ECONOMICS A MATHEMATICAL HANDOUT Fall 3 abd β abc γ abc µ abc Mark Machia A. CALCULUS REVIEW Derivatives, Partial Derivatives ad the Chai Rule You should already kow what a derivative is. We ll use the epressios ƒ () or dƒ()/d 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(*)/d. 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(, )/ + 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()), the 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: ƒ () or d ƒ()/d ad it is obtaied by differetiatig the fuctio ƒ() twice with respect to (if you wat the value of ƒ ( ) at some poit *, do t substitute i * util after you ve differetiated twice). If the material i this sectio is ot already familiar to you, you may have trouble o the A midterms ad fial.
A secod partial derivative of a fuctio of two or more variables is aalogous, i.e., we will use the epressios: ƒ (, ) or ƒ(, )/ to deote differetiatig twice with respect to (ad ƒ(, )/ for twice with respect to ). We get a cross partial derivative whe we differetiate first with respect to ad the with respect to. We will deote this with the epressios: ƒ (, ) or ƒ(, )/ Here s a strage ad woderful result: if we had differetiated i the opposite order, that is, first with respect to ad the with respect to, we would have gotte the same result. I other words, we have ƒ (, ) ƒ (, ) or equivaletly ƒ(, )/ ƒ(, )/. 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 () d 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( ))) gh ( ( )) g ( h ( )) h ( ) so dgh ( ( )) g ( h ( )) h ( ) d d 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: dƒ ƒ(,..., ) ƒ(,..., ) d +... + d or i equivalet otatio: dƒ ƒ (,..., ) d + + ƒ (,..., ) d Eco A Fall 3 Mathematical Hadout
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 ƒ( ) ƒ ( ) absolute chage i d 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 E y,, is give by: E y, ( y y) ( ) proportioate chage i y y proportioate chage i y If we cosider very small chages i (ad hece i y), y/ becomes dy/d ƒ (), so we get that the elasticity of y with respect to is give by: ( y y) ( ) y dy Ey, f ( ) y d 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. Recall that sice the percetage chage i a variable is simply times its proportioal chage, elasticity is also as the ratio of the percetage chage i y to the percetage chage i : E y, y ( ) ( ) y ( ) y y % chage i y ( ) % chage i A useful ituitive iterpretatio: Sice we ca rearrage the above equatio as % chage i y E y, % chage i we see that E y, serves as the coversio factor betwee the percetage chage i ad the percetage chage i y. Although 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 7, the slope of this curve is obviously 7, but its elasticity is : dy E y, 7 d y 7 That is, if y is eactly proportioal to, the elasticity of y with respect to will always be oe, regardless of the coefficiet of proportioality. Eco A Fall 3 3 Mathematical Hadout
Costat Slope Fuctios versus Costat Elasticity Fuctios Aother way to see that slope ad elasticity are differet measures is to cosider the simple fuctio ƒ() 3 + 4. Although ƒ( ) has a costat slope, it does ot have a costat elasticity: dƒ( ) 4 Eƒ( ), 4 d ƒ( ) 3+ 4 3+ 4 which is obviously ot costat as chages. However, some fuctios do have a costat elasticity for all values of, amely fuctios of the form ƒ() c β, for ay costats c > ad β. Sice it ivolves takig to a fied power β, this fuctio ca be called a power fuctio. Derivig its elasticity gives: ( β ) dƒ( ) β c Eƒ( ), β β d ƒ( ) c Coversely, if a fuctio ƒ( ) has a costat elasticity, it must be a power fuctio. I summary: liear fuctios all have a costat slope: power fuctios all have a costat elasticity: ƒ( ) + β dƒ( ) d ƒ( ) c β E ƒ( ), β β 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 simply ƒ(, ) k, where we may or may ot wat to solve for. For eample, the level curves of a cosumer s utility fuctio are just his or her idifferece curves (defied by the equatio U(, ) u ), ad the level curves of a firm s productio fuctio are just the isoquats (defied by the equatio ƒ(l,k) Q ). The slope of a level curve is idicated by the otatio: d d ƒ(, ) k ƒ 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 ). To calculate this slope, recall the vector of chages (, ) will keep us o this level curve if ad oly if it satisfies the equatio: ƒ ƒ (, ) + ƒ (, ) which implies that ad will accordigly satisfy: d d so that i the limit we have: ƒ(, ) ƒ(, ) ƒ(, ) k Eco A Fall 3 4 Mathematical Hadout
d ƒ(, ) d ƒ(, ) ƒ(, ) k 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. slope ƒ(, )/d ƒ(, )/d 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: d ƒ (,..., ) i d j i d d ƒ (,..., ) j ƒ(,..., ) k j ƒ i Give ay level curve (or level surface) correspodig to the value k, its better-tha set is the set of all poits at which the fuctio yields a higher value tha k, ad its worse-tha set is the set of all poits at which the fuctio yields a lower value tha k. D. OPTIMIZATION #: 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 is a lot easier if oe kows what is systematic about such problems. Eco A Fall 3 5 Mathematical Hadout
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(,..., ;,..., ),..., Cosider the followig oe-parameter maimizatio problem ma ƒ(,..., ; ),..., (It s oly for simplicity that we assume just oe parameter. All of our results will apply to the geeral case of may parameters,..., m.) 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: φ( ) ƒ(,..., ; ) ƒ( ( ),..., ( ); ) ad φ() 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 the parameter(s), our problem becomes: ma ƒ(,..., ; ),..., subject to g (,..., ; ) c (Note that we ow have a additioal parameter, amely the costat c.) 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 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 m Eco A Fall 3 6 Mathematical Hadout
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 (,..., ) ƒ (,..., ) 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 i 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 (*) < at the optimum value of the cotrol variable. If there are two cotrol variables, it turs out that the coditios are: ƒ ( *, *) < ƒ ( *, *) < ad ƒ (, ) ƒ (, ) > * * * * ƒ (, ) ƒ (, ) 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 (*) >. If there are two cotrol variables, the coditios are: ƒ ( *, *) > ƒ ( *, *) > ad ƒ (, ) ƒ (, ) > * * * * ƒ (, ) ƒ (, ) (yes, this last determiat really is supposed to be positive). Eco A Fall 3 7 Mathematical Hadout
First Order Coditios for Costraied Optimizatio Problems (VERY importat) The first order coditios for the two-variable costraied optimizatio problem: ma ƒ(, ) subject to g(, ) c, are easy to see from the followig diagram ( *, *) g(, ) c level curves of ƒ(, ) The poit ( *, *) 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: d d g(, ) d d g (, ) g(, ) c g 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 λ. d d d d g ƒ g(, ) ƒ (, ) g (, ) ƒ (, ) * * λ g * * ƒ(, ) (, ) * * λ g * * ƒ(, ) (, ) Eco A Fall 3 8 Mathematical Hadout
To summarize, we have that the first order coditios for the costraied maimizatio problem: ma ƒ(, ), subject to g (, ) c are that the solutios ( *, *) satisfy the equatios * * λ g * * * * * * λ g g ( * *, ) c ƒ(, ) (, ) ƒ(, ) (, ) for some scalar λ. 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 ( *, *) ) λ ( g ( *, *), g ( *, *) ) ad also that the costrait g( *, *) c be satisfied. 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 ƒ(,..., ),..., subject to 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: di d d d i for all i ad j j g j ƒ g (,..., ) ƒ (,..., ) j j for all i ad j g (,..., ) ƒ (,..., ) i i Agai, the oly way to esure that these ratios will be equal for all i ad j is to have: * * * * λ g ƒ (,..., ) (,..., ) * * * * λ g ƒ (,..., ) (,..., ) * * * * λ g ƒ (,..., ) (,..., ) Eco A Fall 3 9 Mathematical Hadout
To summarize: the first order coditios for the costraied maimizatio problem: ma ƒ(,..., ),..., subject to g (,..., ) c are that the solutios ( *,..., *) satisfy the equatios: * * * * ƒ (,..., ) λ g (,..., ) * * * * ƒ (,..., ) λ g (,..., ) * * * * ƒ (,..., ) λ 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 ( *,..., *) ) λ ( 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 λ. 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 λ). However, aother way to obtai these equatios is to costruct the Lagragia fuctio: L(,...,,λ) ƒ(,..., ) + λ [c g(,..., )] (where λ is called the Lagragia multiplier). The, if we calculate the partial derivatives L/,..., L/ ad L/ λ ad set them all equal to zero, we get the equatios: L( *,..., * * * * *, λ) ƒ (,..., ) λ g(,..., ) L( *,..., *, λ) ƒ ( *,..., * ) λ g ( *,..., * ) * * * * * * L(,...,, λ) ƒ (,..., ) λ g(,..., ), ) L ( *,..., *, λ) λ c g( *,... * 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 λ times the ormal vector of g(,..., ), ad the costrait g(,..., ) c be satisfied. We will ot do secod order coditios for costraied optimizatio: they are a royal pai. Eco A Fall 3 Mathematical Hadout
E. SCALE PROPERTIES OF FUNCTIONS A fuctio ƒ(,..., ) is said to ehibit costat returs to scale if: ƒ(λ,...,λ ) λ ƒ(,..., ) for all,..., ad all λ > That is, if multiplyig all argumets by λ leads to the value of the fuctio beig multiplied by λ. Fuctios that ehibit costat returs to scale are also said to be homogeeous of degree. A fuctio ƒ(,..., ) is said to be scale ivariat if: ƒ(λ,...,λ ) ƒ(,..., ) for all,..., ad all λ > I other words, if multiplyig all the argumets by λ leads to o chage i the value of the fuctio. Fuctios that ehibit scale ivariace are also said to be homogeeous of degree. Say that ƒ(,..., ) is homogeeous of degree oe, so that we have ƒ(λ,...,λ ) λ ƒ(,..., ). Differetiatig this idetity with respect to λ yields: i ad settig λ the gives: ƒ ( λ,..., λ ) ƒ(,..., ) for all,...,, all λ > i i i ƒ (,..., ) ƒ(,..., ) for all,..., i 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. To see this, take the idetity ƒ(λ,...,λ ) λ ƒ(,..., ) ad this time differetiate with respect to i, to get: λ ƒ i (λ,...,λ ) λ ƒ i (,..., ) for all,..., ad λ > or equivaletly: ƒ i (λ,...,λ ) ƒ i (,..., ) for all,..., ad λ > which establishes our result. I other words, if a productio fuctio ehibits costat returs to scale (i.e., is homogeeous of degree ), the margial products of all the factors will be scale ivariat (i.e., homogeeous of degree ). F. OPTIMIZATION #: COMPARATIVE STATICS OF SOLUTION FUNCTIONS Havig obtaied 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). Cosider a simple maimizatio problem with a sigle cotrol variable ad sigle parameter ma ƒ( ; ) Eco A Fall 3 Mathematical Hadout
For a give value of, recall that the solutio * is the value that satisfies the first order coditio ƒ( *; ) Sice the values of ecoomic parameters ca (ad do) chage, we have defied the solutio fuctio *() as the formula that specifies the optimal value * for each value of. Thus, for each value of, the value of *() satisfies the first order coditio for that value of. So we ca basically plug the solutio fuctio *() ito the first order coditio to obtai the idetity ƒ( *( ); ) We refer to this as the idetity versio of the first order coditio. Comparative statics is the study of how chages i a parameter affect the optimal value of a cotrol variable. For eample, is *() a icreasig or decreasig fuctio of? How sesitive is *() to chages i? To lear this about *(), we eed to derive its derivative d *()/d. The easiest way to get d *()/d would be to solve the first order coditio to get the formula for *() itself, the differetiate it with respect to to get the formula for d *()/d. But sometimes first order coditios are too complicated to solve. Are we up a creek? No: there is aother approach, implicit differetiatio, which always gets the formula for the derivative d *()/d. I fact, it ca get the formula for d *()/d eve whe we ca t get the formula for the solutio fuctio *() itself! Implicit differetiatio is straightforward. Sice the solutio fuctio *() satisfies the idetity ƒ( *( ); ) we ca just totally differetiate this idetity with respect to, to get ƒ( *( ); ) d*( ) ƒ( *( ); ) + d d*( ) ƒ( *( ); ) ƒ( *( ); ) ad solve to get d For eample, let s go back to that troublesome problem ma e, with first order coditio * e *. Its solutio fuctio * *() satisfies the first order coditio idetity *( ) *( ) e So to get the formula for d *()/d, totally differetiate this idetity with respect to : *( ) *( ) *( ) d d *( ) + e d d d*( ) *( ) ad solve, to get *( ) d e Eco A Fall 3 Mathematical Hadout
Comparative Statics whe there are Several Parameters Implicit differetiatio also works whe there is more tha oe parameter. Cosider the problem ma ƒ( ;, β ) with first order coditio ƒ( *;, β ) Sice the solutio fuctio * *(,β ) satisfies this first order coditio for all values of ad β, we have the idetity ƒ( *(, β);, β) β, Note that the optimal value * *(,β ) is affected by both chages i as well as chages i β. To derive *(,β )/, we totally differetiate the above idetity with respect to, ad the solve. If we wat *(,β )/ β, we totally differetiate the idetity with respect to β, the solve. For eample, cosider the maimizatio problem Sice the first order coditio is ma a l( ) β [*] β * its solutio fuctio *(,β ) will satisfy the idetity [*(,β )] β *(,β ),β To get *(,β )/, totally differetiate this idetity with respect to : *(, β) *(, β) [ *( β, )] [ *( β, )] β β, ad solve to get: *( β, ) [ *( β, )] [ *(, β)] + β O the other had, to get *(,β )/ β, totally differetiate the idetity with respect to β : *(, β) *(, β) [ *(, β)] *(, β) β β β ad solve to get: *(, β) *(, β) β β β [ *(, )] + Eco A Fall 3 3 Mathematical Hadout
Comparative Statics whe there are Several Cotrol Variables Implicit differetiatio also works whe there is more tha oe cotrol variable, ad hece more tha oe equatio i the first order coditio. Cosider the eample ma ƒ(, ; ) The first order coditios are that * ad * solve the pair of equatios ƒ(, ; ) ƒ(, ; ) ad so the solutio fuctios * *() ad * *() satisfy the pair of idetities, ƒ( ( ), ( ); ) ƒ( ( ), ( ); ) ad To get *()/ ad *()/, totally differetiate both of these idetities respect to, to get ( ) ( ) ( ) ( ) ( ) ƒ ( ), ( ); ƒ ( ), ( ); ƒ ( ), ( ); + + ( ) ( ) ( ) ( ) ( ) ƒ ( ), ( ); ƒ ( ), ( ); ƒ ( ), ( ); + + This is a set of two liear equatios i the two derivatives *()/ ad *()/, ad we ca solve for *()/ ad *()/ by substitutio, or by Cramer s Rule, or however. Comparative Statics of Equilibria Implicit differetiatio is t restricted to optimizatio problems. It also allows us to derive how chages i the parameters affect the equilibrium values i a ecoomic system. Cosider a simple market system, with supply ad demad ad supply fuctios Q D D(P,I) ad Q S S(P,w) where P is market price, ad the parameters are icome I ad the wage rate w. Naturally, the equilibrium price is the value P e solves the equilibrium coditio D(P e,i) S(P e,w) It is clear that the equilibrium price fuctio, amely P e P e (I,w), must satisfy the idetity D( P e (I,w), I ) S( P e (I,w), w ) I,w So if we wat to determie how a rise i icome affects equilibrium price, totally differetiate the above idetity with respect to I, to get e e e ( (, ), ) ( (, ), ) ( (, ), ) e DP Iw I e P ( I, w) DP Iw I S P Iw w P ( I, w) + e wi, e P I I P I the solve to get Eco A Fall 3 4 Mathematical Hadout
e P ( I, w) I DP ( e( Iw, ), I) I SP ( e( Iw, ), w) DP ( e( Iw, ), I) Pe Pe I class, we ll aalyze this formula to see what it implies about the effect of chages i icome upo equilibrium price i a market. For practice, see if you ca derive the formula for the effect of chages i the wage rate upo the equilibrium price. Summary of the Use of Implicit Differetiatio to Obtai Comparative Statics Results The approach of implicit differetiatio is straightforward, yet robust ad powerful. It is used etesively i ecoomic aalysis, ad always cosists of the followig four steps: STEP : Obtai the first order coditios for the optimizatio problem, or the equilibrium coditios of the system. STEP : Covert these coditios to idetities i the parameters, by substitutig i the solutio fuctios. STEP 3: Totally differetiate these idetities with respect to the parameter that is chagig. STEP 4: Solve for the derivatives with respect to that parameter. G. OPTIMIZATION #3: COMPARATIVE STATICS OF OPTIMAL VALUES 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 φ() by substitutig the solutios *(),..., *() back ito the objective fuctio, i.e.: * * φ( ) ƒ( ( ),..., ( ); ) Thus, we could simply differetiate with respect to to get: * * * dφ ( ) ƒ( ( ),..., ( ); ) d ( ) d d * * * ƒ( ( ),..., ( ); ) d( ) + d * * ƒ( ( ),..., ( ); ) + Eco A Fall 3 5 Mathematical Hadout
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 ƒ/... ƒ/ at the optimum, all of these first terms are zero, so that we just get: * * dφ ( ) 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 affect 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. This also works for costraied maimizatio problems. Cosider the problem ma ƒ(,..., ; ) subject to g(,..., ; ) c,..., Oce agai, we get the optimal value fuctio by pluggig the optimal values of the cotrol variables (amely *(),..., *()) ito the objective fuctio: * * φ( ) ƒ( ( ),..., ( ); ) Note that sice these values must also satisfy the costrait, we also have: * * c g( ( ),..., ( ); ) so we ca multiply by λ() ad add to the previous equatio to get: φ() ƒ( *(),..., *();) + λ() [c g( *(),..., *();)] which is the same as if we had plugged the optimal values *(),..., *() ad λ () directly ito the Lagragia formula, or i other words: φ() L( *(),..., *(),λ ();) ƒ( *(),..., *();) + λ() [c g( *(),..., *();)] Now if we differetiate the above idetity with respect to, we get: * * * dφ ( ) L( * ( ),..., ( ), λ ( ); ) d( ) d d * * * L( ( ),..., * ( ), λ ( ); ) d( ) + d * * L( * ( ),..., ( ), λ ( ); ) dλ* ( ) + λ d * * L( * ( ),..., ( ), λ ( ); ) + Eco A Fall 3 6 Mathematical Hadout
But oce agai, sice the first order coditios for the costraied maimizatio problem are L/ L/ L/ λ, all but the last of these right had terms are zero, so we get: dφ ( ) L( ( ),..., ( ), λ ( ); ) d * * * 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 λ. A very helpful thig to kow. H. 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: a a a a a a if A we defie a a A that is, the product alog the dowward slopig diagoal (a a ), mius the product alog the upward slopig diagoal (a a ). I the 3 3 case: if A (i.e., recopy the first two colums). The we defie: a a a a a a a a a a a the first form a a a a a a a a a a a a a 3 3 3 3 3 3 33 3 3 33 3 3 A a a a 33 + a a 3 a 3 + a 3 a a 3 a 3 a a 3 a a 3 a 3 a a a 33 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. 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 Eco A Fall 3 7 Mathematical Hadout
a a a c a a a c for some matri of coefficiets A ad vector of costats C a a a c Note that the first subscript i the coefficiet a ij refers to its row ad the secod subscript refers to its colum (thus, a ij is the coefficiet of j i the i th equatio). We ow give Cramer s Rule for solvig liear systems. The solutios to the liear system: a + a c a + a c are simply: c a a c c a a c ad * * a a a a a a a a The solutios to the 3 3 system: are simply: a + a + a c 3 3 a + a + a c 3 3 a + a + a c 3 3 33 3 3 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 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. Eco A Fall 3 8 Mathematical Hadout