PRELIMINARY MATHEMATICS LECTURE 5

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1 SCHOOL OF ORIENTAL AND AFRICAN STUDIES UNIVERSITY OF LONDON DEPARTMENT OF ECONOMICS 5 / MSc Ecoomics PRELIMINARY MATHEMATICS LECTURE 5 Course website: OPTIMIZATION. Readig Secod (-order) derivative, the derivative o a derivative, measures the gradiet o the irst derivative (d/). This helps us i idetiig the shape o the curve. Further it provides a suiciet coditio i testig whether a statioar poit o a curve is a maimum (i.e. a peak o a hill) or a miimum poit (bottom o a valle). This techique is applied to simple optimisatio problems. Materials covered i this lecture are oud i the ollowig tetbooks: Chiag, A. C. ad K. Waiwright (5) Fudametal Methods o Mathematical Ecoomics, Forth Editio. McGraw-Hill. Chapters 9 ad. Dowlig, E. T. () Itroductio to Mathematical Ecoomics, Third Editio. McGraw-Hill.. Relative etrema o a uctio o oe choice variable (), ''() ' A B I the above diagram, A ad B represet turig poits o the curve ( ). A is a maimum turig poit ad B is a miimum turig poit. The value o at poit A is kow as the local maima (maimum) ad the value o at poit B is called the local miima (miimum). Note that the slope o the curve ( ), measured b d ( ), is at both turig poits. This ca be called the irst derivative test or a relative etremum or statioar poits. What coditio eables us to distiguish betwee the two tpes o turig poits? Beore the maimum turig poit the sig o ' is positive ad ater the maimum turig poit the sig is egative. Beore the miimum turig poit the sig o ' is egative ad ater the maimum turig poit the sig is positive.

2 To see this, let us cosider 6 d At a turig poit is equal to ero (irst derivative test or relative etremum) d Let d Whe, (this is the value at which the irst order coditio is satisied) ( ) 6( ) 9 8 There is a turig poit at (, ). Is this turig poit a local maimum or miimum? To establish this, we eed to id what d is happeig to the sig o just beore ad just ater the poit where. Let.9 Let. d d d 6 (.9) 6. < d 6 (.) 6. d We ca see that at the poit (, ), or ' is chagig rom egative through to positive ad thereore there is a miimum turig poit. (), ''() ' A B. Secod derivative test. Let us cosider

3 d 6 From the graph o 6, at both the poit o local maima ad local miima, the value o is equal to ero. However the slope/gradiet o the curve 6 is dieret at the two poits. At poit A, which is the poit o local maima, the slope o the curve is egative, while at d poit B, which is the poit o local miima, the slope is positive. The slope o is d measured b the derivative o kow as the secod derivative, deoted as. Thereore we ma guess that at the local maima o the origial curve, d ad < ad at the local miima, d ad ' (), ''() ' (), ''() A B Let us see this with the eample o.

4 d 6 d I order to id the statioar poit(s), let us set 6 8 ( )( ) Solvig this, we have two solutios Whe ad Whe ( ) ( ) ( ) ( ) ( ) ( ) There are two statioar poits at poits (, 8) ad (, 8). To see i the two poits are local maima or local miima, we derive the secod derivative d Let us check i the sig o is positive or egative at poit (, 8). To do this we evaluate ( ) at the critical poit. ( ) 6( ) (ote sig) ( ) < Thereore we ma sa that the poit (, 8) is a local maima. Similarl to check i the sig o at poit (, 8), let us evaluate ( ) 6( ) ( ) Thereore the poit (, 8) is a local miima. Both o the coclusios are cosistet with the graphs. So ar we saw that whe the sig o is positive the statioar poit is a local miima, ad whe the sig o is egative the statioar poit is a local maima.

5 . Ilectio/ileio poits. What about whe the sig o is equal to ero? Let us cosider the uctio (which we discussed i sectio o this ote). Recall that the irst derivative was d d Let 6 6 Whe, ( ) ( ) ( ) To see i the poit (, ) is local maima, local miima, or the poit o ileio we derive the secod derivative 6 6 Evaluate this at, 6 ( ) 6 The sig o the secod derivative is either positive or egative. I this particular case, the statioar poit is either local miima or maima, but a poit o ilectio/ ileio (but ot alwas so as we shall see below). (), ''() ' A I we graphicall cosider the origial ad irst order derivative, we ca see that the poit o ileio is the poit where the sig o the slope o the irst derivative chages (i this case, rom egative to positive).. Necessar versus suiciet coditios. I the previous sectio we oud that a poit o ilectio ca be characterised b 5

6 d ad However it is also possible to have a local maima or a local miima with Hece, the obtaied secod order coditio or the local maima, d ad < ad at the local miima, d ad is a secod order suiciet coditio but ot ecessar because it is also possible to have a local maima or a local miima with. 5. th derivative test or relative etrema or ilectio poits. I whe evaluated at a statioar poit, i order to veri whether we have a relative etrema (local maima or miima) or ilectio poits, we cotiue dieretiatig util a o-ero higher-order derivative is obtaied. I whe evaluated at the statioar poit the i is a odd umber, we have a poit o ilectio i is a eve umber AND < we have a local maima i is a eve umber AND we have a local miima Eample : (rom Thomas 999: 5-6) Cosider ( ) The irst order ecessar coditio is obtaied as d ( ) Net check the secod order suiciet coditio: ( ) Evaluatig at,. Because we caot eclude the possibilit that the uctio is at a local maima or local miima, we carr out the th derivative test to check i this is a relative etrema or ilectio poits. Third order derivative: ( ), at, Forth order derivative: 6

7 Because the orthth order derivative is o-ero, whereb the o-ero is obtaied at a order which is a eve umber, ad because the o-ero derivative is positive, we have a local miima. This is coirmed b drawig a diagram: Eample : 6 Let us cosider The irst order ecessar coditio is obtaied as d 6 5 Net check the secod order suiciet coditio: Evaluatig at,. Because we caot eclude the possibilit that the uctio is at a local maima or local miima, we carr out the th derivative test to check i this is a relative etrema or ilectio poits. Third order derivative:, at, Forth order derivative: 6, at, 5 5 Fith order derivative: 7, at, Sith order derivative: 7 6 Because the sith order derivative is o-ero, whereb the o-ero is obtaied at a order which is a eve umber, ad because the o-ero derivative is positive, we have a local miima. This is coirmed b drawig a diagram: 7

8 To recap: How to evaluate statioar poits Step. First order coditio Equate the irst derivative to ero ad id all statioar poits. Step. Secod order suiciet coditios Evaluate the secod order derivative at each statioar poits. I < we have a local maima I we have a local miima Step. I at a statioar poit the secod order derivative is ero, carr o dieretiatig util a o-ero higher order derivative is oud. I is a odd umber, we have a poit o ilectio I is a eve umber AND < we have a local maima i is a eve umber AND we have a local miima Optioal th derivative test, Talor epasio ad relative etremum (C.. Chiag 98: 6-6) A uctio () attais a relative maimum (miimum) value at i () ( ) is egative (positive) or values o i the immediate eighbourhood o, both to its let ad to its right. ( ) ( ) ( ) () () ( ) ( ) ( ) O O 8

9 Followig a mathematical theorem kow as Talor s theorem, the uctio () ca be epaded aroud as a Talor series (with the Lagrage orm o the remaider). ( ( ) ) ( ( ) ) ( ) ( ) ( )( ) ( ) ( ) ( p) L!!! A shorthad smbol! (read: actorial ) deied as! ( ) ( ) () () () ( a positive iteger) ( ) ( ) I ( ) ; ( ) ( ), the epasio reduces to ( ) ( ) ( p) ( ) where p is a umber betwee ad, ad accordigl is close to. Thereore, (p) will have the same sig as ( ), whereas ( ) is positive. Accordigl, A relative maimum o () i ( ) ; ( ) < A relative miimum o () i ( ) ; ( ) which is the secod-order suiciet coditios or relative etrema. I ( ) ( ) ; ( ) ( ), the epasio reduces to ( ) ( ) ( p) ( ) 6 Agai because p is close to, the sig o (p) is idetical with that o ( ). But the sig o ( ) is egative to the let o but positive to the right o. This violates the deiitio o a relative etremum, but sice it is a critical value (statioar poit), it must give a ilectio poit. More geerall, i ( ) ( ) (N ) ( ) ; (N ) ( ), the the Talor series will reduce to ( N ) ( ) ( ) ( p) ( ) N N! The sig o (N) (p) is the same as (N) ( ). The sig o ( ) N will var i N is odd, ad is positive i N is eve. Accordigl i N is odd, the sig o i () ( ) will chage as we pass through the poit ad thereb violates the deiitio o a relative etremum, which meas that must be a poit o ilectio. Whe N is eve, the sig o i () ( ) will ot chage rom the let o to it right, ad thereore we ca establish the statioar value ( ) as a relative maimum or miimum, depedig o whether (N) ( ) is egative or positive. This gives rise to the th-derivative test as a geeral method to veri relative etremum o a uctio with oe idepedet variable. 6. Optimum values o uctios cotaiig two or more choice variables,, the secod order (direct) partial derivative sigiies that the uctio has bee partiall dieretiated with respect to oe o the idepedet variables twice while the other idepedet variable has bee held costat Give a uctio (, ) The otatio with a double subscript sigiies that the primitive uctio is dieretiated partiall with respect to twice. The alterative otatio d that o but with the partial derivatives smbol. resembles 9

10 The cross partial derivatives sigiies that the uctio has bee partiall dieretiated with respect to oe o the idepedet variables ad the i tur partiall dieretiated with respect to aother idepedet variable. The two cross partial derivatives are idetical with each other, as log as the two cross partial derivatives are both cotiuous (Youg s theorem). Eample For each o the ollowig uctios, id,,,, ad i) ii) First-order coditios, suiciec ad ecessit, Recall that or a uctio o a sigle variable such as

11 () a poit or which d ad was a local miima. Similarl, a poit or which d was a local maima. ad < Now cosider a uctio o two variables (, ) Whe does have a local miima? Etrapolatig our argumets o local miima o a sigle variable uctio, we ma epect to have a local miima whe it is ot possible to obtai a smaller value o b makig slight chages i the values o a. There are three was i which we ca make slight chages i a: (a) we ca keep costat ad make var; (b) we ca keep costat ad make var; (c) we ca make both a var at the same time. I has a local miima at poit a a b, the we should be able to chage the values o a b a o the methods (a), (b), or (c), aet ot make the value o smaller. Let us irst cosider case (a). I this case, the value o is held costat. At a poit where ad (ote the chage i otatio ) we caot obtai a smaller b varig aloe. Similarl, i we cosider case (b), whe ad we caot obtai a smaller b varig aloe. Case (c) is slightl complicated. The coditio that must be satisied here is or. Hece, the secod order coditios are as ollows: A poit at which the irst partial derivatives are ero is: a local miima i,, ad a local maima i <, <, ad

12 Note that i < the we have either a maima or a miima but what is called a saddle poit this is a equivalet to the poit o ileio o the sigle variable uctio. To recap: The procedure or idig ad idetiig local optima is as ollows: Step : Fid the two irst partial derivatives ad set them equal to ero Step : Solve the resultig pair o simultaeous equatios to determie the statioar poits Step : Fid the secod partial derivatives ad evaluate them at the statioar poit(s). Check the secod order coditio to determie whether we have a local maima, miima, or either. Mathematical proo: (optioal) I the two-variable case, the total dieretial is d d We ca also derive the secod-order partial derivatives as d d ( d) ( d) ( d) d ( d) ( d)d ( d) ( d)d d d d d d I d, we have a local miima I d <, we have a local maima We shall see the ial epressio above as quadratic orms, to ideti critera or determiig whether the sigs are alwas positive or egative or arbitrar values o ad d, ot both ero. d ( ) ( ) d d d d d From the last epressio, d d ( ) ( ) d (positive deiite), i d < (egative deiite), i < d d ( ) ( ) ad ( ) ad ( )

13 must be positive or both cases, ad hece must be Note that ( ) positive (sice it must be greater tha the squared term ( ). Thereore ad must take idetical algebraic sig. 8. Determiatal test I the two-variable case, the total dieretial is d d We ca also derive the secod-order total dieretial as d d ( d) ( d) ( d) d ( d) ( d)d ( d) ( d)d d d d d d I d (positive deiite), we have a local miima I d < (egative deiite), we have a local maima Sice the variables a appear ol i squares, this ca be see as a quadratic orm. As discussed i lecture, the coditio or d to be positive/ egative deiite ma be stated b use o determiats. The quadratic orm ca be epressed i matri orm as: d [ d] d The determiat with the secod order partial derivatives as its elemets is called a Hessia determiat. H The pricipal miors i this case are: H H Hece the secod order coditios are: d (positive deiite), i ad, the local miima. d < (egative deiite), i < ad, the local maima. I we cosider a uctio o three choice variables,, ( ), The Hessia determiat is

14 H Ad the pricipal miors are H H H d (positive deiite), i H, H ad H, the local miima. d < (egative deiite), i H, H ad H, the local maima. < < 9. Ecoomic applicatios. 9. Proit maimisig irm with good Total ad margial reveue uctios P A demad uctio is give b or P.. We ca derive the total. reveue uctio such that TR P (.). Margial reveue is deied as the gradiet or the derivative o the total reveue uctio: dtr MR. 6 d Now let us suppose that we are iterested i the level o output, which maimises total dtr reveue. To id the statioar poit, let, d To check that the statioar poit at is a local maima, let us obtai the secod order coditio, d TR.6 < d Sice the sig o the secod derivative is egative, we ca coclude that the poit o the origial total reveue curve is its local maima. Note that this coclusio is cosistet with the graph o total reveue. What would be the value o the maimum total reveue? Evaluatig the total reveue uctio at, TR. ( ).( )

15 What is the level o the reveue maimisig price P? Because the total reveue is deied b TR P, the level o output that maimises total reveue is, ad the maimum total reveue is TR, P( ) P 6 Average, total ad margial cost uctios The average cost o a irm is give b: 5 AC What is the level o output at which average costs (AC) are at a miimum? Takig the irst derivative o 5 AC 5 dac d dac Settig d Recall that we derived the total cost uctio: ad the margial cost uctio: TC AC 5 5 dtc MC d From the graph o average ad margial cost uctios, we saw that the margial cost curve cuts the average cost curve at its miimum poit i.e. whe average cost is a miimum, margial cost equals average cost. Let us see this b usig dieretials. At 7.7, AC ( ) ( 7.7) 8... ad ( 7.7) 8. MC. I geeral, i we have TC C () ad is, TC AC, the dieretiatio o AC with respect to 5

16 dac d At the statioar poit (local miima), dtc ( ) TC d [quotiet rule] dtc d ( ) TC dtc TC [multipl throughout b ] d dtc ( MC) TC MC d TC MC AC Hece whe average cost is miimum, MC AC. Proit uctio I ma eoclassical ecoomic models, the irms objective is to maimise proit. Give the total reveue ad the total cost uctios o the irm, we ma deie irm s proit uctio. ( p q) ( AC q) TR TC Let us itroduce the same eample we have bee discussig where the irm s total reveue (TR) is TR P ad the total cost (TC) is (.). TC AC 5 5. Set up the proit uctio as ( 5 ) TR TC Fid the irst derivative o with respect to d Let d d.6 6 d

17 .65 To coirm that at. 65 the origial proit uctio has a local maima, let us check d the sig o, d d.6 < d Sice the sig o the secod derivative is egative, we ma coclude that the proit uctio has a local maima at. 65. Aswer: The level o output that maimises proits is.65. What would be the value o proits at this output level? Sice the proit uctio is deied as ( 5 ) TR TC. ad the level o output that maimises proits is. 65 ( 5 ) (.65) 6(.65) 5 ( ) Aswer: The maimum value o proits is What is the proit-maimisig price P? The proit uctio ca be writte as 5 TR TC P AC(). Substitutig AC ( 5 ) P 5 P Net, substitutig the level o output that maimises proit. 65, the maimum proit , P (.65) 5 (.65) (.65) (.65) P (.65) P P.65 P 6.65 Aswer: The proit maimisig price is Proit maimisig irm with goods Let us cosider a irm that is a price taker, produces ad sells two goods, goods ad goods i perectl competitive markets, ad has a proit uctio: 6 What would be the level o output or goods ad satisig the irst order coditio or maimum proit? To obtai the irst order coditios 7

18 6 8 Settig both coditio equal to ero, 6 () Rewrite as 8 () 8 6 Epress this i terms o matrices Use Cramer s rule Note: Obtaiig solutio without usig matri algebra Solvig the two equatios simultaeousl, irst add equatios () ad () to obtai Substitute ito oe o the origial uctios ( ) Hece the level o output/suppl or good must be ad or good be i order to satis the irst order coditios. Let us cosider whether the statioar poit we oud at ad is a local maima or a local miima. Derivig the secod partial derivatives 8

19 9 < 8 < The Hessia matri ca be writte as 8 H < H H Hece the proit is at local maima at ad Note: usig the o-matri versio o the secod order coditio, we ca cosider the coditio, ( )( ) 8 ( ) 6 Sice < ad 6, we ca veri that all secod order coditios or local maima are satisied.