuctio, i.e., the highest attaiable level of profits give curret iput ad output prices. I ecoomics we are iterested both i how the optimal values of the cotrol variables ad the optimal attaiable value vary with the parameters. I other words, we will be iterested i differetiatig both the solutio fuctios ad the optimal value fuctio with respect to the parameters. Before we ca do this, however, we eed to kow how to solve ucostraied or costraied optimizatio problems. First Order Coditios for Ucostraied Optimizatio Problems The first order coditios for the ucostraied optimizatio problem: ma,..., ƒ(,...,) are simply that each of the partial derivatives of the objective fuctio be zero at the solutio values ( *,..., * ), i.e. that: ƒ( *,..., * ) = 0 ƒ( *,..., * ) = 0 The ituitio is that if you wat to be at a moutai top (a maimum) or the bottom of a bowl (a miimum) it must be the case that o small chage i ay cotrol variable be able to move you up or dow. That meas that the partial derivatives of ƒ(,...,) with respect to each of the i s must be zero. Secod Order Coditios for Ucostraied Optimizatio Problems If our optimizatio problem is a maimizatio problem, the secod order coditio for this solutio to be a local maimum is that ƒ(,...,) be a weakly cocave fuctio of (,...,) (i.e., a moutai top) i the locality of this poit. Thus, if there is oly oe cotrol variable, the secod order coditio is that ƒ (*) < 0 at the optimum value of the cotrol variable. If there are two cotrol variables, it turs out that the coditios are: ƒ( *, * ) < 0 ƒ( *, * ) < 0 ad ƒ (, ) ƒ (, ) * * * * ƒ (, ) ƒ (, ) 0 Whe we have a miimizatio problem, the secod order coditio for this solutio to be a local miimum is that ƒ(,...,) be a weakly cove fuctio of (,...,) (i.e., the bottom of a bowl) i
the locality of this poit. Thus, if there is oly oe cotrol variable, the secod order coditio is that ƒ (*) > 0. If there are two cotrol variables, the coditios are: ƒ( *, * ) > 0 ƒ( *, * ) > 0 ad ƒ (, ) ƒ (, ) * * * * ƒ (, ) ƒ (, ) 0 (yes, this last determiat really is supposed to be positive). First Order Coditios for Costraied Optimizatio Problems (VERY importat) The first order coditios for the two-variable costraied optimizatio problem: ma, subject to are easy to see from the followig diagram ƒ(,) g(,) = c ( *, * ) g(,) = c level curves of ƒ(,) 0 The poit ( *, * ) i the diagram is clearly ot a ucostraied maimum, sice icreasig both ad would move you to a higher level curve for ƒ(,). However, this chage is ot legal sice it does ot satisfy the costrait it would move you off of the level curve g(,) = c. I order to stay o the level curve, we must joitly chage ad i a maer which preserves the value of g(,). That is, we ca oly tradeoff agaist at the legal rate: g(, ) g (, ) g(, ) c g 0 The coditio for maimizig ƒ(,) subject to g(,) = c is that o tradeoff betwee ad at this legal rate be able to raise the value of ƒ(,). This is the same as sayig that the level curve of the costrait fuctio be taget to the level curve of the objective fuctio. I other words, the tradeoff rate which preserves the value of g(,) (the legal rate) must be the same as the tradeoff rate that preserves the value of ƒ(,). We thus have the coditio:
which implies that: which is i tur equivalet to: for some scalar. g0 ƒ0 g(, ) f(, ) g (, ) f (, ) ƒ(,) = g(,) ƒ(,) = g(,) To summarize, we have that the first order coditios for the costraied maimizatio problem: ma ƒ(, ), subject to are that the solutios ( *, * ) satisfy the equatios g(,) = c ƒ( *, * ) = g( *, * ) ƒ( *, * ) = g( *, * ) g( *, * ) = for some scalar. A easy way to remember these coditios is simply that the ormal vector to ƒ(,) at the optimal poit ( *, * ) must be a scalar multiple of the ormal vector to g(,) at the optimal poit ( *, * ), i.e. that: ( ƒ( *, * ), ƒ( *, * ) ) = ( g( *, * ), g( *, * )) ad also that the costrait g( *, * ) = c be satisfied. c This same priciple eteds to the case of several variables. I other words, the coditios for ( *,..., * ) to be a solutio to the costraied maimizatio problem: ma,..., subject to ƒ(,...,) g(,...,) = c is that o legal tradeoff betwee ay pair of variables i ad j be able to affect the value of the objective fuctio. I other words, the tradeoff rate betwee i ad j that preserves the value of g(,...,) must be the same as the tradeoff rate betwee i ad j that preserves the value of ƒ(,...,). We thus have the coditio: or i other words, that: i i for ay i ad j j g0 j ƒ0 g (,..., ) ƒ (,..., ) j j for ay i ad j g (,..., ) ƒ (,..., ) i i 3
Agai, the oly way to esure that these ratios will be equal for ay i ad j is to have: ƒ(,...,) = g(,...,) ƒ(,...,) = g(,...,) ƒ(,...,) = g(,...,) To summarize: the first order coditios for the costraied maimizatio problem: ma,..., subject to are that the solutios ( *,..., * ) satisfy the equatios: ƒ(,...,) g(,...,) = c ƒ( *,..., * ) = g( *,..., * ) ƒ( *,..., * ) = g( *,..., * ) ƒ( *,..., * ) = g( *,..., * ) ad the costrait: g( *,..., * ) = c Oce agai, the easy way to remember this is simply that the ormal vector of ƒ(,...,) be a scalar multiple of the ormal vector of g(,...,) at the optimal poit, i.e.: ( ƒ( *,..., * ),..., ƒ( *,..., * ) ) = ( g( *,..., * ),..., g( *,..., * ) ) ad also that the costrait g( *,..., * ) = c be satisfied. Lagragias The first order coditios for the above costraied maimizatio problem are just a system of + equatios i the + ukows,..., ad. Persoally, I suggest that you get these first order coditios the direct way by simply settig the ormal vector of ƒ(,...,) to equal a scalar multiple of the ormal vector of g(,...,) (with the scale factor ). However, aother way to obtai these equatios is to costruct the Lagragia fuctio: L(,...,,) ƒ(,...,) + [c g(,...,)] (where is called the Lagragia multiplier). The, if we calculate the partial derivatives L/,...,L/ ad L/ ad set them all equal to zero, we get the equatios: L( *,..., *,)/ = ƒ( *,..., * ) g( *,..., * ) = 0 L( *,..., *,)/ = ƒ( *,..., * ) g( *,..., * ) = 0 L( *,..., *,)/ = c g( *,..., * ) = 0 4
But these equatios are the same as our origial + first order coditios. I other words, the method of Lagragias is othig more tha a roudabout way of geeratig our coditio that the ormal vector of ƒ(,...,) be times the ormal vector of g(,...,), ad the costrait g(,...,) = c be satisfied. See the stadard tets for secod order coditios for costraied optimizatio. G. COMPARATIVE STATICS OF SOLUTION FUNCTIONS: IMPLICIT DIFFERENTIATION Havig arrived at the first order coditios for a costraied or ucostraied optimizatio problem, we ca ow ask how the optimal values of the cotrol variables chage whe the parameters chage (for eample, how the optimal quatity of a commodity will be affected by a price chage or a icome chage). The easiest way to do this would be to actually solve the first order coditios for the solutios, that is, solve for the fuctios: * = * () * = * ()... * = * () the just differetiate these fuctios with respect to the parameter to obtai the derivatives: ( ) ( ) ( )... d d d However, first order coditios are usually much too complicated to solve. Are we up a creek? No: there is aother approach kow as implicit differetiatio, which always works. The idea behid implicit differetiatio is very simple. Sice the optimal values: * = * () * = * ()... * = * () come from the first order coditios, it stads to reaso that the derivatives of these optimal values will come from the derivatives of the first order coditios, where i each case, we are talkig about derivatives with respect to the parameter. Let s do it for the simplest of cases. Cosider the ucostraied maimizatio problem: ma (, ) with the sigle cotrol variable, the sigle parameter, ad where (,)/ < 0. We kow that the solutio * = *() solves the first order coditio: (*,) 0 If we wat to fid */, we would simply differetiate this first order coditio with respect to the parameter. Before doig this, it s useful to remid ourselves of eactly which variables will be affected by a chage i, or i other words, those variables which deped upo. This ca be doe by drawig arrows over these variables, or alteratively, by represetig all fuctioal depedece eplicitly, that is: Differetiatig with respect to yields: ( *, ) 0 or alteratively ( *( ), ) 0 5
*( ) ( *( ), ) ( *( ), ) 0 d Note that we get oe term for each arrow we have draw o the left (or each appearace of o the right). Rearragig ad solvig for the derivative *()/, we obtai: d *( ) ( *( ), ) ( *( ), ) d ( *( ), ) ( *( ), ) Let s take aother eample, this time with two cotrol variables ad two parameters. For the geeral ucostraied maimizatio problem: ma, ƒ(,;,) the solutios * = * (,) ad * = * (,) solve the first order coditios: ƒ( *, * ;,) = 0 ƒ( *, * ;,) = 0 If, say, we wated to fid the derivative * /, we would differetiate these first order coditios with respect to the parameter. Before doig this, use arrows to idetify the variables that will be affected by a chage i. I this eample, they are the solutios * ad *, as well as itself (the parameter agai remais fied). We thus get: ƒ (, ; ; ) 0 ƒ (, ; ; ) 0 Differetiatig these two equatios with respect to gives us: ƒ (, ;, ) ƒ (, ;, ) ƒ 3(, ;, ) 0 ƒ (, ;, ) ƒ (, ;, ) ƒ 3(, ;, ) 0 or equivaletly: ƒ (, ;, ) ƒ (, ;, ) ƒ 3(, ;, ) ƒ (, ;, ) ƒ (, ;, ) ƒ 3(, ;, ) This is a set of two liear equatios i the two terms * / ad * /, ad hece ca be solved by substitutio, or by Cramer s Rule (see below), to obtai: ad ƒ (, ;, ) ƒ (, ;, ) ƒ (, ;, ) ƒ (, ;, ) ƒ (, ;, ) ƒ (, ;, ) ƒ (, ;, ) ƒ (, ;, ) 3 3 ƒ (, ;, ) ƒ (, ;, ) ƒ (, ;, ) ƒ (, ;, ) ƒ (, ;, ) ƒ (, ;, ) ƒ (, ;, ) ƒ (, ;, ) 3 3 6
We could do this same procedure i a costraied maimizatio problem as well, provided we remember to differetiate all + of the first order coditios, icludig the last oe (the costrait equatio). This procedure will always work i gettig the derivatives of the optimal values of the cotrol variables with respect to ay parameter. To use it, just remember that it cosists of three steps: STEP : Derive the first order coditio(s) for the maimizatio or miimizatio problem. As usual, the eact form of the first order coditios depeds o the umber of cotrol variables i the problem, ad o whether it is a ucostraied or costraied optimizatio problem. STEP : Differetiate the first order coditio(s) with respect to the parameter that is chagig. Before doig this, it s always useful to draw arrows over those variables that will chage as a result of the chage i the parameter. These will geerally cosist of all of the cotrol variables, the Lagragia multiplier if we are workig with a costraied optimizatio problem, ad the parameter itself. STEP 3: Solve for the desired derivative(s). If there is a sigle cotrol variable, this cosists of solvig for the sigle derivative (say) */. If there is more tha oe cotrol variable, this will ivolve solvig a system of liear equatios for the derivatives * /, * / etc. H. COMPARATIVE STATICS OF OPTIMAL VALUE FUNCTIONS: THE ENVELOPE THEOREM The fial questio we ca ask is how the optimal attaiable value of the objective fuctio varies whe we chage the parameters. This has a surprisig aspect to it. I the ucostraied maimizatio problem: ma ƒ(,...,;),..., recall that we get the optimal value fuctio () by substitutig the solutios (),...,() back ito the objective fuctio, i.e.: () ƒ((),...,();) Thus, we could simply differetiate with respect to to get: d ( ) ƒ ( ),..., ( ); ( ) d d ƒ ( ),..., ( ); ( ) d ƒ ( ),..., ( ); where the last term is obviously the direct effect of upo the objective fuctio. The first terms are there because a chage i affects the optimal i values, which i tur affect the objective fuctio. All i all, this derivative is a big mess. However, if we recall the first order coditios to this problem, we see that sice / =... = / = 0 at the optimum, all of these first terms are zero, so that we just get: 7
d ( ) ƒ( ( ),..., ( ); ) d This meas that whe we evaluate how the optimal value fuctio is affected whe we chage a parameter, we oly have to cosider that parameter s direct effect o the objective fuctio, ad ca igore the idirect effects caused by the resultig chages i the optimal values of the cotrol variables. If we keep this i mid, we ca save a lot of time. The evelope theorem also applies to costraied maimizatio problems. Cosider the problem ma,..., subject to ƒ(,...,;) g(,...,;) = c Oce agai, we get the optimal value fuctio by pluggig the optimal values of the cotrol variables (amely (),...,()) ito the objective fuctio: () ƒ((),...,();) Note that sice these values must also satisfy the costrait, we also have: c g((),...,();) 0 so we ca multiply by () ad add to the previous equatio to get: () ƒ((),...,();) + ()[c g((),...,();)] which is the same as if we had plugged the optimal values (),...,() ad () directly ito the Lagragia formula, or i other words: () L((),...,(),();) ƒ((),...,();) + ()[c g((),...,();)] Remember, eve though it ivolves ad the costrait, the above equatio is still the formula for the optimal value fuctio (i.e. the highest attaiable value of ƒ(,...,;) subject to the costrait). Now if we differetiate the above idetity with respect to, we get: d ( ) L ( ),..., ( ), ( ); ( ) d d L L L ( ),..., ( ), ( ); ( ) d ( ),..., ( ), ( ); d ( ) d ( ),..., ( ), ( ); But oce agai, sice the first order coditios for the costraied maimizatio problem are that L/ = = L/ = L/ = 0, all but the last of these right had terms are zero, so we get: 8
d d L( ( ),..., ( ), ( ); ) ( ) I other words, we oly have to take ito accout the direct effect of o the Lagragia fuctio, ad ca igore the idirect effects due to chages i the optimal values of the i s ad. A very helpful thig to kow. 9