Mathematcal Economcs MEMF e ME Flomena Garca Topc 2 Calculus Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca
. Unvarate Calculus Calculus Functons : X Y y ( gves or each element X one element y Y Real unctons : R R, where R s the set o real numbers In economcs: cost uncton, producton uncton, demand uncton,... Doman D o a uncton: the set o nterest on whch t s dened Eample: ( s dened on non-negatve numbers: D {: } Composte uncton: (g( Eample: (+ 2 s composed o (y y 2 and g( + Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 2
Inverse uncton: (g( and g((y y Inverse uncton s denoted by - Eample: Demand uncton q (p - 2p Inverse demand uncton p - (q Lmts When s close to a, ( s close to b Denton: lm a ( b when or any ε >, there ests δ > such that ( - b < ε or any such that - a < δ Eample: lm / 2 Eample: lm / Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 3
Unvarate Calculus Calculus Contnuty: ( s contnuous at lm ( (. ( s contnuous on D t s contnuous at every pont n D Intermedate Value Theorem: ( s contnuous on [a,b], (a (b. Then or any y between (a and (b there ests c between a and b such that (c y. Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 4
Derentaton Consder uncton (. F. The dervatve o ( at s '( d d ( : lm ( ( + ( + When the lmt ests, ( s derentable at. Eample: ( 2 at Appromaton by dervatve Close to, ( ( + ( ( - 5 4 3 2 slope2-2 2 In economcs: margnal cost, margnal product, margnal revenue Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 5
Dervatve s tsel a uncton: or each '( lm ( + ( Eamples: ( k, ( k k- ( ln, ( / - Rules o derentaton Sum-derence: ( ± g ± g Product: ( g g + g ' g Quotent: ( / g' 2 g g' Chan: ((g( (g( g Eample: [(2 + 2 ] Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 6
Propertes Derentable unctons are contnuous Not all contnuous unctons are derentable Eample: s not derentable at I ( s contnuous, then ( s contnuously derentable, C The dervatve o ( s the second dervatve o (, ( Further dervatves are constructed n a smlar ashon I all dervatves o ( are contnuous, C Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 7
Concavty and convety ( 2 (-λ( +λ( 2 ( ((-λ +λ 2 2 ( 2 ((-λ +λ 2 (-λ( +λ( 2 ( 2 ( s conve (( - λ + λ 2 ( - λ( + λ( 2 ( s concave (( - λ + λ 2 ( - λ( + λ( 2 I ( s C 2 : I ( then ( s conve I ( then ( s concave Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 8
Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 9 Calculus Integraton Prmtve o ( s such uncton F( that F ( ( Eample: ( k, Integraton by parts Eample: Prmtve s also denoted by d ( C k d k k + + + Prmtve s determned up to a constant ( g g + g + g g g ' ' ' ( + g g g ' ' g g g ' ' d ln
Dente Integral ( ( - + + ( n ( n - n- ( ( ( lm ( : b a ( d a b n b The Fundamental Theorem o Calculus ( d F( b F( a a (Newton-Lebnz ormula In economcs: consumer surplus, measures o nequalty, probablty, stock and low varables Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca
Optmsaton Calculus s the global mamum o ( ( ( or all n D s the global mnmum o ( ( ( or all n D s the local mamum o ( ( ( or all n a small nterval around Theorem C. I s an nteror local mamum or mnmum, then ( 8 6 4 2 > < decreasng < - 2 3 4 5 8 6 ncreasng > < > 4 2-2 3 4 5 Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca
Ponts where ( (along wth ponts where ( does not est are crtcal ponts o (. Theorem C 2. (. I ( <, then s a local mamum. I ( >, then s a local mnmum. Eample: ( 3-3 2 + 4 y 5 25 - -5 5-25 -5 Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 2
Theorem C. (. s the unque soluton o (. I s local mamum, then s global mamum Eample: ( 3-3 2-9 + 5 on Theorem C 2. (. I ( < or all (( s concave, then s global mamum Eample: ( - 2 + 2-3 Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 3
General approach to mamsaton o C uncton on an nterval. Fnd all crtcal ponts o ( by solvng ( 2. Fnd values o ( at these ponts 3. Fnd values o ( at the ends o the nterval I ends are not ncluded, nd lm ( as approaches the ends 4. Choose the hghest value rom 2 and 3 I the hghest value s at the nterval end that s not ncluded, there s no mamum I the ends are ncluded n the nterval, then mamum (and mnmum ests Eample: ( 2-4 + 5 on [,3 Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 4
Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 5
2. Multvarate Calculus Calculus Functons o n varables y (,, n F: R n R In economcs: Lmts Utlty uncton u(,y Producton uncton y(k,l K α L -α lm a ( b or any ε >, there ests δ > such that ( - b < ε or any such that - a < δ. (,, n, a (a,,a n Contnuty ( s contnuous at lm ( ( ( s contnuous on D t s contnuous at every pont n D Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 6
Derentaton Consder uncton (,, n. F (,, n. + j The partal dervatve o wth respect to ( : lm h (,..., ( +,..., + n ( / gves the slope o the tangent lne to the graph o the uncton n the hyperplane parallel to as and z In economcs: Margnal product o labour: Y/L Propertes Partal dervatves are unctons Margnal utlty: u/ I all partal dervatves o ( est and contnuous, then ( s contnuously derentable (C. Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 7
Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 8 Calculus Rules o derentaton (,, n and (t,,t m Appromaton The same as or ordnary derentaton Total derental The chan rule n n t t t + +... Total dervatve (gradent n n n n + + + + + (... ( (,..., ( (,..., ( ( ( D n ( + ( + D( n n d d d + +... :
Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 9 Calculus Second order dervatves / (,, n s a uncton o n varables Hessan j j 2 Eample: ( 4 2 2-2 22 - - 2 2 2 2 2 2 2 n n n H D L M O M L Theorem I C 2, then j j 2 2 cross-partals
Concavty and Convety Conve set D s a conve set or any,y D, (-λ+λy D, or any λ [,] D y y Conve set D Non-conve set ( s conve on conve D (( - λ + λ 2 ( - λ( + λ( 2 ( s concave on conve D (( - λ + λ 2 ( - λ( + λ( 2 Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 2
Theorem C 2. s concave on conve D when D 2 s negatve semdente on D s conve on conve D when D 2 s postve semdente on D Eample (,y /3 y /3 on D { >, y > } Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 2
Implct uncton y (,, n eplct uncton F(,, n,y mplct uncton It may be dcult to solve F(,, n,y to get y (,, n Eample: y 3 - y + ln Suppose we are nterested n y/ Theorem F(,, n,y. F C. F(,y. F/y(,y. Then there ests y (,, n such that F(,, n, (,, n and y F / ( F / y(, y, y ( Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 22
Optmsaton (,, n s the global mamum o ( ( ( or all n D (,, n s the local mamum o ( ( ( or all n small neghbourhood o Theorem C. I s an nteror local mamum or mnmum, then / ( or all. Theorem C 2. / ( or all. I D 2 ( s negatve dente, then s a local mamum. I D 2 ( s postve dente, then s a local mnmum. I D 2 ( s ndente, then s nether local mamum nor local mnmum Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 23
Theorem C 2. / ( or all. I s concave (D 2 s negatve semdente on D, then s a global mamum. I s conve (D 2 s postve semdente on D, then s a global mnmum. General approach to mamsaton o C uncton wthout constrants. Fnd all crtcal ponts o ( by solvng / ( or all. 2. Fnd values o ( at these ponts 3. Fnd values o ( at the boundary o D 4. Choose the hghest value rom 2 and 3 There may be no mamum Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 24
Eample: (, 2 4 2 2-2 22 - - D { [,], 2 [,]} Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 25
Constraned optmsaton ma (,, n s.t. g (,, n h j (,, n,,m j,,k Eamples ma u(, 2 s.t. p + p 2 2 I utlty mamsaton subject to budget constrant prot mamsaton subject to producton possbltes welare mamsaton subject to ndvduals reacton Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 26
Lagrangean method Lagrangean λ, µ j are Lagrangean multplers L Mamse the Lagrangean ma (,, n s.t. g (,, n h j (,, n m k ( + λ g ( + µ h j,,m j,,k Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 27 j j ( Idea: represent common tangent hyperplane o the set dened by the constrants and the level curves o uncton At mamum there s no way to move along constrants wthout decreasng the value o Interpretaton o the multplers: shadow prce o the constrant
Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 28 Calculus Equalty constrants Lagrangean ma (,, n s.t. h j (,, n j,,k + k j j j h L ( ( µ Jacoban matr o the set o constrants / / / / h h h h Dh k k n L M O M L Assume k n Rows o Dh as vectors h / + + m m m h h α α α α K K Dh has rank m
Theorem,h j C. I s a local mamum or mnmum and rank Dh( k then there est µ j such that, L ( L, µ or all (, µ h j ( or all j µ j Eample ma (,y 2 - y s.t. 2 + y 2 Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 29
Inequalty constrants ma (,, n s.t. g (,, n,,m Bndng constrant g ( Non-bndng constrant g ( > Lagrangean L m ( + λ g ( Theorem,h j C. s a local mamum. g (,, g k ( and g k+ ( >,, g m ( >. rank Dg,..,k ( k. L Then there est λ such that: (, λ g ( or all λ g ( or all λ or all [At mnmum λ ] or all Complementary slackness condton Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 3
Eample ma (,y 2 - y s.t. 2 + y 2 Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 3
Med constrants ma (,, n s.t. g (,, n h j (,, n,,m j,,k Lagrangean Theorem,h j C. s a local mamum. L ( [Techncal condton on rank o a Jacobean]. Then there est λ,µ j such that: L, λ m or all g ( or all λ g ( or all λ or all Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 32 k ( + λ g ( + µ h j j j ( h j ( or all j
Eample ma (,y 2 - y s.t. 2 + y 2 + y Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 33
Sucent condtons or local mamum to be global mamum L s concave Can be wrtten usng concavty o and h General approach to mamsaton o C uncton wth constrants. Form the Lagrangean 2. Fnd all crtcal ponts by solvng the appoprate system o equatons and nequaltes 3. Fnd values o ( at these ponts 4. Choose the hghest value rom Step 3 (boundary o D s taken care o by some o the equatons n the system Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 34
Comparatve statcs ( ; a d da / a / F (, 2 ; a F 2 (, 2 ; a Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 35
Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 36 Calculus The envelope theorem ma (, 2 ; a s.t. h (, 2 ; a 2 2 + + h h h µ µ Lagrangean ;, ( ;, ( 2 2 a h a L µ + FOC Dervatve dl / da
The envelope theorem ma (, 2 ; a s.t. h (, 2 ; a Let, 2 be a soluton FOC h + µ, 2 h + µ 2, h Value uncton V ( a ( ( a, ( a; 2 a Dervatve dv / da Dervatve dh / da Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 37
The envelope theorem ma (,, n ; a s.t. h j (,, n ;a j,,k Suppose (a s the soluton [Techncal condtons on smoothness o the soluton] V a ( ( a; a L a ( ( a, µ ( a; a V V( ;a Eample: ma u(, 2 s.t. p + p 2 2 I V( ; V( ;2 a, Mathematcal Economcs - www.seg.utl.pt/~garca/economa_matematca 38